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} )

in exercises 4–6, find the values of the variables. write your answers in simplest form.
4.

5.

6.

1. you build a two - person tent, as shown. how many square feet of material is

needed to make the tent, assuming the tent has a floor?

Explanation:

Exercise 1 (45°-45°-90° Triangle)

Step1: Recall 45-45-90 side rules

In a 45°-45°-90° triangle, legs are equal ($x_1=x_2=x$), hypotenuse $y = x\sqrt{2}$, so $x=\frac{y}{\sqrt{2}}=\frac{y\sqrt{2}}{2}$.

Step2: Calculate $y$ when $x=5$

$y=5\sqrt{2}$

Step3: Calculate $x$ when $y=4\sqrt{2}$

$x=\frac{4\sqrt{2}\cdot\sqrt{2}}{2}=\frac{4\cdot2}{2}=4$

Step4: Calculate $y$ when $x=\sqrt{2}$

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

Step5: Calculate $x$ when $y=24$

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

Step1: Recall 30-60-90 side rules

In a 30°-60°-90° triangle: shorter leg $b$, longer leg $a=b\sqrt{3}$, hypotenuse $c=2b$; so $b=\frac{a}{\sqrt{3}}=\frac{a\sqrt{3}}{3}$, $a=\frac{c\sqrt{3}}{2}$, $b=\frac{c}{2}$.

Step2: Calculate $b,c$ when $a=11$

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

Step3: Calculate $a,c$ when $b=9$

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

Step4: Calculate $a,b$ when $c=16$

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

Step5: Calculate $a,c$ when $b=5\sqrt{3}$

$a=5\sqrt{3}\cdot\sqrt{3}=15$, $c=10\sqrt{3}$

Step1: Identify 30-60-90/45-45-90 rules

• 45°-45°-90°: Two equal sides, hypotenuse = side$\cdot\sqrt{2}$
• 30°-60°-90°: Sides in ratio $1:\sqrt{3}:2$

Step2: Analyze part a ($5,10,5\sqrt{3}$)

Ratio: $5:5\sqrt{3}:10 = 1:\sqrt{3}:2$ → 30°-60°-90°

Step3: Analyze part b ($7,7,7\sqrt{3}$)

Does not match either ratio → neither

Step4: Analyze part c ($6,6,6\sqrt{2}$)

Two equal sides, hypotenuse $6\sqrt{2}=6\cdot\sqrt{2}$ → 45°-45°-90°

Answer:

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

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Exercise 2 (30°-60°-90° Triangle)