Question

pythagorean theorem
find the missing length.
1.
(right triangle with legs 48, x and hypotenuse 80)
(right triangle with legs x, 84 and hypotenuse 91)
(right triangle with legs 45, 28 and hypotenuse x)
4.
(right triangle with legs x, 40 and hypotenuse 41)
5.
(right triangle with legs x, 6 and hypotenuse 8)
6.
(right triangle with legs 10, x and hypotenuse 26)
7.
(right triangle with legs x, 36 and hypotenuse 39)
8.
(right triangle with legs x, 24 and hypotenuse 25)
9.
(right triangle with legs 20, x and hypotenuse 29)
10.
(right triangle with legs x, 27 and hypotenuse 36)
11.
(right triangle with legs 24, 10 and hypotenuse x)
12.
(right triangle with legs x, 11 and hypotenuse 61)
13.
(right triangle with legs 15, 20 and hypotenuse x)
14.
(right triangle with legs 9, 15 and hypotenuse x)
15.
(right triangle with legs 16, 30 and hypotenuse x)

Explanation:

Step1: Recall Pythagorean Theorem

For a right triangle with legs $a,b$ and hypotenuse $c$, $a^2 + b^2 = c^2$.
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Problem 1

Step1: Identify sides, solve for $x$

Legs: 48, $x$; Hypotenuse: 80.
$x^2 = 80^2 - 48^2$

Step2: Calculate values

$x^2 = 6400 - 2304 = 4096$

Step3: Find square root

$x = \sqrt{4096} = 64$

Problem 2

Step1: Identify sides, solve for $x$

Legs: $x$, 84; Hypotenuse: 91.
$x^2 = 91^2 - 84^2$

Step2: Calculate values

$x^2 = 8281 - 7056 = 1225$

Step3: Find square root

$x = \sqrt{1225} = 35$

Problem 3

Step1: Identify sides, solve for $x$

Legs: 45, 28; Hypotenuse: $x$.
$x^2 = 45^2 + 28^2$

Step2: Calculate values

$x^2 = 2025 + 784 = 2809$

Step3: Find square root

$x = \sqrt{2809} = 53$

Problem 4

Step1: Identify sides, solve for $x$

Legs: $x$, 40; Hypotenuse: 41.
$x^2 = 41^2 - 40^2$

Step2: Calculate values

$x^2 = 1681 - 1600 = 81$

Step3: Find square root

$x = \sqrt{81} = 9$

Problem 5

Step1: Identify sides, solve for $x$

Legs: $x$, 6; Hypotenuse: 8.
$x^2 = 8^2 - 6^2$

Step2: Calculate values

$x^2 = 64 - 36 = 28$

Step3: Find square root

$x = \sqrt{28} = 2\sqrt{7} \approx 5.29$

Problem 6

Step1: Identify sides, solve for $x$

Legs: 10, $x$; Hypotenuse: 26.
$x^2 = 26^2 - 10^2$

Step2: Calculate values

$x^2 = 676 - 100 = 576$

Step3: Find square root

$x = \sqrt{576} = 24$

Problem 7

Step1: Identify sides, solve for $x$

Legs: $x$, 36; Hypotenuse: 39.
$x^2 = 39^2 - 36^2$

Step2: Calculate values

$x^2 = 1521 - 1296 = 225$

Step3: Find square root

$x = \sqrt{225} = 15$

Problem 8

Step1: Identify sides, solve for $x$

Legs: $x$, 24; Hypotenuse: 25.
$x^2 = 25^2 - 24^2$

Step2: Calculate values

$x^2 = 625 - 576 = 49$

Step3: Find square root

$x = \sqrt{49} = 7$

Problem 9

Step1: Identify sides, solve for $x$

Legs: 20, $x$; Hypotenuse: 29.
$x^2 = 29^2 - 20^2$

Step2: Calculate values

$x^2 = 841 - 400 = 441$

Step3: Find square root

$x = \sqrt{441} = 21$

Problem 10

Step1: Identify sides, solve for $x$

Legs: 27, $x$; Hypotenuse: 36.
$x^2 = 36^2 - 27^2$

Step2: Calculate values

$x^2 = 1296 - 729 = 567$

Step3: Find square root

$x = \sqrt{567} = 9\sqrt{7} \approx 23.81$

Problem 11

Step1: Identify sides, solve for $x$

Legs: 24, 10; Hypotenuse: $x$.
$x^2 = 24^2 + 10^2$

Step2: Calculate values

$x^2 = 576 + 100 = 676$

Step3: Find square root

$x = \sqrt{676} = 26$

Problem 12

Step1: Identify sides, solve for $x$

Legs: 11, $x$; Hypotenuse: 61.
$x^2 = 61^2 - 11^2$

Step2: Calculate values

$x^2 = 3721 - 121 = 3600$

Step3: Find square root

$x = \sqrt{3600} = 60$

Problem 13

Step1: Identify sides, solve for $x$

Legs: 15, 20; Hypotenuse: $x$.
$x^2 = 15^2 + 20^2$

Step2: Calculate values

$x^2 = 225 + 400 = 625$

Step3: Find square root

$x = \sqrt{625} = 25$

Problem 14

Step1: Identify sides, solve for $x$

Legs: 9, 15; Hypotenuse: $x$.
$x^2 = 9^2 + 15^2$

Step2: Calculate values

$x^2 = 81 + 225 = 306$

Step3: Find square root

$x = \sqrt{306} = 3\sqrt{34} \approx 17.49$

Problem 15

Step1: Identify sides, solve for $x$

Legs: 16, 30; Hypotenuse: $x$.
$x^2 = 16^2 + 30^2$

Step2: Calculate values

$x^2 = 256 + 900 = 1156$

Step3: Find square root

$x = \sqrt{1156} = 34$
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Answer:

1. $x=64$
2. $x=35$
3. $x=53$
4. $x=9$
5. $x=2\sqrt{7} \approx 5.29$
6. $x=24$
7. $x=15$
8. $x=7$
9. $x=21$
10. $x=9\sqrt{7} \approx 23.81$
11. $x=26$
12. $x=60$
13. $x=25$
14. $x=3\sqrt{34} \approx 17.49$
15. $x=34$