Question

the lengths of two sides of a right triangle are given. find the length of the third side. 13. a = 9, c = 41; 14. a = 40, c = 58; 15. b = 56, c = 65; 16. b = 70, c = 74; 17. a = 13, b = 84; 18. a = 16, b = 63

Explanation:

For all right triangles, we use the Pythagorean theorem: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse (longest side). We will solve for the unknown side in each case.

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Problem 13:

Step1: Identify hypotenuse (c=41)

Unknown side is $b$. Rearrange theorem:
$b^2 = c^2 - a^2$

Step2: Substitute values

$b^2 = 41^2 - 9^2 = 1681 - 81 = 1600$

Step3: Solve for b

$b = \sqrt{1600} = 40$

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Problem 14:

Step1: Identify hypotenuse (c=58)

Unknown side is $b$. Rearrange theorem:
$b^2 = c^2 - a^2$

Step2: Substitute values

$b^2 = 58^2 - 40^2 = 3364 - 1600 = 1764$

Step3: Solve for b

$b = \sqrt{1764} = 42$

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Problem 15:

Step1: Identify hypotenuse (c=65)

Unknown side is $a$. Rearrange theorem:
$a^2 = c^2 - b^2$

Step2: Substitute values

$a^2 = 65^2 - 56^2 = 4225 - 3136 = 1089$

Step3: Solve for a

$a = \sqrt{1089} = 33$

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Problem 16:

Step1: Identify hypotenuse (c=74)

Unknown side is $a$. Rearrange theorem:
$a^2 = c^2 - b^2$

Step2: Substitute values

$a^2 = 74^2 - 70^2 = 5476 - 4900 = 576$

Step3: Solve for a

$a = \sqrt{576} = 24$

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Problem 17:

Step1: Find hypotenuse c

Use Pythagorean theorem:
$c^2 = a^2 + b^2$

Step2: Substitute values

$c^2 = 13^2 + 84^2 = 169 + 7056 = 7225$

Step3: Solve for c

$c = \sqrt{7225} = 85$

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Problem 18:

Step1: Find hypotenuse c

Use Pythagorean theorem:
$c^2 = a^2 + b^2$

Step2: Substitute values

$c^2 = 16^2 + 63^2 = 256 + 3969 = 4225$

Step3: Solve for c

$c = \sqrt{4225} = 65$

Answer:

1. $b=40$
2. $b=42$
3. $a=33$
4. $a=24$
5. $c=85$
6. $c=65$