Question

in exercises 1 and 2, copy and complete the table. write your answers in simplest form.
1.

2.

1. the side lengths of a triangle are given. determine whether each triangle is

a ( 45^circ\text{-}45^circ\text{-}90^circ ) triangle, a ( 30^circ\text{-}60^circ\text{-}90^circ ) triangle, or neither.

a. ( 5, 10, 5sqrt{3} )
b. ( 7, 7, 7sqrt{3} )
c. ( 6, 6, 6sqrt{2} )

for problems 4 - 7, solve for the value of each variable.
4.

5.

6.

7.

Explanation:

Exercise 1

This is a 45-45-90 right triangle, where hypotenuse $y = x\sqrt{2}$, and $x = \frac{y}{\sqrt{2}} = \frac{y\sqrt{2}}{2}$.

Step1: Find y when x=5

$y = 5\sqrt{2}$

Step2: Find x when y=$4\sqrt{2}$

$x = \frac{4\sqrt{2}}{\sqrt{2}} = 4$

Step3: Find y when x=$\sqrt{2}$

$y = \sqrt{2} \times \sqrt{2} = 2$

Step4: Find x when y=24

$x = \frac{24}{\sqrt{2}} = 12\sqrt{2}$

This is a 30-60-90 right triangle: $a = b\sqrt{3}$, $b = \frac{a}{\sqrt{3}} = \frac{a\sqrt{3}}{3}$, $c = 2b = \frac{2a}{\sqrt{3}} = \frac{2a\sqrt{3}}{3}$.

Step1: Find b,c when a=11

$b = \frac{11\sqrt{3}}{3}$, $c = \frac{22\sqrt{3}}{3}$

Step2: Find a,c when b=9

$a = 9\sqrt{3}$, $c = 18$

Step3: Find a,b when c=16

$b = 8$, $a = 8\sqrt{3}$

Step4: Find a,c when b=$5\sqrt{3}$

$a = 15$, $c = 10\sqrt{3}$

• 45-45-90: Sides are $s, s, s\sqrt{2}$
• 30-60-90: Sides are $s, s\sqrt{3}, 2s$

Part a: 5,10,$5\sqrt{3}$

Match $s=5, 2s=10, s\sqrt{3}=5\sqrt{3}$

Part b: 7,7,$7\sqrt{3}$

Does not match either ratio

Part c: 6,6,$6\sqrt{2}$

Match $s=6, s=6, s\sqrt{2}=6\sqrt{2}$

Answer:

$x$	5	4	$\sqrt{2}$	$12\sqrt{2}$

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Exercise 2