Examples of Larger Constructions

You’re going to construct a triangle $△ A B C$ and a quadrilateral $□ A B C D$ with the following specifications:

1.: $∠ A$ is $60$ °, the line $A B$ is $5 cm$ long, and $∠ B$ is $90$ °. Construct the triangle $△ A B C$ . What kind of triangle is this? How large is $∠ C$ ?
2.: The area of the triangle $△ A B C$ in Exercise 1 is half the area of the rectangle $□ A B C D$ . How long are the sides in the rectangle? How long is the diagonal?

Example 1

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:

Example 2

Continue the construction by constructing the quadrilateral $□ A B C D$ :

Construct a normal on the line $B C$ at $C$ .
Construct a normal on $A B$ at $A$ . Name the intersection between the two normals $D$ . You have now constructed the rectangle $□ A B C D$ .
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 $2 \cdot 5 cm = 10 cm$ .
To find the height of the triangle, you use the Pythagorean theorem: $\begin{array}{l} a^{2} + b^{2} & = c^{2} \\ B C^{2} + 5^{2} & = 1 0^{2} \\ B C^{2} + 25 & = 100 \\ B C & = 75 \\ \sqrt{B C^{2}} & = \sqrt{75} \\ B C & \approx 8.67 cm \end{array}$
You’ve found that the sides are $5 cm$ and $8.67 cm$ , and the diagonal is $10 cm$ .

Description of construction of a rectangle