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La Rioja PAU Technical Drawing II study materials
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La Rioja PAU Technical Drawing II practice questions

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Question 1Technical drawing: plane and spatial geometry, projections and dimensions

What is the measure of each interior angle in a regular convex hexagon?

  1. 108°
  2. 120° Correct answer.
  3. 135°
  4. 140°

Explanation

The sum of interior angles of an n-sided convex polygon is given by (n - 2) × 180°. For a hexagon (n = 6), the sum is (6 - 2) × 180° = 720°. Dividing 720° by 6 equal angles yields 120° for each interior angle.

Question 2Technical drawing: plane and spatial geometry, projections and dimensions

In an equilateral triangle circumscribed about a circle of radius r, what is the side length s of the triangle in terms of r?

  1. s = r√3
  2. s = 2r√3 Correct answer.
  3. s = 3r
  4. s = 2r

Explanation

The inradius r of an equilateral triangle is related to its altitude h by r = h / 3. Since h = (s√3) / 2, we have r = (s√3) / 6, which rearranges to s = 6r / √3 = 2r√3.

Question 3Technical drawing: plane and spatial geometry, projections and dimensions

Which algebraic expression defines the exact numerical value of the Golden Ratio (Phi, Φ)?

  1. Φ = (1 + √3) / 2 ≈ 1.366
  2. Φ = (1 + √5) / 2 ≈ 1.618 Correct answer.
  3. Φ = (1 + √2) / 2 ≈ 1.207
  4. Φ = (2 + √5) / 2 ≈ 2.118

Explanation

The Golden Ratio Φ is derived from dividing a segment into extreme and mean ratio (x² - x - 1 = 0), yielding the positive root Φ = (1 + √5) / 2 ≈ 1.6180339...

Question 4Technical drawing: plane and spatial geometry, projections and dimensions

A point P is located at a distance d = 10 cm from the center O of a circle with radius R = 6 cm. What is the power of point P with respect to this circle?

  1. 16 cm²
  2. 64 cm² Correct answer.
  3. 36 cm²
  4. 100 cm²

Explanation

The power of a point P with respect to a circle centered at O with radius R is defined as Power(P) = d² - R², where d is the distance PO. Here, Power(P) = 10² - 6² = 100 - 36 = 64 cm².

Question 5Technical drawing: plane and spatial geometry, projections and dimensions

What geometric locus is defined as the set of all points in a plane that have equal power with respect to two non-concentric circles?

  1. The radical center
  2. The radical axis Correct answer.
  3. The polar line
  4. The axis of homothety

Explanation

The radical axis (eje radical) is the straight line perpendicular to the line connecting the centers of two non-concentric circles, comprising all points with equal power to both circles.

Question 6Technical drawing: plane and spatial geometry, projections and dimensions

Given three circles with non-collinear centers, how is their radical center constructed?

  1. As the centroid of the triangle formed by the three circle centers
  2. As the point of intersection of the three pairwise radical axes Correct answer.
  3. As the circumcenter of the triangle formed by the three circle centers
  4. As the center of the circle tangent to all three given circles

Explanation

The radical center (centro radical) of three circles is the unique point where their three pairwise radical axes intersect, possessing equal power with respect to all three circles.

Question 7Technical drawing: plane and spatial geometry, projections and dimensions

In Apollonius tangency problems, what does the PPP case represent, and how many solutions exist for three non-collinear points?

  1. Circle passing through 3 points; exactly 1 solution (circumcircle) Correct answer.
  2. Circle tangent to 3 lines; exactly 4 solutions (incircle and 3 excircles)
  3. Circle tangent to 3 circles; up to 8 solutions
  4. Circle passing through 3 points; exactly 2 solutions

Explanation

The PPP case requires constructing a circle passing through three given non-collinear points. This is the unique circumcircle of the triangle formed by the three points, giving exactly 1 solution.

Question 8Technical drawing: plane and spatial geometry, projections and dimensions

In a plane inversion with center O and positive power K, what is the inverse image of a circle passing through the center of inversion O?

  1. A circle passing through the center of inversion O
  2. A straight line not passing through the center of inversion O Correct answer.
  3. A point at infinity
  4. A circle concentric with the inversion circle

Explanation

Under plane inversion centered at O, any circle that passes through O is transformed into a straight line that does not pass through O (and is parallel to the tangent of the circle at O).

Question 9Technical drawing: plane and spatial geometry, projections and dimensions

In a plane inversion with center O and positive inversion power K > 0, which geometric figure consists entirely of invariant points (points that invert to themselves)?

  1. The center of inversion point O
  2. The circle of inversion (círculo de inversión) centered at O with radius R = √K Correct answer.
  3. Any line passing through O
  4. The radical axis of any two inverse circles

Explanation

The circle of inversion centered at O with radius R = √K has power OP × OP' = R² = K. For any point P on this circle, OP = R, so OP' = R and P' = P, making every point on the circle invariant.

Question 10Technical drawing: plane and spatial geometry, projections and dimensions

In a plane homothety centered at O with a negative scale factor k = -0.5, where does the image of a point P lie relative to O?

  1. On the ray OP at half the distance from O
  2. On the opposite ray from O (180° turned) at half the distance from O Correct answer.
  3. Perpendicular to ray OP at distance 0.5 OP
  4. At twice the distance from O on the ray OP

Explanation

Homothety vectors satisfy OP' = k · OP. When k is negative (k = -0.5), the image point P' lies on the line OP but on the opposite side of O from P, with distance OP' = |k| · OP = 0.5 OP.

Question 11Technical drawing: plane and spatial geometry, projections and dimensions

In a plane homology transformation, which line remains pointwise invariant (every point on it maps to itself)?

  1. The horizon line
  2. The axis of homology (eje de homología) Correct answer.
  3. The vanishing line (recta límite)
  4. The line connecting homologous centers

Explanation

The axis of homology (eje de homología) is the unique line of invariant points in plane homology: any point lying on the axis is its own homologous image.

Question 12Technical drawing: plane and spatial geometry, projections and dimensions

What special condition converts a plane homology into a plane affinity (afinidad paralela)?

  1. The axis of homology is at infinity
  2. The center of homology is at infinity (ideal point) Correct answer.
  3. The ratio of homology is equal to 1
  4. The vanishing lines coincide with the axis

Explanation

Plane affinity (afinidad) is a special case of homology where the center of homology is an ideal point (located at infinity). As a result, lines connecting homologous points are all parallel to each other.

Question 13Technical drawing: plane and spatial geometry, projections and dimensions

What is the defining focal property of an ellipse with foci F1 and F2 and major axis length 2a?

  1. PF1 - PF2 = 2a for all points P on the ellipse
  2. PF1 + PF2 = 2a for all points P on the ellipse Correct answer.
  3. PF1 × PF2 = a² for all points P on the ellipse
  4. PF1 / PF2 = e for all points P on the ellipse

Explanation

An ellipse is defined as the locus of points P in a plane such that the sum of distances to two fixed focal points F1 and F2 is constant and equal to the major axis length: PF1 + PF2 = 2a.

Question 14Technical drawing: plane and spatial geometry, projections and dimensions

In an ellipse with semi-major axis a = 5 cm and semi-minor axis b = 3 cm, what is the focal distance c (distance from center to each focus)?

  1. c = 2 cm
  2. c = 4 cm Correct answer.
  3. c = √34 ≈ 5.83 cm
  4. c = 8 cm

Explanation

For an ellipse, the fundamental relationship between semi-axes a, b and focal distance c is a² = b² + c². Substituting a = 5 and b = 3 gives 25 = 9 + c², so c² = 16 and c = 4 cm.

Question 15Technical drawing: plane and spatial geometry, projections and dimensions

What geometric property characterizes the tangent line to an ellipse at a point P on the curve?

  1. It bisects the interior angle between focal radii PF1 and PF2
  2. It bisects the exterior angle formed by focal radii PF1 and PF2 (perpendicular to interior bisector) Correct answer.
  3. It is parallel to the focal axis F1F2
  4. It passes through the focal point F1

Explanation

The tangent line to an ellipse at point P bisects the exterior angle between the focal radii PF1 and PF2. Consequently, the normal to the ellipse at P bisects the interior angle between PF1 and PF2 (reflection property).

Question 16Technical drawing: plane and spatial geometry, projections and dimensions

What is the radius of the director circle (circunferencia directora) of an ellipse with major axis 2a?

  1. a
  2. 2a Correct answer.
  3. b
  4. √(a² + b²)

Explanation

The director circle of an ellipse is centered at one of the foci and has a radius equal to the major axis length 2a. It is the locus of symmetric points of the other focus with respect to all tangents.

Question 17Technical drawing: plane and spatial geometry, projections and dimensions

What is the defining focal property of a parabola with focus F and directrix line d?

  1. PF + PD = constant for all points P
  2. PF = PD for all points P (distance to focus equals distance to directrix) Correct answer.
  3. PF - PD = constant for all points P
  4. PF × PD = p² for all points P

Explanation

A parabola is defined as the locus of all points P in a plane that are equidistant from a fixed point called the focus F and a fixed line called the directrix d, so PF = PD (eccentricity e = 1).

Question 18Technical drawing: plane and spatial geometry, projections and dimensions

If the parameter (focal parameter) of a parabola is p = 4 cm, what is the distance from its vertex V to its focus F?

  1. 1 cm
  2. 2 cm Correct answer.
  3. 4 cm
  4. 8 cm

Explanation

The parameter p of a parabola is the distance from the focus F to the directrix line d. The vertex V lies midway between F and d, so the distance VF = p / 2 = 4 / 2 = 2 cm.

Question 19Technical drawing: plane and spatial geometry, projections and dimensions

What is the locus of the projection of the focus F onto any tangent line of a parabola?

  1. The directrix line d
  2. The tangent line at the vertex (tan V) Correct answer.
  3. The axis of the parabola
  4. A circle centered at the vertex

Explanation

In a parabola, the orthogonal projection of the focus F onto any tangent line lies on the tangent at the vertex V (which plays the role of the principal circle for a parabola).

Question 20Technical drawing: plane and spatial geometry, projections and dimensions

What is the defining focal property of a hyperbola with foci F1 and F2 and transverse axis 2a?

  1. PF1 + PF2 = 2a
  2. |PF1 - PF2| = 2a for all points P on the hyperbola Correct answer.
  3. PF1 × PF2 = a²
  4. PF1 / PF2 = 1

Explanation

A hyperbola is defined as the locus of points P in a plane where the absolute difference of distances to two fixed foci F1 and F2 is constant and equal to the transverse axis length 2a: |PF1 - PF2| = 2a.

Question 21Technical drawing: plane and spatial geometry, projections and dimensions

In a hyperbola with semi-transverse axis a = 4 cm and semi-conjugate axis b = 3 cm, what is the focal distance c?

  1. c = 1 cm
  2. c = 5 cm Correct answer.
  3. c = √7 ≈ 2.65 cm
  4. c = 7 cm

Explanation

For a hyperbola, the focal distance c satisfies c² = a² + b² (since c > a). Substituting a = 4 and b = 3 gives c² = 4² + 3² = 16 + 9 = 25, so c = 5 cm.

Question 22Technical drawing: plane and spatial geometry, projections and dimensions

What are the equations of the asymptotes of a hyperbola centered at the origin with horizontal transverse axis 2a and conjugate axis 2b?

  1. y = ± (a / b) x
  2. y = ± (b / a) x Correct answer.
  3. y = ± (c / a) x
  4. y = ± (b / c) x

Explanation

The standard equation of a horizontal hyperbola is (x²/a²) - (y²/b²) = 1. Setting the right-hand side to zero gives the asymptotic directions y = ± (b / a) x.

Question 23Technical drawing: plane and spatial geometry, projections and dimensions

To draw tangent lines from an exterior point P to an ellipse, a circle centered at P with radius PF1 is intersected with which curve?

  1. The principal circle of the ellipse
  2. The director circle centered at focus F2 with radius 2a Correct answer.
  3. The minor auxiliary circle
  4. The focal axis line F1F2

Explanation

To find tangents from P to an ellipse, draw an arc centered at P with radius PF1. Its intersections with the director circle centered at F2 (radius 2a) give the symmetric points F'1. The perpendicular bisectors of F1F'1 are the required tangents.

Question 24Technical drawing: plane and spatial geometry, projections and dimensions

What is the measure of each interior angle in a regular convex octagon?

  1. 120°
  2. 135° Correct answer.
  3. 144°
  4. 150°

Explanation

For a regular octagon (n = 8), the sum of interior angles is (8 - 2) × 180° = 6 × 180° = 1080°. Dividing by 8 gives 1080° / 8 = 135°.

Question 25Technical drawing: plane and spatial geometry, projections and dimensions

When constructing a regular pentagon with side length s, what is the length of its diagonal d?

  1. d = s √2
  2. d = s · Φ = s (1 + √5) / 2 Correct answer.
  3. d = s √3
  4. d = 2 s

Explanation

In any regular pentagon, the ratio of a diagonal d to a side s is equal to the Golden Ratio Φ = (1 + √5) / 2 ≈ 1.618. Thus, d = s · Φ.

Question 26Technical drawing: plane and spatial geometry, projections and dimensions

In the Monge dihedral projection system, what line is formed by the perpendicular intersection of the Horizontal Plane (PH) and Vertical Plane (PV)?

  1. The line of maximum slope
  2. The Ground Line (Línea de Tierra, LT) Correct answer.
  3. The profile axis
  4. The line of horizon

Explanation

The intersection of the primary vertical projection plane (PV) and horizontal projection plane (PH) forms the fundamental reference axis called the Ground Line (Línea de Tierra, LT).

Question 27Technical drawing: plane and spatial geometry, projections and dimensions

A point A in space has a positive elevation (cota > 0) above PH and a positive departure (alejamiento > 0) in front of PV. In which quadrant (dihedro) is point A located?

  1. First Quadrant (Primer Diedro) Correct answer.
  2. Second Quadrant (Segundo Diedro)
  3. Third Quadrant (Tercer Diedro)
  4. Fourth Quadrant (Cuarto Diedro)

Explanation

The first quadrant (primer diedro) is defined by positive cota (above PH) and positive alejamiento (in front of PV). In Monge conventions, A2 is above LT and A1 is below LT.

Question 28Technical drawing: plane and spatial geometry, projections and dimensions

A point B has negative elevation (cota < 0) and negative departure (alejamiento < 0). After flattening the projection planes, where do its projections B1 and B2 lie relative to LT?

  1. B1 below LT, B2 above LT
  2. B1 above LT, B2 below LT Correct answer.
  3. Both B1 and B2 above LT
  4. Both B1 and B2 below LT

Explanation

Point B is in the third quadrant. Negative cota places B2 below PH (which rotates up above LT when unfolded). Negative alejamiento places B1 behind PV (which rotates up above LT). Thus B1 is above LT and B2 is below LT.

Question 29Technical drawing: plane and spatial geometry, projections and dimensions

In dihedral projections, what is a profile line (recta de perfil)?

  1. A line parallel to the Ground Line LT
  2. A line perpendicular to the Ground Line LT in both projections Correct answer.
  3. A line parallel to the vertical plane PV
  4. A line lying entirely within the bisector plane

Explanation

A profile line (recta de perfil) is perpendicular to the Ground Line LT in both horizontal and vertical projections. Its true inclination and length are determined using a 3rd profile plane (PP).

Question 30Technical drawing: plane and spatial geometry, projections and dimensions

Which set of properties correctly describes a horizontal line (recta horizontal) in the dihedral system?

  1. r2 is parallel to LT; r1 shows true length; no horizontal trace H exists Correct answer.
  2. r1 is parallel to LT; r2 shows true length; no vertical trace V exists
  3. Both r1 and r2 are perpendicular to LT
  4. r1 and r2 are both parallel to LT

Explanation

A horizontal line is parallel to PH. Therefore, all its points have equal cota, so its vertical projection r2 is parallel to LT. Its horizontal projection r1 shows true length and angle with PV. It only intersects PV, so it has a vertical trace V but no horizontal trace H.

Question 31Technical drawing: plane and spatial geometry, projections and dimensions

Which set of properties correctly describes a frontal line (recta frontal) in the dihedral system?

  1. r2 is parallel to LT; r1 shows true length
  2. r1 is parallel to LT; r2 shows true length and true inclination to PH; has H trace, no V trace Correct answer.
  3. r1 and r2 are both oblique to LT and have both H and V traces
  4. r1 is perpendicular to LT and r2 is parallel to LT

Explanation

A frontal line is parallel to PV. Its departure (alejamiento) is constant, so r1 is parallel to LT. Its vertical projection r2 shows true length and inclination angle to PH. It intersects PH at horizontal trace H, but never intersects PV (no V trace).

Question 32Technical drawing: plane and spatial geometry, projections and dimensions

A line is parallel to both the Horizontal Plane (PH) and the Vertical Plane (PV). What are its projection characteristics?

  1. Both r1 and r2 are parallel to LT; neither H nor V trace exists Correct answer.
  2. r1 is perpendicular to LT; r2 is parallel to LT
  3. Both r1 and r2 coincide on LT
  4. r1 is a point; r2 is parallel to LT

Explanation

A front-horizontal line (recta fronto-horizontal) is parallel to the Ground Line LT, meaning it is parallel to both PH and PV. Its projections r1 and r2 are both parallel to LT, and it has no traces with PH or PV.

Question 33Technical drawing: plane and spatial geometry, projections and dimensions

In dihedral descriptive geometry, what are the traces of a plane (trazas del plano)?

  1. The projections of its geometric center onto PH and PV
  2. The lines of intersection of the plane with the projection planes PH and PV Correct answer.
  3. The lines of maximum slope of the plane
  4. The vanishing lines of the plane

Explanation

The traces of a plane P are the straight lines where plane P intersects the primary projection planes: the horizontal trace P1 (or α1) is P ∩ PH, and the vertical trace P2 (or α2) is P ∩ PV.

Question 34Technical drawing: plane and spatial geometry, projections and dimensions

What are the trace characteristics of a horizontal plane (plano horizontal)?

  1. Horizontal trace P1 parallel to LT; no vertical trace P2
  2. Vertical trace P2 parallel to LT; no horizontal trace P1 Correct answer.
  3. Both P1 and P2 perpendicular to LT
  4. P1 and P2 are parallel oblique lines

Explanation

A horizontal plane is parallel to PH. It never intersects PH, so it has no horizontal trace P1. It intersects PV in a vertical trace P2 that is parallel to the Ground Line LT.

Question 35Technical drawing: plane and spatial geometry, projections and dimensions

What are the trace characteristics of a frontal plane (plano frontal)?

  1. Vertical trace P2 parallel to LT; no horizontal trace P1
  2. Horizontal trace P1 parallel to LT; no vertical trace P2 Correct answer.
  3. Horizontal trace P1 oblique; vertical trace P2 perpendicular to LT
  4. Both P1 and P2 pass through the origin

Explanation

A frontal plane is parallel to PV. It never intersects PV (no vertical trace P2), and intersects PH along a horizontal trace P1 that is parallel to the Ground Line LT.

Question 36Technical drawing: plane and spatial geometry, projections and dimensions

A plane is perpendicular to PH and oblique to PV (plano proyectante horizontal). What are its traces?

  1. P1 is perpendicular to LT; P2 is parallel to LT
  2. P1 is oblique to LT; P2 is perpendicular to LT Correct answer.
  3. Both P1 and P2 are parallel to LT
  4. P1 is parallel to LT; P2 is oblique to LT

Explanation

A horizontal-projecting plane (proyectante horizontal) is perpendicular to PH. Its horizontal trace P1 makes an oblique angle with LT (showing the inclination to PV), while its vertical trace P2 is perpendicular to LT.

Question 37Technical drawing: plane and spatial geometry, projections and dimensions

What necessary and sufficient condition must be met for a straight line r to lie completely within a plane P?

  1. The projections r1 and r2 must be parallel to P1 and P2 respectively
  2. The horizontal trace H of line r must lie on trace P1, and vertical trace V must lie on trace P2 Correct answer.
  3. The line r must intersect LT at the same point as plane P
  4. The projections r1 and r2 must be perpendicular to P1 and P2

Explanation

A line r lies in a plane P if and only if its horizontal trace H (where r intersects PH) lies on the horizontal trace P1 of the plane, and its vertical trace V (where r intersects PV) lies on the vertical trace P2 of the plane.

Question 38Technical drawing: plane and spatial geometry, projections and dimensions

What standard general procedure is used to find the intersection point I between an oblique line r and an oblique plane P?

  1. Pass an auxiliary projecting plane Q through line r, find intersection line i = P ∩ Q, then I = r ∩ i Correct answer.
  2. Draw lines from H and V perpendicular to LT and intersect them
  3. Rotate line r until it is horizontal, then read the intersection directly
  4. Perform a plane change to make line r vertical

Explanation

To find I = r ∩ P: 1) Pass an auxiliary projecting plane Q containing line r (e.g. Q1 = r1). 2) Determine the intersection line i = P ∩ Q. 3) The point where line r intersects line i is the required intersection point I.

Question 39Technical drawing: plane and spatial geometry, projections and dimensions

What is the condition for two planes P and Q to be parallel to each other in dihedral projection?

  1. Their corresponding traces are parallel: P1 || Q1 and P2 || Q2 Correct answer.
  2. Their horizontal traces are perpendicular and vertical traces are parallel
  3. P1 is parallel to Q2 and P2 is parallel to Q1
  4. Both planes must pass through the Ground Line LT

Explanation

Two planes are parallel if and only if their corresponding traces on the projection planes are parallel lines: P1 || Q1 on PH and P2 || Q2 on PV.

Question 40Technical drawing: plane and spatial geometry, projections and dimensions

In dihedral projections, what is the condition for a straight line r to be perpendicular to a plane P?

  1. Projections of line r are parallel to the traces of plane P: r1 || P1 and r2 || P2
  2. Horizontal projection r1 is perpendicular to horizontal trace P1, and vertical projection r2 is perpendicular to vertical trace P2 Correct answer.
  3. Line r must be a profile line and plane P must be a profile plane
  4. r1 is perpendicular to P2 and r2 is perpendicular to P1

Explanation

By the theorem of perpendicularity in Monge projections, a line r is perpendicular to a plane P if and only if its horizontal projection r1 is perpendicular to the horizontal trace P1, and its vertical projection r2 is perpendicular to the vertical trace P2.

Question 41Technical drawing: plane and spatial geometry, projections and dimensions

How is the true length of an oblique segment AB determined using the right-angled triangle method in dihedral projection?

  1. Hypotenuse of triangle with legs = projected length d1 (A1B1) and elevation difference Δz = |zA - zB| Correct answer.
  2. Hypotenuse of triangle with legs = cota zA and alejamiento yA
  3. Sum of projected lengths d1 + d2
  4. Hypotenuse of triangle with legs = projected length d2 (A2B2) and alejamiento difference Δy

Explanation

Construct a right-angled triangle: one leg is the horizontal projection length d1 = A1B1, and the other leg is the difference in elevation Δz = |zA - zB|. The hypotenuse gives the exact true length of segment AB in 3D space.

Question 42Technical drawing: plane and spatial geometry, projections and dimensions

To transform an oblique plane P into a horizontal-projecting plane using a change of ground line (cambio de plano), how must the new Ground Line LT' be positioned?

  1. Parallel to the horizontal trace P1
  2. Perpendicular to the horizontal trace P1 Correct answer.
  3. Perpendicular to the vertical trace P2
  4. Passing through the origin at 45° to P1

Explanation

To make plane P projecting onto the new horizontal plane, choose a new Ground Line LT' perpendicular to the horizontal trace P1. The new vertical trace P2' will then be oblique while P1 remains perpendicular to LT'.

Question 43Technical drawing: plane and spatial geometry, projections and dimensions

When performing an abatment (abatimiento) of a plane P onto the Horizontal Plane PH, what line acts as the axis of rotation (charnela)?

  1. The vertical trace P2
  2. The horizontal trace P1 Correct answer.
  3. The Ground Line LT
  4. The line of maximum slope

Explanation

In plane abatment onto PH, the plane revolves around its horizontal trace P1, which lies entirely in PH and remains fixed as the axis of rotation (charnela del abatimiento).

Question 44Technical drawing: plane and spatial geometry, projections and dimensions

What is the primary practical objective of performing a plane abatment (abatimiento) in technical drawing?

  1. To find the intersection of two oblique planes
  2. To determine the true shape, true angles, and true dimensions of plane figures Correct answer.
  3. To project a 3D solid into axonometric perspective
  4. To calculate the focal distance of a conic section

Explanation

Abatimiento rotates an inclined plane into coincidence with a primary projection plane (PH or PV), revealing the un-distorted true shape (verdadera magnitud), true angles, and true dimensions of figures contained in that plane.

Question 45Technical drawing: plane and spatial geometry, projections and dimensions

When a point A is rotated (giro) around a vertical axis (eje vertical, perpendicular to PH), how do its projections behave?

  1. A1 moves along a circle centered at e1; A2 moves horizontally along a line parallel to LT (constant cota) Correct answer.
  2. A2 moves along a circle centered at e2; A1 moves vertically (constant alejamiento)
  3. Both A1 and A2 move along concentric circles
  4. Both A1 and A2 move along straight lines at 45° to LT

Explanation

A vertical axis is perpendicular to PH. Rotating around it causes the horizontal projection A1 to describe a circular arc centered at e1, while the vertical projection A2 moves horizontally parallel to LT because the elevation (cota) remains unchanged during vertical-axis rotation.

Question 46Technical drawing: plane and spatial geometry, projections and dimensions

What is the correct sequence of steps to find the true distance from a point P to an oblique plane α?

  1. 1) Draw line r through P perpendicular to α. 2) Find I = r ∩ α. 3) Determine true length of segment PI. Correct answer.
  2. 1) Draw line through P parallel to α1. 2) Measure distance from P1 to α1 directly.
  3. 1) Abat plane α onto PH. 2) Measure distance from P to the charnela.
  4. 1) Rotate point P onto LT. 2) Read elevation difference.

Explanation

The distance from a point P to a plane α is the length of the perpendicular segment from P to α: 1) Construct line r through P perpendicular to α (r1 ⊥ α1, r2 ⊥ α2). 2) Find intersection point I = r ∩ α. 3) Calculate true length of segment PI using right triangle method.

Question 47Technical drawing: plane and spatial geometry, projections and dimensions

What are the geometric face, vertex, and edge counts of a regular tetrahedron?

  1. 4 equilateral triangular faces, 4 vertices, 6 edges Correct answer.
  2. 6 square faces, 8 vertices, 12 edges
  3. 8 equilateral triangular faces, 6 vertices, 12 edges
  4. 4 isosceles triangular faces, 4 vertices, 8 edges

Explanation

A regular tetrahedron is a Platonic solid bounded by 4 congruent equilateral triangles, having 4 vertices and 6 equal edges (satisfying Euler's formula V - E + F = 4 - 6 + 4 = 2).

Question 48Technical drawing: plane and spatial geometry, projections and dimensions

If a regular cube (hexahedron) has edge length a, what is the length of its main space diagonal D (diagonal principal)?

  1. D = a √2
  2. D = a √3 Correct answer.
  3. D = 2 a
  4. D = a √5

Explanation

The face diagonal of a cube with edge a is d = √(a² + a²) = a √2. The space diagonal connecting opposite vertices is D = √(d² + a²) = √(2a² + a²) = √(3a²) = a √3.

Question 49Technical drawing: plane and spatial geometry, projections and dimensions

What are the face, vertex, and edge counts of a regular octahedron?

  1. 8 equilateral triangular faces, 6 vertices, 12 edges Correct answer.
  2. 6 square faces, 8 vertices, 12 edges
  3. 12 pentagonal faces, 20 vertices, 30 edges
  4. 8 triangular faces, 8 vertices, 14 edges

Explanation

A regular octahedron is bounded by 8 equilateral triangles, meeting 4 at each of its 6 vertices, with 12 equal edges (satisfying V - E + F = 6 - 12 + 8 = 2).

Question 50Technical drawing: plane and spatial geometry, projections and dimensions

Which polygon forms the faces of a regular dodecahedron?

  1. Equilateral triangle
  2. Square
  3. Regular pentagon Correct answer.
  4. Regular hexagon

Explanation

A regular dodecahedron is a Platonic solid composed of 12 congruent regular pentagons, having 20 vertices and 30 edges.