Try each of these eight original questions before opening its answer. They cover the four lessons in geometric construction reasoning.
Questions
1. To bisect AB = 80 mm, a student sets the compass to 30 mm for the arcs from A and B. What goes wrong, and what is the smallest whole-number radius that works?
Answer
The radius must be more than half of AB, which is 40 mm. With 30 mm the arcs do not meet. The smallest whole-number radius is 41 mm.
2. Explain in one sentence why the compass stays at the same radius for both arcs in the perpendicular bisector construction.
Answer
The same radius makes the meeting point equally far from A and B, and every point equally far from A and B lies on the perpendicular bisector of AB.
3. A circle of radius 20 mm must touch a line L at point T. A student places the centre 20 mm from T along a line slanting at 70° to L. Explain why the circle does not touch L only at T.
Answer
The centre must be on the perpendicular to L at T. On a slanting line the radius OT is not at right angles to L, so the circle crosses L at a second point.
4. Diagram A draws an equilateral triangle on AB = 45 mm using arcs of 45 mm from both ends. Diagram B uses 45 mm from A and 40 mm from B. Which has BC = 45 mm?
Answer
Diagram A. In Diagram B, BC = 40 mm and the triangle is isosceles, not equilateral.
5. A hexagon is inscribed in a circle of radius 35 mm. State the side length and the distance between two opposite vertices.
Answer
The side equals the radius, 35 mm. Opposite vertices are a diameter apart: 2 × 35 = 70 mm.
6. Construct triangle PQR with PQ = 80, PR = 45 and QR = 30 mm. Explain why no such triangle exists.
Answer
The two shorter sides must be longer in total than the longest side: 45 + 30 = 75, which is less than 80. The arcs from P and Q cannot meet.
7. A square ABCD with side 50 mm is constructed: AB = 50, a perpendicular at B with C marked 50 mm from B, a perpendicular at A with D marked 50 mm from A, then DC joined. List the conditions and say whether they are all met.
Answer
Conditions: four sides of 50 mm, four right angles. AB, BC and AD are 50 mm; the angles at A and B are right angles. Since AD and BC are both perpendicular to AB and equal, DC is parallel to AB and equals 50 mm, and all angles are right angles. All conditions are met.
8. In a construction the steps are: (1) draw line AB = 60 mm; (2) arc from A, radius 60; (3) arc from B, radius 55; (4) join the meeting point C to A and B. The aim is an equilateral triangle. Name the first inconsistent line and explain why fixing step 4 is not enough.
Answer
Step 3 is the first inconsistent line: it should be radius 60. Step 4 only joins the point that step 3 already placed wrongly, so changing step 4 leaves C in the wrong position.
If you got these wrong
- Questions 1 to 3: reread explaining a construction using its constraints.
- Question 4: reread comparing construction steps in two diagrams.
- Questions 5 to 7: reread checking a completed construction.
- Question 8: reread finding the first inconsistent line.
Next, move to orthographic views.
If you want a teacher to set new constructions and check your reasons, see online one-to-one Engineering Drawing tuition.