Question

assignment - practice pythagorean theorem and special right triangles period
find the missing side of each triangle. leave your answers in simplest radical form.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)

Explanation:

Step1: Recall Pythagorean theorem

For right triangle: $a^2 + b^2 = c^2$, where $c$ is hypotenuse.
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Problem 1

Step1: Identify hypotenuse and legs

Hypotenuse $c=14$, leg $a=4$, leg $b=x$

Step2: Rearrange for missing leg

$x^2 = c^2 - a^2$
$x^2 = 14^2 - 4^2 = 196 - 16 = 180$

Step3: Simplify radical

$x = \sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5}$

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Problem 2

Step1: Identify legs and hypotenuse

Legs $a=7, b=5$, hypotenuse $c=x$

Step2: Calculate hypotenuse

$x^2 = 7^2 + 5^2 = 49 + 25 = 74$

Step3: Simplify radical

$x = \sqrt{74}$

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Problem 3

Step1: Identify hypotenuse and legs

Hypotenuse $c=16$, leg $a=14$, leg $b=x$

Step2: Rearrange for missing leg

$x^2 = 16^2 - 14^2 = 256 - 196 = 60$

Step3: Simplify radical

$x = \sqrt{60} = \sqrt{4 \times 15} = 2\sqrt{15}$

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Problem 4

Step1: Identify legs and hypotenuse

Legs $a=7, b=10$, hypotenuse $c=x$

Step2: Calculate hypotenuse

$x^2 = 7^2 + 10^2 = 49 + 100 = 149$

Step3: Simplify radical

$x = \sqrt{149}$

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Problem 5

Step1: Identify hypotenuse and legs

Hypotenuse $c=\sqrt{101}$, leg $a=12$, leg $b=x$

Step2: Rearrange for missing leg

$x^2 = c^2 - a^2 = (\sqrt{101})^2 - 12^2 = 101 - 144$

Step3: Correct for valid triangle

*Note: $101 < 144$, so rearrange: $(\sqrt{101})^2 + x^2 = 12^2$
$x^2 = 144 - 101 = 43$
$x = \sqrt{43}$

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Problem 6

Step1: Identify legs and hypotenuse

Legs $a=10, b=8$, hypotenuse $c=x$

Step2: Calculate hypotenuse

$x^2 = 10^2 + 8^2 = 100 + 64 = 164$

Step3: Simplify radical

$x = \sqrt{164} = \sqrt{4 \times 41} = 2\sqrt{41}$

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Problem 7

Step1: Identify legs and hypotenuse

Legs $a=9, b=10$, hypotenuse $c=x$

Step2: Calculate hypotenuse

$x^2 = 9^2 + 10^2 = 81 + 100 = 181$

Step3: Simplify radical

$x = \sqrt{181}$

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Problem 8

Step1: Identify hypotenuse and legs

Hypotenuse $c=4$, leg $a=2\sqrt{2}$, leg $b=x$

Step2: Rearrange for missing leg

$x^2 = 4^2 - (2\sqrt{2})^2 = 16 - 8 = 8$

Step3: Simplify radical

$x = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$

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Problem 9

Step1: Identify hypotenuse and legs

Hypotenuse $c=16$, leg $a=\sqrt{37}$, leg $b=x$

Step2: Rearrange for missing leg

$x^2 = 16^2 - (\sqrt{37})^2 = 256 - 37 = 219$

Step3: Simplify radical

$x = \sqrt{219}$

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Problem 10

Step1: Identify legs and hypotenuse

Legs $a=7, b=9$, hypotenuse $c=x$

Step2: Calculate hypotenuse

$x^2 = 7^2 + 9^2 = 49 + 81 = 130$

Step3: Simplify radical

$x = \sqrt{130}$

Answer:

1. $6\sqrt{5}$
2. $\sqrt{74}$
3. $2\sqrt{15}$
4. $\sqrt{149}$
5. $\sqrt{43}$
6. $2\sqrt{41}$
7. $\sqrt{181}$
8. $2\sqrt{2}$
9. $\sqrt{219}$
10. $\sqrt{130}$