Question

part a
what are the exact side lengths of the triangle shown?
4 in. 45° c
45°
b
b =
inches
c =
inches
part b
what are the exact side lengths of the triangle shown?
30° c
21 cm
60°
a

Explanation:

Step1: Identify triangle type for Part A

This is a 45 - 45 - 90 right - triangle. In a 45 - 45 - 90 triangle, the legs are congruent. Given one leg is 4 inches, so $b = 4$ inches.

Step2: Find hypotenuse for Part A

The ratio of the sides of a 45 - 45 - 90 triangle is $a:a:a\sqrt{2}$, where $a$ is the length of a leg and $a\sqrt{2}$ is the hypotenuse. So $c=4\sqrt{2}$ inches.

Step3: Identify triangle type for Part B

This is a 30 - 60 - 90 right - triangle. The ratio of the sides of a 30 - 60 - 90 triangle is $a:a\sqrt{3}:2a$, where $a$ is the shorter leg, $a\sqrt{3}$ is the longer leg and $2a$ is the hypotenuse. Given the longer leg is 21 cm. If the shorter leg is $a$ and the longer leg is $a\sqrt{3}$, then $a\sqrt{3}=21$, so $a = \frac{21}{\sqrt{3}}=\frac{21\sqrt{3}}{3}=7\sqrt{3}$ cm.

Step4: Find hypotenuse for Part B

Since the hypotenuse $c = 2a$, and $a = 7\sqrt{3}$ cm, then $c = 14\sqrt{3}$ cm.

Answer:

Part A:
$b = 4$ inches
$c = 4\sqrt{2}$ inches
Part B:
$a = 7\sqrt{3}$ cm
$c = 14\sqrt{3}$ cm