Question

9.2 - extra practice
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 $45^\circ$-$45^\circ$-$90^\circ$ triangle, a $30^\circ$-$60^\circ$-$90^\circ$ triangle, or neither.
a. $5, 10, 5\sqrt{3}$
b. $7, 7, 7\sqrt{3}$
c. $6, 6, 6\sqrt{2}$
for problems 4 - 7, solve for the value of each variable.
4.
5.
6.
7.

Explanation:

Exercise 1 (45-45-90 Triangle)

Step1: Recall 45-45-90 ratios

In a 45-45-90 triangle, legs are equal ($x = x$), hypotenuse $y = x\sqrt{2}$, or $x = \frac{y}{\sqrt{2}} = \frac{y\sqrt{2}}{2}$.

Step2: Fill first column ($x=5$)

$y = 5\sqrt{2}$

Step3: Fill second column ($y=4\sqrt{2}$)

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

Step4: Fill third column ($x=\sqrt{2}$)

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

Step5: Fill fourth column ($y=24$)

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

Step1: Recall 30-60-90 ratios

In a 30-60-90 triangle: short leg $a$, long leg $b = a\sqrt{3}$, hypotenuse $c = 2a$.

Step2: Fill first column ($a=11$)

$b = 11\sqrt{3}$, $c = 2 \times 11 = 22$

Step3: Fill second column ($b=9$)

$a = \frac{9}{\sqrt{3}} = 3\sqrt{3}$, $c = 2 \times 3\sqrt{3} = 6\sqrt{3}$

Step4: Fill third column ($c=16$)

$a = \frac{16}{2} = 8$, $b = 8\sqrt{3}$

Step5: Fill fourth column ($b=5\sqrt{3}$)

$a = \frac{5\sqrt{3}}{\sqrt{3}} = 5$, $c = 2 \times 5 = 10$

Step1: 30-60-90 check

Sides follow $a, a\sqrt{3}, 2a$ (smallest, medium, largest).

Step2: 45-45-90 check

Sides follow $a, a, a\sqrt{2}$ (two equal, hypotenuse $a\sqrt{2}$).

Step3: Part a ($5,10,5\sqrt{3}$)

Order: $5, 5\sqrt{3}, 10$. Matches $a, a\sqrt{3}, 2a$ ($a=5$).

Step4: Part b ($7,7,7\sqrt{3}$)

Does not match either ratio set.

Step5: Part c ($6,6,6\sqrt{2}$)

Matches $a, a, a\sqrt{2}$ ($a=6$).

Answer:

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

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Exercise 2 (30-60-90 Triangle)