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Castilla y León PAU Technical Drawing Applied to Plastic Arts and Design II study materials
Castilla y León PAU Technical Drawing Applied to Plastic Arts and Design II study resources

Castilla y León PAU Technical Drawing Applied to Plastic Arts and Design II practice questions

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Question 1Applied technical drawing: geometry, projections and explicit dimensions

In plane geometry applied to technical drawing, what is the fundamental relationship between a circle's radius and a tangent line at the exact point of tangency?

  1. The radius drawn to the point of tangency is strictly perpendicular (forms a 90-degree angle) to the tangent line. Correct answer.
  2. The radius drawn to the point of tangency forms a 45-degree angle with the tangent line.
  3. The tangent line intersects the radius at two distinct points separated by half the radius length.
  4. The distance from the circle center to the tangent line is strictly greater than the radius length.

Explanation

A tangent line touches a circle at exactly one point. The radius drawn from the center of the circle to this point of contact is strictly perpendicular (at a right angle of 90 degrees) to the tangent line. This fundamental geometric principle is essential for resolving tangency problems and smooth curved connections in artistic design.

Question 2Applied technical drawing: geometry, projections and explicit dimensions

In a planar homothety geometric transformation centered at point O with a scaling ratio of k = -2, how are the original figure and its image related?

  1. The image is scaled to twice the linear dimensions of the original, inverted relative to center O, and corresponding sides remain parallel. Correct answer.
  2. The image is reduced to half the original linear size and stays on the same side of center O as the original object.
  3. The image is rotated 90 degrees around center O without any change in linear dimensions or orientation.
  4. The image undergoes an oblique shear distortion into a non-similar parallelogram.

Explanation

A homothety (central similarity) maps points along rays passing through center O. When the scaling factor k is negative (k = -2), the image points lie on rays pointing in the opposite direction from O, producing an inverted figure whose linear dimensions are scaled by |k| = 2. Corresponding line segments remain strictly parallel.

Question 3Applied technical drawing: geometry, projections and explicit dimensions

What is the defining geometric locus definition of an ellipse in plane geometry?

  1. The set of all points in a plane where the sum of the distances to two fixed points (foci F1 and F2) is constant and equal to the major axis length 2a. Correct answer.
  2. The set of all points in a plane equidistant from a fixed point (focus) and a fixed straight line (directrix).
  3. The set of all points in a plane where the absolute difference of the distances to two fixed points is constant.
  4. The set of all points in a plane whose product of distances to two perpendicular lines is constant.

Explanation

An ellipse is defined as the locus of points for which the sum of the distances to two fixed focus points (F1 and F2) remains constant. By convention, this constant sum equals the length of the major axis, 2a (where PF1 + PF2 = 2a).

Question 4Applied technical drawing: geometry, projections and explicit dimensions

Which conic section is geometrically defined as the locus of all points in a plane that are equidistant from a fixed point (focus F) and a fixed line (directrix d)?

  1. Parabola Correct answer.
  2. Ellipse
  3. Hyperbola
  4. Circle

Explanation

A parabola is the conic section formed by the set of all points in a plane that are equidistant from a single focus point F and a directrix line d. Its eccentricity e is exactly equal to 1.

Question 5Applied technical drawing: geometry, projections and explicit dimensions

In the locus definition of a hyperbola, what is the geometric relationship involving any point P on the curve and its two foci F1 and F2?

  1. The absolute difference of the distances |PF1 - PF2| is constant and equal to the real axis length 2a. Correct answer.
  2. The sum of the distances PF1 + PF2 is constant and equal to the transverse axis 2a.
  3. The ratio of the distance PF1 to PF2 is constantly equal to 0.5.
  4. The product of distances PF1 * PF2 is constant and equal to the semi-major axis squared.

Explanation

A hyperbola consists of two open branches defined by the locus of points where the absolute difference between the distances to the two foci, |PF1 - PF2|, remains constant and equals 2a, the distance between the two vertices along the real axis.

Question 6Applied technical drawing: geometry, projections and explicit dimensions

In planar geometric inversion (inversión) centered at point O with power K > 0, if point P maps to point P', what algebraic and directional conditions must P and P' satisfy?

  1. Points O, P, and P' are collinear on the same ray from O, and the product of distances satisfies OP * OP' = K. Correct answer.
  2. Points P and P' lie on a circle centered at O, with OP + OP' = K.
  3. Segment PP' is perpendicular to ray OP, with the distance PP' = sqrt(K).
  4. Points P and P' are reflected across a fixed axis line with OP - OP' = K.

Explanation

Inversion in a plane relative to a center O and power K > 0 transforms point P into point P' such that O, P, and P' lie on the same line (collinear) and the product of their distances from O equals the power of inversion: OP * OP' = K. If P lies inside the inversion circle of radius r = sqrt(K), P' lies outside it.

Question 7Applied technical drawing: geometry, projections and explicit dimensions

To construct the two lines passing through an exterior point P that are tangent to a given circle with center O, which auxiliary geometric construction is used to locate the exact points of tangency?

  1. Draw an auxiliary Thales circle using segment OP as diameter; its intersections with the given circle mark the tangency points. Correct answer.
  2. Draw a concentric circle with twice the radius of the given circle.
  3. Erect a perpendicular bisector to segment OP and intersect it with the given circle.
  4. Draw an equilateral triangle with base OP to locate the points of tangency.

Explanation

By Thales' theorem, any angle inscribed in a semicircle is a right angle (90°). Drawing a circle whose diameter is segment OP creates right angles at the points where this auxiliary circle intersects the original circle. These intersection points are the precise points of tangency T1 and T2.

Question 8Applied technical drawing: geometry, projections and explicit dimensions

What is the condition for two circles with centers O1 and O2 and radii r1 and r2 to be externally tangent (tangentes exteriores)?

  1. The distance between centers O1O2 is equal to the sum of radii, O1O2 = r1 + r2. Correct answer.
  2. The distance between centers O1O2 is equal to the absolute difference of radii, O1O2 = |r1 - r2|.
  3. The distance between centers O1O2 is strictly greater than r1 + r2.
  4. The distance between centers O1O2 is zero.

Explanation

Two circles are externally tangent when they touch at a single point located on the segment connecting their centers. The distance between centers O1O2 exactly equals the sum of their radii (r1 + r2).

Question 9Applied technical drawing: geometry, projections and explicit dimensions

What is the distance between centers O1 and O2 for two circles with radii r1 = 8 cm and r2 = 3 cm to be internally tangent (tangentes interiores)?

  1. 5 cm Correct answer.
  2. 11 cm
  3. 14 cm
  4. 24 cm

Explanation

For internal tangency, one circle lies inside the other, touching at one point. The center distance O1O2 equals the difference between their radii: O1O2 = |r1 - r2| = 8 cm - 3 cm = 5 cm.

Question 10Applied technical drawing: geometry, projections and explicit dimensions

Which of the following geometric properties is preserved in a planar affinity (afinidad) transformation?

  1. Parallelism between corresponding line segments. Correct answer.
  2. Exact side lengths and segment distances.
  3. Interior angles between non-parallel lines.
  4. Circle shapes (all circles remain un-distorted circles).

Explanation

Planar affinity is an affine geometric transformation. It preserves parallelism (parallel lines map to parallel lines) and ratios of distances along parallel lines, but it distorts angles, lengths, and can transform circles into ellipses.

Question 11Applied technical drawing: geometry, projections and explicit dimensions

In a planar central homology (homología) transformation, what is the vanishing line (recta de desaparición RL)?

  1. The locus of original points whose homologue images lie at infinity. Correct answer.
  2. The axis line (eje de homología) where all invariant points reside.
  3. The straight line connecting the center of homology to the origin.
  4. The line segment connecting the two focal points.

Explanation

In central homology, the vanishing line (recta de desaparición) is parallel to the homology axis. It contains all points in the original plane whose homologous image points are projected to infinity (ideal points).

Question 12Applied technical drawing: geometry, projections and explicit dimensions

In an ellipse with semi-major axis a, semi-minor axis b, and semi-focal distance c, which Pythagorean relation connects these three parameters?

  1. a² = b² + c² Correct answer.
  2. b² = a² + c²
  3. c² = a² + b²
  4. a + b = 2c

Explanation

In an ellipse, the distance from a co-vertex (end of the minor axis) to either focus equals the semi-major axis length a. Constructing the right triangle formed by semi-minor axis b, semi-focal distance c, and hypotenuse a yields the fundamental relation a² = b² + c².

Question 13Applied technical drawing: geometry, projections and explicit dimensions

What is the primary purpose of Rytz's construction (método de Rytz) in conic drawing?

  1. To find the principal major and minor axes of an ellipse when a pair of conjugate diameters is given. Correct answer.
  2. To calculate the numerical area of an irregular parabola.
  3. To convert a hyperbola into an equivalent circle through inversion.
  4. To construct a golden spiral inside an equilateral triangle.

Explanation

Rytz's construction is a classic descriptive geometry method used to determine the exact direction and lengths of the principal major and minor axes (2a and 2b) of an ellipse from any given pair of conjugate diameters.

Question 14Applied technical drawing: geometry, projections and explicit dimensions

What characterizes a rectangular hyperbola (hipérbola equilátera) in plane geometry?

  1. Its semi-axes are equal (a = b) and its asymptotes are perpendicular to each other (form a 90° angle). Correct answer.
  2. Its major axis is exactly double its minor axis (a = 2b).
  3. Its asymptotes form a 45° angle with each other and its foci coincide at the origin.
  4. It has no asymptotes and forms a closed oval curve.

Explanation

A rectangular hyperbola has equal semi-axes (a = b). As a consequence, its asymptotes intersect at right angles (90°) and its eccentricity is e = sqrt(2) ≈ 1.414.

Question 15Applied technical drawing: geometry, projections and explicit dimensions

What is the relationship between the tangent line to a parabola at its vertex V and the parabola's main axis of symmetry?

  1. The tangent line at the vertex is strictly perpendicular to the axis of symmetry. Correct answer.
  2. The tangent line at the vertex is parallel to the axis of symmetry.
  3. The tangent line at the vertex forms a 45-degree angle with the directrix.
  4. The tangent line at the vertex coincides with the directrix.

Explanation

The vertex V of a parabola is the point where the curve intersects its axis of symmetry. The tangent line at V is perpendicular to the axis of symmetry and parallel to the directrix line.

Question 16Applied technical drawing: geometry, projections and explicit dimensions

In planar inversion geometry, what is the inverse image of a circle that passes through the center of inversion O?

  1. A straight line not passing through center O. Correct answer.
  2. A smaller concentric circle centered at O.
  3. A parabola whose vertex lies at center O.
  4. A rectangular hyperbola.

Explanation

In plane inversion, any circle that passes through the inversion center O maps into a straight line that does not pass through O. Conversely, lines not passing through O invert into circles passing through O.

Question 17Applied technical drawing: geometry, projections and explicit dimensions

How many general cases of Apollonius' Problem (Problema de Apolonio) exist for constructing circles tangent to three geometric entities (points, lines, or circles)?

  1. 10 cases (ranging from 3 points PPP to 3 circles CCC). Correct answer.
  2. 5 cases.
  3. 3 cases.
  4. 12 cases.

Explanation

Apollonius' problem involves constructing circles tangent to combinations of points (P), lines (L), and circles (C). Taking 3 entities with repetition yields 10 fundamental combinations: PPP, PPL, PPC, PLL, PLC, PCC, LLL, LLC, LCC, and CCC.

Question 18Applied technical drawing: geometry, projections and explicit dimensions

To divide a line segment AB of length L according to the Golden Ratio (sección áurea), what is the ratio of the total length L to the larger segment x?

  1. L / x = (1 + sqrt(5)) / 2 ≈ 1.618 Correct answer.
  2. L / x = sqrt(2) ≈ 1.414
  3. L / x = 2.0
  4. L / x = 1.333

Explanation

The golden section divides a segment such that the ratio of the whole length L to the longer part x equals the ratio of the longer part x to the shorter part (L - x). This ratio phi = (1 + sqrt(5))/2 ≈ 1.618033...

Question 19Applied technical drawing: geometry, projections and explicit dimensions

What fundamental property defines a cyclic quadrilateral (cuadrilátero inscriptible en una circunferencia)?

  1. Its opposite interior angles sum to 180 degrees (supplementary angles). Correct answer.
  2. All four side lengths must be strictly equal.
  3. Its diagonals intersect at a 90-degree right angle.
  4. Its adjacent angles are always equal.

Explanation

A quadrilateral can be inscribed in a circle (cyclic) if and only if its opposite interior angles are supplementary, meaning their sum equals 180 degrees (A + C = 180° and B + D = 180°).

Question 20Applied technical drawing: geometry, projections and explicit dimensions

In an ellipse, what geometric property is possessed by the normal line (perpendicular to the tangent line) at any point P on the curve?

  1. The normal line bisects the angle formed by the focal radii PF1 and PF2. Correct answer.
  2. The normal line passes through the center of the ellipse for all points P.
  3. The normal line is parallel to the major axis.
  4. The normal line intersects the directrix at a 45-degree angle.

Explanation

By the focal properties of conics (optical property of the ellipse), the tangent line at point P bisects the exterior angle between focal radii PF1 and PF2, while the normal line (perpendicular to the tangent) bisects the interior angle formed by PF1 and PF2.

Question 21Applied technical drawing: geometry, projections and explicit dimensions

In standard dihedral (orthographic) projection, what is the horizontal trace (traza horizontal h) of a straight line?

  1. The point where the line pierces or intersects the Horizontal Projection Plane (PH). Correct answer.
  2. The point where the line pierces or intersects the Vertical Projection Plane (PV).
  3. The angle the line forms with the Ground Line (LT).
  4. The projection of the line onto the profile plane.

Explanation

In dihedral projection, the horizontal trace h (or h') of a line is the exact point where the 3D line intersects the Horizontal Projection Plane (PH). At this point, the vertical projection h'' lies on the Ground Line (LT).

Question 22Applied technical drawing: geometry, projections and explicit dimensions

In the dihedral system, how are the traces of a plane alpha defined?

  1. As the intersection lines where plane alpha intersects the Horizontal Plane (alpha_1) and Vertical Plane (alpha_2). Correct answer.
  2. As the two focal points of the plane's boundary curve.
  3. As the distances from the origin to the center of the plane.
  4. As the projection of the plane's normal vector onto the Ground Line.

Explanation

A plane in dihedral geometry is represented by its traces: alpha_1 (horizontal trace) is the line of intersection between plane alpha and the Horizontal Plane (PH), and alpha_2 (vertical trace) is the line of intersection between plane alpha and the Vertical Plane (PV). They meet on the Ground Line (LT).

Question 23Applied technical drawing: geometry, projections and explicit dimensions

In standard European first-angle orthographic projection (used in Spain), where is the top view (planta) placed relative to the front view (alzado)?

  1. Directly below the front view. Correct answer.
  2. Directly above the front view.
  3. To the right of the front view.
  4. To the left of the front view.

Explanation

In European first-angle projection (ISO / UNE standards), the object is positioned between the observer and the projection plane. Consequently, the top view (planta) viewed from above is projected onto the horizontal plane and displayed directly below the front view (alzado) after unfolding.

Question 24Applied technical drawing: geometry, projections and explicit dimensions

What is characteristic of a line parallel to the Ground Line (línea paralela a la Línea de Tierra) in dihedral projection?

  1. Both its horizontal projection r' and vertical projection r'' are parallel to the Ground Line (LT) and it has no traces. Correct answer.
  2. Its horizontal projection is perpendicular to LT while its vertical projection is parallel.
  3. It intersects the Ground Line at 45 degrees.
  4. Its traces lie at the origin of coordinates.

Explanation

A line parallel to the Ground Line runs parallel to both the Horizontal Plane (PH) and Vertical Plane (PV). Therefore, both of its projections (r' and r'') appear as lines parallel to LT, and because it never pierces PH or PV, it has no traces.

Question 25Applied technical drawing: geometry, projections and explicit dimensions

When working with a profile line (recta de perfil) in dihedral projection, why is a third auxiliary profile projection plane required?

  1. Because both main projections (r' and r'') are perpendicular to LT on the same line, making it impossible to determine point positions without a 3rd view. Correct answer.
  2. Because profile lines cannot be projected onto the horizontal plane.
  3. To change the scale of the drawing by a factor of sqrt(2).
  4. Because profile lines are infinitely long curves.

Explanation

A profile line lies in a plane perpendicular to both PH and PV. Its horizontal (r') and vertical (r'') projections overlap along a single line perpendicular to LT. An auxiliary profile projection (3rd view) obtained by revolving a profile plane is necessary to resolve true positions and slopes.

Question 26Applied technical drawing: geometry, projections and explicit dimensions

What distinguishes a horizontal plane (plano horizontal) from a frontal plane (plano frontal) in dihedral projection?

  1. A horizontal plane is parallel to PH (its vertical trace is parallel to LT, no horizontal trace), whereas a frontal plane is parallel to PV (its horizontal trace is parallel to LT, no vertical trace). Correct answer.
  2. A horizontal plane has no vertical trace, while a frontal plane has no horizontal trace.
  3. A horizontal plane is perpendicular to both projection planes.
  4. A frontal plane is inclined at 45 degrees to LT.

Explanation

A horizontal plane (plano horizontal) is parallel to the Horizontal Plane PH; all its points have constant height, so its vertical trace alpha_2 is parallel to LT and it has no horizontal trace. A frontal plane (plano frontal) is parallel to the Vertical Plane PV; all points have constant depth, so its horizontal trace alpha_1 is parallel to LT and it has no vertical trace.

Question 27Applied technical drawing: geometry, projections and explicit dimensions

How can the true length (verdadera magnitud) of an oblique line segment AB be found using the right-triangle construction method in dihedral projection?

  1. Construct a right triangle whose legs are the length of one projection (e.g., A'B') and the difference in coordinates (delta z) of the endpoints; the hypotenuse is the true length. Correct answer.
  2. Multiply the length of the horizontal projection by sqrt(3).
  3. Add the lengths of the horizontal and vertical projections together.
  4. Project segment AB onto a line parallel to the Ground Line at 45 degrees.

Explanation

The true length of an oblique segment AB is the hypotenuse of a right-angled triangle where one leg is the horizontal projection length (A'B') and the other leg is the difference in elevation (delta z = zB - zA) between the endpoints.

Question 28Applied technical drawing: geometry, projections and explicit dimensions

To measure the true distance between two parallel oblique planes alpha and beta in dihedral projection, what procedure is performed?

  1. Draw a perpendicular line r to both planes, find its intersection points P and Q with alpha and beta, and determine the true length of segment PQ. Correct answer.
  2. Measure the distance between their horizontal traces alpha_1 and beta_1 directly along LT.
  3. Sum the angles of their vertical traces and divide by 2.
  4. Rebat plane alpha onto plane beta using an arbitrary axis.

Explanation

The true distance between two parallel planes is measured along a common perpendicular line r. One draws a perpendicular line r (whose projections r' and r'' are perpendicular to the respective plane traces), determines intersection points P (with alpha) and Q (with beta), and calculates the true length of segment PQ.

Question 29Applied technical drawing: geometry, projections and explicit dimensions

What type of conic curve is formed when a right circular cone is intersected by a plane that is parallel to exactly one generator (generatriz) of the cone?

  1. Parabola Correct answer.
  2. Ellipse
  3. Hyperbola
  4. Circle

Explanation

A plane section of a right circular cone is a parabola when the cutting plane is parallel to exactly one generator line of the cone. In terms of angles, the cutting plane inclination angle equals the cone's generator angle.

Question 30Applied technical drawing: geometry, projections and explicit dimensions

What shape is obtained from the plane section of a right circular cylinder when sliced by a plane inclined obliquely to its central axis?

  1. Ellipse Correct answer.
  2. Parabola
  3. Hyperbola
  4. Catenary

Explanation

Cutting a right circular cylinder with an oblique plane (not parallel or perpendicular to the axis) yields a plane section that is a perfect ellipse. The minor axis of the ellipse equals the cylinder's diameter, while the major axis increases with plane inclination.

Question 31Applied technical drawing: geometry, projections and explicit dimensions

In dihedral descriptive geometry, what defines a line of steepest slope (línea de máxima pendiente p) of an oblique plane alpha?

  1. A line contained in plane alpha whose horizontal projection p' is strictly perpendicular to the horizontal trace alpha_1 of the plane. Correct answer.
  2. A line in plane alpha whose vertical projection p'' is parallel to LT.
  3. A line perpendicular to the Vertical Projection Plane.
  4. A line passing through the origin at a 45-degree angle.

Explanation

A line of steepest slope (línea de máxima pendiente) of a plane forms the maximum possible angle with the Horizontal Plane PH. Its horizontal projection p' is strictly perpendicular to the horizontal trace alpha_1 of the plane.

Question 32Applied technical drawing: geometry, projections and explicit dimensions

What property defines a line of maximum inclination (línea de máxima inclinación i) of a plane alpha relative to the Vertical Plane PV?

  1. A line in plane alpha whose vertical projection i'' is strictly perpendicular to the vertical trace alpha_2 of the plane. Correct answer.
  2. A line whose horizontal projection i' is parallel to LT.
  3. A line perpendicular to the Horizontal Plane PH.
  4. A line bisecting the angle between alpha_1 and alpha_2.

Explanation

A line of maximum inclination (línea de máxima inclinación) forms the maximum angle with the Vertical Projection Plane PV. Its vertical projection i'' is strictly perpendicular to the vertical trace alpha_2 of plane alpha.

Question 33Applied technical drawing: geometry, projections and explicit dimensions

How is the intersection line i between two oblique planes alpha and beta constructed in dihedral geometry using their traces?

  1. Find the intersection point H of horizontal traces (alpha_1 and beta_1) and intersection point V of vertical traces (alpha_2 and beta_2); line i connects H and V. Correct answer.
  2. Draw a line parallel to LT through the midpoint of the traces.
  3. Erect perpendicular lines from the origin to the trace intersections.
  4. Rebat both planes onto PH and find their chord of contact.

Explanation

The intersection of two planes alpha and beta is a line i. Its horizontal trace H is the intersection of horizontal traces (alpha_1 ∩ beta_1) and its vertical trace V is the intersection of vertical traces (alpha_2 ∩ beta_2). Connecting H and V defines line i.

Question 34Applied technical drawing: geometry, projections and explicit dimensions

To find the intersection point I between an oblique line r and an oblique plane alpha, which standard auxiliary plane method is employed?

  1. Pass an auxiliary projecting plane beta (e.g. vertical projecting) through line r, find the intersection line i between alpha and beta, and intersect line i with line r to obtain point I. Correct answer.
  2. Draw a sphere centered at the origin that contains line r.
  3. Rotate line r by 90 degrees until it becomes a profile line.
  4. Scale plane alpha by a factor of 0.5 until it intersects line r.

Explanation

The standard general method to intersect a line r with a plane alpha is: (1) Contain line r in an auxiliary projecting plane beta; (2) Find intersection line i = alpha ∩ beta; (3) The intersection of lines r and i yields the required point I = r ∩ alpha.

Question 35Applied technical drawing: geometry, projections and explicit dimensions

Under what condition is a straight line r strictly perpendicular to an oblique plane alpha in dihedral projection?

  1. The projections of line r (r' and r'') are perpendicular to the respective traces of plane alpha (r' ⊥ alpha_1 and r'' ⊥ alpha_2). Correct answer.
  2. The projections of line r are parallel to the plane traces.
  3. Line r forms a 45-degree angle with the Ground Line.
  4. Line r has no horizontal trace.

Explanation

In orthographic dihedral projection, perpendicularity between a line and a plane is directly visible: a line r is perpendicular to a plane alpha if and only if its horizontal projection r' is perpendicular to horizontal trace alpha_1, and its vertical projection r'' is perpendicular to vertical trace alpha_2.

Question 36Applied technical drawing: geometry, projections and explicit dimensions

In a change of projection plane (cambio de plano) in dihedral geometry, what coordinate dimension remains invariant when changing the Vertical Projection Plane while keeping the Horizontal Plane fixed?

  1. The elevation (z-coordinate / cota) of every point remains constant. Correct answer.
  2. The depth (y-coordinate / alejamiento) of every point remains constant.
  3. The x-coordinate distance along the Ground Line remains constant.
  4. The true distance to the origin point increases by 50%.

Explanation

When changing the Vertical Plane (PV → PV1) while keeping PH fixed, the height/elevation (cota z) of every point above PH remains unchanged. The new vertical projections are constructed by projecting perpendicular to the new Ground Line LT1 with unchanged heights z.

Question 37Applied technical drawing: geometry, projections and explicit dimensions

When performing a rotation (giro) of a point P around a vertical axis e (perpendicular to PH) in dihedral projection, what paths do its projections describe?

  1. The horizontal projection P' describes a circular arc centered at e', while the vertical projection P'' moves along a horizontal straight line (constant height z). Correct answer.
  2. The vertical projection P'' describes a circular arc, while P' stays motionless.
  3. Both projections move in straight lines perpendicular to LT.
  4. P' moves along LT while P'' describes an ellipse.

Explanation

A vertical axis e is perpendicular to PH. Rotating point P around e keeps its height z constant. Thus, its horizontal projection P' moves in a true circle around e', while its vertical projection P'' moves horizontally parallel to LT.

Question 38Applied technical drawing: geometry, projections and explicit dimensions

What is the primary objective of plane rebatment (abatimiento de un plano) onto a projection plane in descriptive geometry?

  1. To obtain the true shape, true side lengths, and true interior angles of plane figures contained in an oblique plane. Correct answer.
  2. To calculate the total 3D volume of a polyhedral solid.
  3. To project an oblique line onto the profile plane.
  4. To convert orthographic views into isometric axonometric views.

Explanation

Rebatment (abatimiento) rotates an oblique plane around one of its traces (acting as the hinge axis or charnela) until it coincides with a projection plane (PH or PV). This reveals the true shape (verdadera magnitud) and exact dimensions of figures lying within that plane.

Question 39Applied technical drawing: geometry, projections and explicit dimensions

In technical drawing standards, how are hidden outlines and hidden edges (aristas ocultas) represented in orthographic views?

  1. With thin dashed lines (línea de trazos fina). Correct answer.
  2. With continuous thick lines.
  3. With thin chain dot-dash lines.
  4. With continuous red lines.

Explanation

Under UNE-EN-ISO normalization standards, visible outlines are drawn with continuous thick lines, whereas hidden outlines and hidden edges are drawn using thin dashed lines (línea de trazos).

Question 40Applied technical drawing: geometry, projections and explicit dimensions

What polygon shape is formed by the cross-section of a regular tetrahedron cut by a plane parallel to two of its opposite (skew) edges?

  1. Square or rectangle Correct answer.
  2. Equilateral triangle
  3. Regular hexagon
  4. Trapezoid with 60-degree angles

Explanation

A plane parallel to two opposite skew edges of a regular tetrahedron intersects all four faces along four line segments that form a parallelogram. When the cutting plane is equidistant from both opposite edges, the resulting section is an exact square.

Question 41Applied technical drawing: geometry, projections and explicit dimensions

In topographic elevation representation (sistema de planos acotados), what are contour lines (curvas de nivel)?

  1. Lines connecting terrain points of equal elevation or altitude above a reference datum. Correct answer.
  2. Lines of maximum terrain slope connecting summits to valleys.
  3. The boundary lines between private land parcels.
  4. Projections of artificial roads onto a vertical plane.

Explanation

Contour lines (curvas de nivel) in topographic projections (planos acotados) are isolines formed by intersecting terrain surfaces with horizontal planes at constant vertical intervals (equidistancia); they connect points of identical altitude.

Question 42Applied technical drawing: geometry, projections and explicit dimensions

In planos acotados (topographic projection), if the interval (módulo) between adjacent contour lines of vertical step h = 5 m is measured on paper as d = 10 mm at scale 1:1000, what is the slope (pendiente) of the terrain?

  1. 50% (slope angle approx 26.6°) Correct answer.
  2. 100% (slope angle 45°)
  3. 10% (slope angle 5.7°)
  4. 200%

Explanation

At scale 1:1000, 10 mm on paper represents a real horizontal distance D = 10 mm * 1000 = 10,000 mm = 10 m. The terrain slope is the ratio of vertical rise to horizontal distance: Pendiente = h / D = 5 m / 10 m = 0.50 = 50%.

Question 43Applied technical drawing: geometry, projections and explicit dimensions

In roof design drafting (cubiertas en planos acotados), what is a valley line (limahoya)?

  1. The line of intersection between two roof slopes that collects rainwater runoff. Correct answer.
  2. The highest horizontal edge of a roof structure (cumbrera).
  3. The outer descending edge separating two slopes that sheds water outward (limatesa).
  4. The vertical wall supporting the roof gutter.

Explanation

In roof construction geometry, a limahoya (valley) is the internal intersection line between two roof slopes towards which water drains. Conversely, a limatesa (hip) is an salient intersection line shedding water outward.

Question 44Applied technical drawing: geometry, projections and explicit dimensions

What is the key characteristic of Direct Dihedral Projection (Sistema Diédrico Directo) compared to classic dihedral projection?

  1. It dispenses with the explicit Ground Line (Línea de Tierra) and works directly with relative 3D point positions and reference planes. Correct answer.
  2. It requires three compulsory ground lines angled at 120 degrees.
  3. It only allows representation of spherical objects.
  4. It eliminates the top view entirely.

Explanation

Direct Dihedral Projection (Sistema Diédrico Directo) modernizes descriptive geometry by removing reliance on an arbitrary fixed Ground Line (LT). Position and alignment are established using relative dimensions and spatial constraints directly, aligning with CAD practices.

Question 45Applied technical drawing: geometry, projections and explicit dimensions

When a plane figure contained in an oblique plane is rebatable (abatida) onto the horizontal projection plane, what relationship exists between the original horizontal projection and the rebatable true shape?

  1. They are related by a planar affinity transformation whose axis is the horizontal trace alpha_1 of the plane. Correct answer.
  2. They are related by a pure 90-degree central rotation about the origin.
  3. They are identical in area and side lengths.
  4. They are related by a 3D spherical inversion.

Explanation

The relationship between the orthographic projection of a plane figure on PH and its rebatable (abatida) true shape is a planar affinity transformation. The axis of affinity is the hinge trace alpha_1, and affinity rays are perpendicular to alpha_1.

Question 46Applied technical drawing: geometry, projections and explicit dimensions

In normalized isometric axonometry (isometría normalizada), what is the theoretical foreshortening scale factor (coeficiente de reducción) applied equally along all three principal axes X, Y, and Z?

  1. sqrt(2/3) ≈ 0.816 (often simplified to 0.82 or 1.0 in unreduced drawings) Correct answer.
  2. 0.500
  3. 0.707
  4. 0.943

Explanation

Because all three axes in isometric projection form equal angles (120°) with each other and equal angles (approx 35°16') with the projection plane, the true orthographic reduction coefficient is mu = sqrt(2/3) ≈ 0.8164. In simplified practice, drawing at scale 1:1 (drawing scale 1.0) is permitted.

Question 47Applied technical drawing: geometry, projections and explicit dimensions

What are the angular intervals between the projected coordinate axes X, Y, and Z in isometric axonometry?

  1. 120 degrees between all three adjacent axes (X-Y, Y-Z, Z-X). Correct answer.
  2. 90 degrees between X and Z, and 135 degrees between X-Y and Y-Z.
  3. 90 degrees between all three axes.
  4. 60 degrees between X-Y and 150 degrees between Y-Z.

Explanation

Isometric axonometry distributes the three spatial axes symmetrically in the projection plane. The angles between adjacent axes are equal: 360° / 3 = 120° (with vertical Z axis and X, Y axes angled at 30° to the horizontal baseline).

Question 48Applied technical drawing: geometry, projections and explicit dimensions

In standard Cavalier axonometric perspective (perspectiva caballera), how are the X and Z axes oriented relative to each other in the frontal plane?

  1. Perpendicular to each other at a 90-degree angle (forming an un-distorted frontal plane). Correct answer.
  2. At a 120-degree angle.
  3. At a 135-degree angle.
  4. At a 45-degree angle.

Explanation

Cavalier axonometry is an oblique projection onto a plane parallel to the frontal plane (XZ). Consequently, the X and Z axes remain mutually perpendicular (90° angle), ensuring that true shapes parallel to XZ undergo zero angular distortion.

Question 49Applied technical drawing: geometry, projections and explicit dimensions

What standard reduction coefficient cy is applied to measurements along the receding axis (eje Y de fuga) in normalized Cavalier perspective to reduce depth distortion?

  1. cy = 1/2 (0.5) Correct answer.
  2. cy = 1.0 (no reduction)
  3. cy = 2.0 (double length)
  4. cy = 0.816

Explanation

In standard Cavalier perspective (perspectiva caballera), dimensions parallel to X and Z are drawn full scale (1:1), while dimensions along the receding Y axis are multiplied by a reduction factor cy = 1/2 (0.5) to maintain visual realism.

Question 50Applied technical drawing: geometry, projections and explicit dimensions

What defines dimetric axonometry (dimetría) compared to isometric and trimetric systems?

  1. Two of the three scale reduction coefficients are equal, and two of the angles between coordinate axes are equal. Correct answer.
  2. All three scale reduction coefficients are equal.
  3. All three scale reduction coefficients are different.
  4. It is drawn exclusively without any vertical axis.

Explanation

Dimetric projection (dimetría) is characterized by having two equal scale factors (e.g. cx = cz ≠ cy) and two equal angles between the projection axes. Isometric has 3 equal scale factors, while trimetric has 3 different scale factors.