Question

45-45-90 triangles
date__________ period
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find the missing side lengths. leave your answers as radicals in simplest form.
1)
$x =$
$y =$
$h=2$
2)
$h=8\sqrt{2}$
3)
$h=9$
4)
$h=9\sqrt{2}$
5)
6)
$h=10$
7)
8)
9)
10)
$h=5\sqrt{2}$

Explanation:

Step1: Recall 45-45-90 triangle rules

In a 45-45-90 triangle:

• Legs are equal: $l_1 = l_2$
• Hypotenuse $h = l\sqrt{2}$, leg $l = \frac{h}{\sqrt{2}} = \frac{h\sqrt{2}}{2}$

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Problem 1

Step1: Identify given side (leg)

Given leg $l = \sqrt{2}$

Step2: Calculate hypotenuse $x$

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

Step3: Equal legs, so $y$ = given leg

$y = \sqrt{2}$

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Problem 2

Step1: Identify given leg

Given leg $l = 8$

Step2: Calculate hypotenuse $h$

$h = 8 \times \sqrt{2} = 8\sqrt{2}$

Step3: Equal legs, so $n$ = given leg

$n = 8$

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Problem 3

Step1: Identify given hypotenuse

Given hypotenuse $h = 9\sqrt{2}$

Step2: Calculate leg length $a=b$

$a = b = \frac{9\sqrt{2}}{\sqrt{2}} = 9$

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Problem 4

Step1: Identify given leg

Given leg $l = 9$

Step2: Calculate hypotenuse $h$

$h = 9 \times \sqrt{2} = 9\sqrt{2}$

Step3: Equal legs, so $y$ = given leg

$y = 9$

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Problem 5

Step1: Identify given hypotenuse

Given hypotenuse $h = 2$

Step2: Calculate leg length $x=y$

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

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Problem 6

Step1: Identify given leg

Given leg $l = 5\sqrt{2}$

Step2: Calculate hypotenuse $h$

$h = 5\sqrt{2} \times \sqrt{2} = 5 \times 2 = 10$

Step3: Equal legs, so $a=b$

$a = b = 5\sqrt{2}$

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Problem 7

Step1: Identify given hypotenuse

Given hypotenuse $h = 10$

Step2: Calculate leg length $u=v$

$u = v = \frac{10\sqrt{2}}{2} = 5\sqrt{2}$

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Problem 8

Step1: Identify given leg

Given leg $l = \frac{9\sqrt{2}}{2}$

Step2: Equal legs, so $v$ = given leg

$v = \frac{9\sqrt{2}}{2}$

Step3: Calculate hypotenuse $u$

$u = \frac{9\sqrt{2}}{2} \times \sqrt{2} = \frac{9 \times 2}{2} = 9$

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Problem 9

Step1: Identify given hypotenuse

Given hypotenuse $h = 3\sqrt{5}$

Step2: Calculate leg length $x=y$

$x = y = \frac{3\sqrt{5} \times \sqrt{2}}{2} = \frac{3\sqrt{10}}{2}$

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Problem 10

Step1: Identify given leg

Given leg $l = 5$

Step2: Calculate hypotenuse $x$

$x = 5 \times \sqrt{2} = 5\sqrt{2}$

Step3: Equal legs, so $y$ = given leg

$y = 5$

Answer:

1. $x=2$, $y=\sqrt{2}$
2. $h=8\sqrt{2}$, $n=8$
3. $a=9$, $b=9$
4. $h=9\sqrt{2}$, $y=9$
5. $x=\sqrt{2}$, $y=\sqrt{2}$
6. $h=10$, $a=5\sqrt{2}$, $b=5\sqrt{2}$
7. $u=5\sqrt{2}$, $v=5\sqrt{2}$
8. $u=9$, $v=\frac{9\sqrt{2}}{2}$
9. $x=\frac{3\sqrt{10}}{2}$, $y=\frac{3\sqrt{10}}{2}$
10. $x=5\sqrt{2}$, $y=5$