GKT101: General Knowledge for Teachers – Math

Example 1

Michael is 6 feet tall and is standing outside next to his younger sister. He notices that he can see both of their shadows and decides to measure each shadow. His shadow is 8 feet long and his sister's shadow is 5 feet long. How tall is Michael's sister?

You can answer this question by applying what you know about similar triangles.

The sun creates shadows at the same angle for both Michael and his sister. Assuming they are both standing up straight and making right angles with the ground, similar triangles are created.

[Figure 7]

Corresponding sides are proportional because the triangles are similar.

$\begin{aligned}\frac{\text { Michael's Height }}{\text { Sister's Height }} &=\frac{\text { Length of Michael's Shadow }}{\text { Length of Sister's Shadow }} \\\frac{6 ft }{\text { Sister's Height }} &=\frac{8 ft }{5 ft } \\\text { Sister's Height } &=\frac{6 \cdot 5}{8} \\&=3.75 ft\end{aligned}$

His sister is $3.75$ feet tall.

Example 2

Prove that all isosceles right triangles are similar.

Example 3

Find the measures of the angles of an isosceles right triangle. Why are isosceles right triangles called $45-45-90$ triangles?

• An isosceles right triangle is called a $45-45-90$ triangle because those are its angle measures.

Example 4

Use the Pythagorean Theorem to find the missing side of an isosceles right triangle whose legs are each length $x$.

[Figure 9]

The missing side is the hypotenuse of the right triangle, labeled $c$.

By the Pythagorean Theorem,

$\begin{aligned}A B^{2} &=A C^{2}+B C^{2} \\c^{2} &=x^{2}+x^{2} \\c^{2} &=2 x^{2} \\c &=\sqrt{2 x^{2}} \\c &=x \sqrt{2}\end{aligned}$

• The ratio of the sides of any isosceles right triangle will be $x: x: x \sqrt{2}$ which simplifies to $1: 1: \sqrt{2}$.

Example 5

Find the missing sides of the right triangle below without using the Pythagorean Theorem.

[Figure 10]