Geometric Construction — Challenge
With only a compass and a straightedge you can copy, bisect and build exactly — and the angle facts tell you when a figure is even possible.
What these problems cover
These questions use 14 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Complement — Everything except the event. P(not A) = 1 - P(A).
- Vertex — A flat surface; a line where two faces meet; a corner point.
- Compass — Tools for drawing exact circles and lines — no measuring allowed.
- Complementary — Two angles summing to 90° / to 180°.
- Supplementary — Two angles summing to 90° / to 180°.
- Vertical angles — The opposite pair formed by two crossing lines — always equal.
- Base — The number being multiplied. In 53 the base is 5.
- Arc — Part of the edge; a "pizza slice" bounded by two radii.
- Transversal — A line that crosses two or more other lines.
- Co-interior angles — Between the parallels, same side. They add to 180°.
- Exterior angle — The angle formed by extending one side of a triangle.
- Parallel — Same slope, different intercept.
- Segment — Two endpoints and everything between them. Written AB.
- Limit — The value a function approaches as the input approaches some number.
Problem 1
Which two tools are allowed in a classical geometric construction?
- Compass and straight edge
- Protractor and ruler
- A calculator
- Set square and protractor
- Compass and protractor
Worked answer
Compass and straight edge
A classical construction allows only a compass, for drawing circles, and an unmarked straight edge, for drawing lines.
A ruler's markings and a protractor's degrees are not permitted, because the point is to build figures from geometry rather than from measurement.
This restriction is why some constructions, such as trisecting a general angle, are impossible.
Problem 2
Bisecting an angle produces two angles that are what?
- Equal
- Supplementary
- Complementary
- Right angles
- Unequal
Worked answer
Equal
To bisect means to cut into two equal parts.
So the two resulting angles are equal, each half of the original.
They are complementary only in the special case where the original angle was 90 degrees.
Problem 3
Two parallel lines are cut by a transversal. One interior angle is 100°. What is the interior angle on the same side of the transversal?
- 100°
- 90°
- 70°
- 180°
- 80°
Worked answer
80°
Same-side interior angles sum to 180°.
The missing angle is 180°-100°=80°.
Problem 4
Triangle ABC has AB=AC and ∠ B=19°. What is ∠ A?
- 142°
- 19°
- 38°
- 71°
- 161°
Worked answer
142°
The equal sides imply ∠ C=∠ B=19°.
Then ∠ A=180°-19°-19°=142°.
Problem 5
The perpendicular bisector of a segment passes through which point of it?
- Its midpoint
- One endpoint
- Neither endpoint nor midpoint
- A point one third along
- Its longest point
Worked answer
Its midpoint
A bisector divides the segment into two equal halves, so it must cross at the middle.
Perpendicular means it crosses at a right angle.
So the perpendicular bisector passes through the midpoint at 90 degrees.
Problem 6
An isosceles trapezoid has parallel bases AB and CD, with ∠ A=95°. What is ∠ C?
- 85°
- 95°
- 90°
- 47.5°
- 170°
Worked answer
85°
The equal base angles give ∠ B=95°.
Angles B and C lie on the same side of a transversal through parallel bases, so C=180°-95°=85°.
Problem 7
What is true of every point on the perpendicular bisector of segment AB?
- It is the same distance from A as from B
- It is closer to A
- It is closer to B
- It lies on segment AB
- It is the midpoint
Worked answer
It is the same distance from A as from B
Take any point P on the bisector and join it to A and to B.
The two triangles formed share the segment from P to the midpoint, have equal halves of AB, and both have a right angle there.
So the triangles are congruent by SAS, which makes PA = PB. This is why the bisector is the set of points equidistant from A and B.
Problem 8
A circle has centre O. Points B,C,D lie on it, ∠ BOC=45°, and D lies on the major arc BC. What is ∠ BDC?
- 45°
- 90°
- 22.5°
- 67.5°
- 135°
Worked answer
22.5°
The minor arc BC measures 45°.
The inscribed angle intercepting it is half that: 22.5°.
Problem 9
To construct an equilateral triangle on a given segment, what radius should the two arcs have?
- The length of the segment
- Half the segment
- Twice the segment
- Any convenient length
- The distance to the midpoint
Worked answer
The length of the segment
Set the compass to the length of the segment and draw an arc centred at each endpoint.
Where the arcs cross is a point whose distance to both endpoints equals the segment's length.
All three sides are then equal, so the triangle is equilateral. A different radius would give an isosceles triangle instead.
Problem 10
Bisecting a right angle gives an angle of how many degrees?
- 30
- 45
- 60
- 90
- 180
Worked answer
45
A right angle is 90 degrees.
Bisecting halves it: 902 = 45 degrees.
Bisecting again would give 22.5 degrees, and this halving can be repeated indefinitely with compass and straight edge.