Examples of Larger Constructions
You’re going to construct a triangle △ABC and a quadrilateral □ABCD with the following specifications:
What kind of triangle is this? How large is ∠C?
How long are the sides in the rectangle? How long is the diagonal?
Make an auxiliary figure, which is a small figure of what you’re going to construct. Mark the angles, lengths, and any other information.
Begin the construction:
Draw a long line.
Mark point A and measure 5cm along the line to point B.
Construct 60° in A.
Construct 90° in B.
Name the intersection between ∠A and ∠B C.
This is a right triangle.
∠C = 180° − 90° − 60° = 30°.
Continue the construction by constructing the quadrilateral □ABCD:
Construct a normal on the line BC in C.
Construct a normal on AB in A. Name the intersection between the two normals D. You have now constructed the rectangle □ABCD.
Because the rectangle is made up of two “30°-60°-90°” triangles, the hypotenuse is twice the length of the shortest leg. That means the diagonal is 2 ⋅ 5cm = 10cm.
To find the height of the triangle, you use the Pythagorean theorem: a2 + b2 = c2 BC2 + 52 = 102 BC2 + 25 = 100 BC = 75 BC2 = 75 BC = 8.67cm
You’ve found that the sides are 5cm and 8.67cm, and the diagonal is 10cm.