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MATSEC IM Engineering Drawing and Graphical Communication study materials
MATSEC IM Engineering Drawing and Graphical Communication study resources

MATSEC IM Engineering Drawing and Graphical Communication practice questions

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Question 1Drawing standards, conventions and graphical literacy

In a first-angle orthographic projection layout, where is the view from above placed relative to the front view?

  1. To the left of the front view
  2. Below the front view Correct answer.
  3. Above the front view
  4. To the right of the front view

Explanation

In first-angle projection the object is imagined between the observer and the plane, so the top view is projected onto the plane beneath the object and appears below the front view. Third-angle projection reverses this and places the top view above the front view.

Question 2Drawing standards, conventions and graphical literacy

A drawing carries the symbol of a truncated cone with the wide end facing the viewer's left. Which projection arrangement does this symbol denote?

  1. Third-angle projection
  2. Fourth-angle projection
  3. First-angle projection Correct answer.
  4. Second-angle projection

Explanation

The standard first-angle symbol shows the frustum's small end facing the viewer with the circular end view drawn to the right of the cone shape. This matches the arrangement where the object sits between observer and plane.

Question 3Drawing standards, conventions and graphical literacy

On a technical drawing, a visible outline of an object is normally shown using which line type?

  1. A dashed thin line
  2. A chain thin line
  3. A continuous thin line
  4. A continuous thick line Correct answer.

Explanation

Visible edges and outlines are drawn with continuous thick lines so they dominate the drawing and read clearly against thinner construction and dimension lines. Line thickness hierarchy is a core convention of graphical communication.

Question 4Drawing standards, conventions and graphical literacy

Which line type is used to indicate the position of a cutting plane on a drawing that includes a section view?

  1. Chain thin line with thick ends and arrows Correct answer.
  2. Continuous thick line
  3. Continuous wavy thin line
  4. Dashed thick line

Explanation

A cutting plane is shown by a chain thin line whose ends and direction changes are thickened, with arrows indicating the viewing direction. This convention tells the reader exactly where the object is imagined to be cut.

Question 5Drawing standards, conventions and graphical literacy

Hatching lines within a sectional view are conventionally drawn at which angle to the principal outlines?

  1. 60 degrees
  2. 30 degrees
  3. 45 degrees Correct answer.
  4. 90 degrees

Explanation

Section hatching is drawn as thin continuous lines inclined at 45 degrees to the principal outlines of the part, spaced evenly. This angle gives good contrast with outlines and is easy to reproduce with standard drawing instruments.

Question 6Drawing standards, conventions and graphical literacy

Two adjacent parts appear in the same sectional view. How should their hatching be differentiated?

  1. Outline one part with a dashed line
  2. Use the same hatching but add shading
  3. Change the direction or spacing of the hatch lines Correct answer.
  4. Omit hatching on the smaller part

Explanation

Adjacent components in a section are hatched in opposite directions or with different spacing so the boundary between separate parts is unmistakable. This prevents the reader from misreading two parts as one solid piece.

Question 7Drawing standards, conventions and graphical literacy

What does the abbreviation 'A3' specify on a drawing sheet?

  1. The number of views required
  2. The drawing scale, 3:1
  3. The trimmed sheet size, 420 mm by 297 mm Correct answer.
  4. The paper weight in grams per square metre

Explanation

ISO A-series designations define trimmed sheet dimensions in a halving sequence, with A3 measuring 420 mm by 297 mm. The A0 sheet has an area of one square metre and each subsequent size halves the area.

Question 8Drawing standards, conventions and graphical literacy

A drawing is labelled 'Scale 1:5'. A feature measures 30 mm on the drawing. What is its real length?

  1. 35 mm
  2. 6 mm
  3. 150 mm Correct answer.
  4. 30 mm

Explanation

A ratio of 1:5 means one unit on the drawing represents five units on the object, so a drawing length is multiplied by five. Thirty millimetres on paper therefore corresponds to 150 mm in reality.

Question 9Drawing standards, conventions and graphical literacy

Which scale is appropriate for a detail drawing of a small machined component that must be enlarged for clarity?

  1. 1:2
  2. 1:50
  3. 2:1 Correct answer.
  4. 1:100

Explanation

A scale written with the larger number first, such as 2:1, is an enlargement and suits small parts that need to be shown bigger than life. Reduction scales are used for large objects such as buildings or site plans.

Question 10Drawing standards, conventions and graphical literacy

Which projection method places the side view to the right of the front view, showing the left-hand side of the object?

  1. First-angle projection Correct answer.
  2. Isometric projection
  3. Third-angle projection
  4. Oblique projection

Explanation

In first-angle projection the view seen from the left is projected onto a plane on the far side and so appears to the right of the front view. Third-angle places the left-side view on the left.

Question 11Drawing standards, conventions and graphical literacy

In isometric drawing, the three principal axes are conventionally drawn at which angles to the horizontal?

  1. 45 degrees, 45 degrees and vertical
  2. 15 degrees, 15 degrees and vertical
  3. 60 degrees, 60 degrees and vertical
  4. 30 degrees, 30 degrees and vertical Correct answer.

Explanation

Isometric axes are set at 30 degrees above the horizontal on each side with the third axis vertical, giving three equal angles of 120 degrees between axes. Measurements along these axes are drawn to true length in isometric drawing.

Question 12Drawing standards, conventions and graphical literacy

A dimension is given as 42 with a tolerance of plus or minus 0.2 mm. Which statement correctly interprets this?

  1. The acceptable size ranges from 42 mm to 42.2 mm
  2. The acceptable size ranges from 41.8 mm to 42 mm
  3. The acceptable size ranges from 41.8 mm to 42.2 mm Correct answer.
  4. The acceptable size is exactly 42 mm

Explanation

A bilateral tolerance of plus or minus 0.2 mm allows the actual size to deviate equally above and below the nominal value. The permitted limits are therefore 41.8 mm and 42.2 mm inclusive.

Question 13Drawing standards, conventions and graphical literacy

Which symbol on a drawing indicates that a surface must be produced by a machining operation?

  1. A surface texture symbol with a bar added to the tick Correct answer.
  2. A circle inscribed in the tick
  3. A plain tick without any bar
  4. A dashed triangle

Explanation

In surface texture notation, a tick with a horizontal bar added means material removal by machining is required. The plain tick denotes that machining is optional and the circle variant prohibits removal.

Question 14Drawing standards, conventions and graphical literacy

A drawing uses the symbol of a circle with a diagonal cross through it placed before a dimension. What does this symbol convey?

  1. The dimension is a square side
  2. The dimension is a counterbore depth
  3. The dimension is a diameter Correct answer.
  4. The dimension is a radius

Explanation

A circle crossed by a diagonal line is the standard diameter symbol and precedes the numerical value. The radius symbol is an R, and a square is indicated by a small square outline.

Question 15Drawing standards, conventions and graphical literacy

On a drawing, a dimension line terminates at an arrow touching an extension line. Which practice keeps the drawing to standard?

  1. Dimension lines are chain lines running along the object outline
  2. Dimension lines are thin and placed outside the object outline where possible Correct answer.
  3. Dimension lines are dashed and cross the object outline
  4. Dimension lines are thick and drawn inside the object outline

Explanation

Dimension lines are thin continuous lines drawn between extension lines and kept clear of the object's visible outlines wherever space allows. This separation stops dimensioning from being confused with the geometry itself.

Question 16Drawing standards, conventions and graphical literacy

A drawing shows a centre line extending slightly beyond the outline of a circular hole. What is the purpose of this projection?

  1. To indicate the centre of the feature clearly Correct answer.
  2. To show the hole is threaded
  3. To mark a surface finish requirement
  4. To indicate a hidden edge behind the hole

Explanation

Centre lines are extended a short distance past the feature outline so the centre point is unambiguous and can be used for setting out and measurement. It is a drafting convention rather than a manufacturing instruction.

Question 17Drawing standards, conventions and graphical literacy

A drawing contains a view labelled 'Section A-A'. What does the second letter indicate?

  1. The direction of viewing along the cutting plane Correct answer.
  2. The sheet number on which the section appears
  3. The scale of the section view
  4. The material of the component

Explanation

The letters on a cutting plane line identify the section and the arrows at the ends show the direction in which the cut is viewed. The same letters label the resulting section view so the reader can match it to the cut.

Question 18Drawing standards, conventions and graphical literacy

An assembly drawing includes a parts list. Which information is conventionally included in such a list?

  1. Surface finish of every face
  2. The drawing scale only
  3. Item number, description, quantity and material Correct answer.
  4. Tolerance of every dimension

Explanation

A parts list identifies each component of an assembly by item number and gives description, quantity and often material or specification. It links balloon references on the drawing to the components shown.

Question 19Drawing standards, conventions and graphical literacy

A drawing shows two views of the same object, and a feature appears as a rectangle in one view and a circle in the other. What does this indicate?

  1. The feature is a sphere
  2. The feature is a cylinder Correct answer.
  3. The feature is a cone
  4. The feature is a rectangular prism

Explanation

A circular profile in one view combined with a rectangular profile in a perpendicular view describes a cylinder whose axis is normal to the circular view. Reading paired views together is the basis of interpreting multiview drawings.

Question 20Drawing standards, conventions and graphical literacy

A dimension value is placed above a horizontal dimension line in a drawing. What is the standard convention for reading it?

  1. It reads from the bottom of the sheet, with figures upright Correct answer.
  2. It reads along the dimension line at any orientation
  3. It reads from the top of the sheet, upside down
  4. It reads from the right-hand edge of the sheet

Explanation

Dimension figures are placed above the dimension line and oriented so they read from the bottom or right-hand edge of the sheet. This keeps all text legible without rotating the drawing.

Question 21Geometric construction and plane geometry

In a standard drafting construction, which pair of instruments is used to bisect a straight line segment of about 80 mm accurately?

  1. A pair of compasses and a straightedge Correct answer.
  2. A flexicurve and a ruling pen
  3. A 45° set square and a T-square
  4. A protractor and a French curve

Explanation

Bisecting a line with classical construction requires drawing equal arcs from both endpoints and joining their intersections, which needs compasses plus a straightedge. A set square, protractor or flexicurve cannot produce the equal-radius arcs the construction depends on.

Question 22Geometric construction and plane geometry

A line AB is 90 mm long. Using compasses set to a radius greater than half of AB, arcs are struck from A and from B and the intersections joined. What is the perpendicular distance from AB to each arc intersection if the compass radius is 60 mm?

  1. 45 mm
  2. 30 mm
  3. 60 mm
  4. 39.7 mm Correct answer.

Explanation

Each intersection lies on a circle of radius 60 mm centred at an endpoint, so its distance from the midpoint is the perpendicular height of an isosceles triangle with sides 60, 60 and base 90. Half the base is 45 mm, giving height √(60² − 45²) = √1575 ≈ 39.7 mm.

Question 23Geometric construction and plane geometry

Which construction produces a line perpendicular to a given straight line at a point lying on that line?

  1. Bisect the given line and join the midpoint to any external point
  2. Join the point to the ends of the line and bisect the resulting angle
  3. Strike equal arcs on the line either side of the point, then intersect arcs of equal radius from those two marks Correct answer.
  4. Draw a 30° line through the point and add 60° with a set square

Explanation

Marking equal distances either side of the point creates two symmetric centres; equal-radius arcs from them intersect on the perpendicular through the point. The symmetry guarantees the joining line is at 90° to the original.

Question 24Geometric construction and plane geometry

A perpendicular is to be dropped from an external point P to a line, but P is so close to the line that arcs of convenient radius cannot be struck from P. Which approach is correct?

  1. Erect a perpendicular at a distant point and draw a parallel through P
  2. Choose any two points on the line, strike arcs from them through P, and join their intersections to P Correct answer.
  3. Draw a line through P parallel to the given line and bisect the gap
  4. Use the given line as a chord and construct a circle centred on P

Explanation

When P is too near the line, two arbitrary points on the line are used as centres for arcs passing through P. Joining P to the second intersection of those arcs gives the perpendicular, because P and the intersection are equidistant from both centres.

Question 25Geometric construction and plane geometry

An angle of 60° is to be constructed with compasses alone from a straight ray. What compass setting is used to strike the arc that cuts the ray?

  1. A radius equal to the chord of the finished angle
  2. Any radius, provided it is kept unchanged for the second arc Correct answer.
  3. A radius equal to twice the first arc length
  4. A radius of exactly 30 mm

Explanation

The construction relies on an equilateral triangle: the same radius is stepped from the vertex along the ray and then from that point to cut the first arc. Any radius works as long as it is not altered between the two arc strikes.

Question 26Geometric construction and plane geometry

To construct an angle of 75° using only compasses and a straightedge, which combination of standard constructions is used?

  1. Construct 60° and add half of a bisected 30° Correct answer.
  2. Construct 45° and add a bisected 30°
  3. Construct 60° and bisect the residual 30° twice
  4. Construct 90° and bisect the 15° between 60° and 90°

Explanation

A 30° angle is obtained by bisecting 60°, and bisecting that 30° gives 15°; adding this 15° to 60° yields 75°. The other routes give 45°, 67.5°, 52.5° or 75° only by coincidence of arithmetic, not by valid construction steps.

Question 27Geometric construction and plane geometry

Which statement correctly describes the bisection of an angle using compasses?

  1. The bisector joins the vertex to the midpoint of any chord across the angle
  2. The bisector is the line joining the two arc intersections on the arms
  3. The bisector is the locus of points equidistant from the two arms of the angle Correct answer.
  4. The bisector is found by measuring half the angle with a protractor only

Explanation

Every point on the bisector is the same perpendicular distance from both arms, which is why equal arcs from the arms meet on it. This equidistance property is the geometric definition underlying the construction.

Question 28Geometric construction and plane geometry

Three lines are to be drawn through a point on a sheet such that each is at 60° to the next, forming a symmetric three-line fan. What total angle is swept from the first line to the third?

  1. 180°
  2. 120° Correct answer.
  3. 240°
  4. 60°

Explanation

Two successive 60° steps separate the first and third lines, giving 120°. The third line is therefore not collinear with the first, which would require 180°.

Question 29Geometric construction and plane geometry

In constructing a triangle given its three side lengths, which instrument sequence gives the third vertex directly?

  1. Draw one side, then strike arcs of the other two lengths from its endpoints Correct answer.
  2. Draw one side, then construct 60° at each endpoint
  3. Draw one side, then bisect it and erect a perpendicular
  4. Draw one side, then erect perpendiculars at both endpoints

Explanation

The third vertex must be at a fixed distance from each endpoint, so arcs of the two remaining side lengths are struck from the endpoints. Their intersection satisfies both distances simultaneously.

Question 30Geometric construction and plane geometry

A triangle has sides 70 mm, 50 mm and 40 mm. When the 70 mm side is drawn as the base, how far from the base line does the third vertex lie?

  1. About 35.0 mm
  2. About 40.0 mm
  3. About 28.6 mm Correct answer.
  4. About 20.0 mm

Explanation

Using the altitude formula h = (2/A)√(s(s−a)(s−b)(s−c)) with semiperimeter 80 mm gives area ≈ 1000 mm². Dividing twice the area by the 70 mm base gives an altitude of about 28.6 mm.

Question 31Geometric construction and plane geometry

Which set of three lengths can form a triangle by construction?

  1. 35 mm, 45 mm, 70 mm Correct answer.
  2. 20 mm, 30 mm, 55 mm
  3. 30 mm, 40 mm, 80 mm
  4. 25 mm, 25 mm, 50 mm

Explanation

A triangle exists only when the sum of any two sides exceeds the third. For 35, 45 and 70 the sums are 80, 105 and 115, all greater than the opposite side, so the arcs intersect.

Question 32Geometric construction and plane geometry

In the construction of a regular hexagon inscribed in a circle, what compass setting is used to step around the circumference?

  1. Half the radius of the circle
  2. The side of an inscribed square
  3. The radius of the circle Correct answer.
  4. The diameter of the circle

Explanation

A regular hexagon divides the circle into six equilateral triangles, so each side equals the radius. Stepping the radius around the circumference therefore marks the six vertices exactly.

Question 33Geometric construction and plane geometry

A regular pentagon is to be constructed inside a circle of 50 mm radius. Which method is a valid compass construction for locating the vertices?

  1. Draw five tangents at 72° and use their intersections
  2. Divide the diameter into five equal parts and join to the centre
  3. Step off the radius around the circle five times
  4. Bisect the radius, then use the resulting chord length to step around the circle Correct answer.

Explanation

The standard pentagon construction bisects a radius and uses a compass arc from that midpoint to fix the chord length equal to the pentagon side. Stepping this chord five times closes exactly on the starting vertex.

Question 34Geometric construction and plane geometry

How many sides has a regular polygon whose exterior angle is 24°?

  1. 12
  2. 15 Correct answer.
  3. 18
  4. 24

Explanation

The exterior angles of any convex polygon sum to 360°, so the number of sides equals 360 ÷ 24 = 15. The interior angle of 156° confirms a fifteen-sided figure.

Question 35Geometric construction and plane geometry

Which construction gives the centre of a circle from three points lying on its circumference?

  1. Draw the perpendicular bisectors of two chords joining the points Correct answer.
  2. Join each point to the midpoint of the opposite chord
  3. Draw tangents at two of the points and bisect the angle between them
  4. Join the three points and bisect the largest angle

Explanation

The perpendicular bisector of any chord passes through the centre, so two such bisectors intersect at the circle's centre. Two chords are sufficient because their bisectors are not parallel.

Question 36Geometric construction and plane geometry

A circle of radius 40 mm is to have a tangent drawn from an external point 100 mm from its centre. What is the length of the tangent from that point to the point of contact?

  1. About 91.7 mm Correct answer.
  2. About 80.0 mm
  3. About 107.7 mm
  4. About 60.0 mm

Explanation

The radius to the point of contact is perpendicular to the tangent, forming a right triangle with hypotenuse 100 mm and one leg 40 mm. The tangent length is √(100² − 40²) = √8400 ≈ 91.7 mm.

Question 37Geometric construction and plane geometry

When constructing a common external tangent to two equal circles, which line is drawn first?

  1. A circle centred midway between the two centres
  2. A line joining the two centres Correct answer.
  3. A chord across one circle parallel to the other centre
  4. A line through one centre parallel to the required tangent

Explanation

Joining the centres establishes the axis of symmetry, and the external tangent is then drawn parallel to this line at a distance equal to the radius. For equal circles the tangent is therefore parallel to the centre line.

Question 38Geometric construction and plane geometry

Which conic is produced when a plane cuts a right circular cone parallel to exactly one generator?

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

Explanation

A section parallel to a single generator has an eccentricity of one, which defines a parabola. Sections at smaller inclination to the base give ellipses, and steeper sections cutting both nappes give hyperbolas.

Question 39Geometric construction and plane geometry

An ellipse has a major axis of 120 mm and a minor axis of 80 mm. What is the distance between its two foci?

  1. About 44.7 mm
  2. About 120.0 mm
  3. About 89.4 mm Correct answer.
  4. About 63.2 mm

Explanation

For an ellipse the focal distance from the centre is √(a² − b²), with semi-axes 60 mm and 40 mm. That gives √(3600 − 1600) = √2000 ≈ 44.7 mm from the centre, so the separation between foci is about 89.4 mm.

Question 40Geometric construction and plane geometry

In the concentric-circle method for drawing an ellipse, what determines the spacing of the construction lines?

  1. The number of equal divisions chosen on the two circles Correct answer.
  2. The tangent angle at the ends of the major axis
  3. The focal distance between the two foci
  4. The length of the minor axis only

Explanation

The method divides both the major-axis circle and the minor-axis circle into the same number of equal sectors. Corresponding horizontal and vertical construction lines then intersect at points on the ellipse; more divisions give a smoother curve.

Question 41Geometric construction and plane geometry

Which construction is used to inscribe a circle inside a given triangle?

  1. Draw the altitudes and use their intersection as the centre
  2. Bisect two of the triangle's sides and use the intersection as the centre
  3. Bisect two of the triangle's angles and use the intersection as the centre Correct answer.
  4. Drop a perpendicular from the largest angle to the opposite side

Explanation

The incentre is equidistant from all three sides, and that property belongs to the angle bisectors. Dropping a perpendicular from the incentre to any side then gives the radius of the inscribed circle.

Question 42Geometric construction and plane geometry

A circle is to be drawn through the three vertices of an obtuse triangle. Where does its centre lie?

  1. Outside the triangle, on the side of the longest side Correct answer.
  2. At the intersection of the three altitudes
  3. Exactly on the longest side
  4. Inside the triangle

Explanation

The circumcentre is the intersection of the perpendicular bisectors of the sides. For an obtuse triangle the circumcentre falls outside the triangle, beyond the longest side, because the obtuse angle subtends more than a semicircle.

Question 43Geometric construction and plane geometry

Two circles of radii 30 mm and 20 mm have centres 60 mm apart. Which statement about their common tangents is correct?

  1. There are three common tangents because the circles nearly touch
  2. There are no common tangents because the circles are separate
  3. There are four common tangents, two external and two internal Correct answer.
  4. There are only two common tangents, both external

Explanation

Because the centre distance of 60 mm exceeds the sum of the radii (50 mm), the circles are separate and do not intersect. Separate circles always have four common tangents: two external and two crossing internal tangents.

Question 44Geometric construction and plane geometry

Which construction transfers the exact shape of an irregular figure to a new position on the drawing sheet?

  1. Trace the figure with a flexicurve and redraw freehand
  2. Construct similar triangles on each edge of the figure
  3. Draw a grid of squares over the figure and reproduce corresponding squares elsewhere Correct answer.
  4. Bisect each side of the figure and join the midpoints

Explanation

A square grid provides a coordinate framework, so each vertex can be located by counting squares in the new position. Reproducing the same count in each cell preserves the shape exactly.

Question 45Geometric construction and plane geometry

A regular octagon is inscribed in a circle of 60 mm diameter. What is the angle subtended at the centre by one side?

  1. 72°
  2. 45° Correct answer.
  3. 60°
  4. 30°

Explanation

The circle is divided into as many equal sectors as the polygon has sides, so each central angle is 360° ÷ 8 = 45°. The diameter affects the side length but not the central angle.

Question 46Geometric construction and plane geometry

A straight line AB of length 90 mm is to be divided into five equal parts using only a straightedge, a pair of compasses and a set square. Which construction sequence is correct?

  1. Draw a ray from A at any convenient angle, step off five equal divisions with the compasses, join the last division to B, then draw parallels to that joining line through the other division points Correct answer.
  2. Draw a ray from A, step off five equal divisions, then join the first division point to B and transfer that direction to the remaining points
  3. Bisect AB, bisect each half, then bisect the two outer quarters to obtain five equal parts
  4. Construct the perpendicular bisector of AB and step the compasses along AB from that midpoint in both directions

Explanation

The standard construction for proportional division is to set off equal intercepts along an inclined ray from one end, join the final intercept to the far end of the given line, and then draw lines parallel to that join through each intercept. The parallels cut AB in the same ratio as the ray, giving five equal parts. The parallel lines must be drawn with the set square slid along a straightedge to keep them truly parallel.

Question 47Geometric construction and plane geometry

A line PQ is 100 mm long. It is required to locate the point R on PQ such that PR:RQ = 3:2. What is the length PR?

  1. 40 mm
  2. 66.7 mm
  3. 50 mm
  4. 60 mm Correct answer.

Explanation

The ratio 3:2 means the line is divided into 3 + 2 = 5 equal parts, each 100 mm ÷ 5 = 20 mm. PR corresponds to three of these parts, so PR = 3 × 20 mm = 60 mm, leaving RQ = 40 mm. The construction would step five equal intercepts along an inclined ray and draw parallels to divide PQ in that ratio.

Question 48Geometric construction and plane geometry

A line AB is 80 mm long and a point P lies 30 mm above its midpoint. A circular arc of radius 50 mm centred on P is to be drawn and its points of intersection with AB located. Which statement about this construction is correct?

  1. The arc is tangent to AB because the perpendicular distance equals the radius
  2. The arc does not meet AB because the perpendicular distance is less than the radius
  3. The arc cuts AB at two points because the perpendicular distance from P to AB is less than the radius Correct answer.
  4. The arc cuts AB at exactly one point, which is the midpoint of AB

Explanation

The number of intersections depends on comparing the perpendicular distance from the centre to the line with the radius. Here the distance is 30 mm and the radius is 50 mm, so the arc crosses the line at two points, each 40 mm from the foot of the perpendicular since √(50² − 30²) = 40 mm. Both intersections lie within the 80 mm line, 40 mm either side of the midpoint.

Question 49Geometric construction and plane geometry

An arc of radius 65 mm is to be drawn through two given points A and B that are 50 mm apart. How is the centre of this arc located?

  1. Set the compasses to 65 mm and strike arcs from A and from B; their intersection is the centre Correct answer.
  2. Draw the perpendicular bisector of AB and mark the centre where it crosses AB
  3. Bisect the angle at A and set off 65 mm along the bisector to find the centre
  4. Erect a perpendicular at A of length 65 mm and use its end as the centre

Explanation

The centre of any arc of given radius passing through two points must be equidistant from both, so it lies at the intersection of two arcs of that radius struck from A and from B. Because 65 mm exceeds half of AB, the two arcs meet at two points, giving two possible centres on opposite sides of AB; either may be chosen depending on which way the arc is to curve.

Question 50Geometric construction and plane geometry

A regular heptagon is to be constructed on a given side AB using the general polygon-on-a-side method. Which description of the construction is correct?

  1. Construct an equilateral triangle on AB and use its apex as the centre of the circumscribing circle
  2. Bisect AB, erect an angle of 360° ÷ 7 at the midpoint, and step the side length from the bisector
  3. Erect the perpendicular bisector of AB, construct an angle of 360° ÷ 14 at A, mark where its arm meets the bisector as a centre, then step the side length around that centre Correct answer.
  4. Draw a semicircle on AB, divide it into seven equal parts, and use the second division point as the centre

Explanation

In the general method the centre of the circumscribing circle is found by erecting the perpendicular bisector of the side and constructing at one end an angle equal to half the exterior angle, that is 180° ÷ 7 or 360° ÷ 14. The arm of this angle meets the perpendicular bisector at the required centre, from which the remaining vertices are stepped off with the compasses set to AB. For a heptagon the half-exterior angle is about 25.7°.