Question

7.
8.
9.
10.

Explanation:

All these are isosceles right triangles (45-45-90 triangles), where the two legs are equal, and the hypotenuse is $\text{leg} \times \sqrt{2}$.

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Problem 7

Step1: Identify triangle type

This is a 45-45-90 triangle, so legs $u = v$, hypotenuse $= 8\sqrt{2}$.

Step2: Solve for legs

$\text{Hypotenuse} = u\sqrt{2} = 8\sqrt{2}$
$u = \frac{8\sqrt{2}}{\sqrt{2}} = 8$, so $v = u = 8$

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Problem 8

Step1: Identify triangle type

This is a 45-45-90 triangle, so leg $a = 20$, hypotenuse $b = a\sqrt{2}$.

Step2: Solve for hypotenuse

$b = 20\sqrt{2}$

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Problem 9

Step1: Identify triangle type

This is a 45-45-90 triangle, so leg $MN = PM = 5\sqrt{2}$, hypotenuse $NP = MN\sqrt{2}$.

Step2: Calculate hypotenuse

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

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Problem 10

Step1: Identify triangle type

This is a 45-45-90 triangle, hypotenuse $=7$, legs $m = n = \frac{\text{hypotenuse}}{\sqrt{2}}$.

Step2: Rationalize to find legs

$m = n = \frac{7}{\sqrt{2}} = \frac{7\sqrt{2}}{2}$

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Problem 11

Step1: Identify triangle type

This is a 45-45-90 triangle, hypotenuse $=10$, legs $AB = BC = \frac{\text{hypotenuse}}{\sqrt{2}}$.

Step2: Rationalize to find legs

$AB = BC = \frac{10}{\sqrt{2}} = 5\sqrt{2}$

Answer:

1. $u=8$, $v=8$
2. $a=20$, $b=20\sqrt{2}$
3. $MN=5\sqrt{2}$, $NP=10$
4. $m=\frac{7\sqrt{2}}{2}$, $n=\frac{7\sqrt{2}}{2}$
5. $AB=5\sqrt{2}$, $BC=5\sqrt{2}$