Find the missing side length x in each right triangle using the Pythagorean theorem.

This image contains a worksheet with five problems, all asking to find the length of a side marked 'x' in different right-angled triangles. The solution for each problem involves applying the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The formula is $a^2 + b^2 = c^2$.

I will solve each problem one by one.

Problem 1

- Given: Two legs of the triangle are 15 cm and 8 cm. The side marked 'x' is the hypotenuse.
- Apply Pythagorean theorem:
$x^2 = 15^2 + 8^2$
$x^2 = 225 + 64$
$x^2 = 289$
$x = \sqrt{289}$
$x = 17$
- Answer: x = 17 cm

Problem 2

- Given: One leg is 9 cm, and the hypotenuse is 20 cm. The side marked 'x' is the other leg.
- Apply Pythagorean theorem:
$x^2 + 9^2 = 20^2$
$x^2 + 81 = 400$
$x^2 = 400 - 81$
$x^2 = 319$
$x = \sqrt{319}$
$x \approx 17.86$ (rounded to two decimal places)
- Answer: x ≈ 17.86 cm

Problem 3

- Given: Two legs of the triangle are 6.5 cm and 10.2 cm. The side marked 'x' is the hypotenuse.
- Apply Pythagorean theorem:
$x^2 = 6.5^2 + 10.2^2$
$x^2 = 42.25 + 104.04$
$x^2 = 146.29$
$x = \sqrt{146.29}$
$x \approx 12.09$ (rounded to two decimal places)
- Answer: x ≈ 12.09 cm

Problem 4

- Given: One leg is 8.4 cm, and the hypotenuse is 10.5 cm. The side marked 'x' is the other leg.
- Apply Pythagorean theorem:
$x^2 + 8.4^2 = 10.5^2$
$x^2 + 70.56 = 110.25$
$x^2 = 110.25 - 70.56$
$x^2 = 39.69$
$x = \sqrt{39.69}$
$x = 6.3$
- Answer: x = 6.3 cm

Problem 5

- Given: One leg is 5 cm, and the hypotenuse is 13 cm. The side marked 'x' is the other leg.
- Apply Pythagorean theorem:
$x^2 + 5^2 = 13^2$
$x^2 + 25 = 169$
$x^2 = 169 - 25$
$x^2 = 144$
$x = \sqrt{144}$
$x = 12$
- Answer: x = 12 cm

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Final Answers:

1. x = 17 cm
2. x ≈ 17.86 cm
3. x ≈ 12.09 cm
4. x = 6.3 cm
5. x = 12 cm