Question

8 practice 8 (from unit 4, lesson 1)
here are two triangles, each with an unknown side length.
find the unknown side in each triangle using any method. check your answers using a different method.
type your answers in the boxes.
x = units
y = units
how did i do?

Explanation:

Step1: Find \( x \) using Pythagorean theorem

The small triangle is a right triangle with legs 3 and \( x \), hypotenuse 5. By Pythagorean theorem: \( 3^2 + x^2 = 5^2 \)
\( 9 + x^2 = 25 \)
\( x^2 = 25 - 9 = 16 \)
\( x = \sqrt{16} = 4 \)

Step2: Check \( x \) using similarity (optional, for checking)

The two triangles are similar (same angles, right triangles with a common acute angle). The ratio of legs: \( \frac{3}{9} = \frac{1}{3} \), \( \frac{x}{12} \) should also be \( \frac{1}{3} \), so \( x = 12 \times \frac{1}{3} = 4 \), matches.

Step3: Find \( y \) using Pythagorean theorem

The large triangle is right-angled with legs 12 and 9, hypotenuse \( y \). By Pythagorean theorem: \( 12^2 + 9^2 = y^2 \)
\( 144 + 81 = y^2 \)
\( y^2 = 225 \)
\( y = \sqrt{225} = 15 \)

Step4: Check \( y \) using similarity (optional, for checking)

Ratio of corresponding sides: \( \frac{5}{y} = \frac{3}{9} = \frac{1}{3} \), so \( y = 5 \times 3 = 15 \), matches.

Answer:

\( x = 4 \) units
\( y = 15 \) units