Question

find the missing side of the following right triangles using the pythagorean theorem
$a^2 + b^2 = c^2$
show your work to justify your answer!!!
22)
triangle with legs 17m, x and hypotenuse 22m
23)
triangle with hypotenuse 6.9cm, leg 6.4cm and leg x
24)
triangle with leg 3m, leg x and hypotenuse 7m

Explanation:

Problem 22:

Step1: Identify sides (a, b, c)

In a right triangle, \( c \) is the hypotenuse (longest side). Here, \( a = 17 \, \text{m} \), \( c = 22 \, \text{m} \), \( b = x \). Use Pythagorean theorem: \( a^2 + b^2 = c^2 \). Rearrange to solve for \( b \): \( b^2 = c^2 - a^2 \).

Step2: Substitute values

\( b^2 = 22^2 - 17^2 = 484 - 289 = 195 \). Then \( b = \sqrt{195} \approx 13.96 \, \text{m} \) (or keep as \( \sqrt{195} \)).

Step1: Identify sides (a, b, c)

Hypotenuse \( c = 6.9 \, \text{cm} \), one leg \( b = 6.4 \, \text{cm} \), other leg \( a = x \). Use \( a^2 + b^2 = c^2 \). Rearrange: \( a^2 = c^2 - b^2 \).

Step2: Substitute values

\( a^2 = 6.9^2 - 6.4^2 = 47.61 - 40.96 = 6.65 \). Then \( a = \sqrt{6.65} \approx 2.58 \, \text{cm} \) (or keep as \( \sqrt{6.65} \)).

Step1: Identify sides (a, b, c)

Hypotenuse \( c = 7 \, \text{m} \), one leg \( a = 3 \, \text{m} \), other leg \( b = x \). Use \( a^2 + b^2 = c^2 \). Rearrange: \( b^2 = c^2 - a^2 \).

Step2: Substitute values

\( b^2 = 7^2 - 3^2 = 49 - 9 = 40 \). Then \( b = \sqrt{40} = 2\sqrt{10} \approx 6.32 \, \text{m} \).

Answer:

\( x \approx 13.96 \, \text{m} \) (or \( \sqrt{195} \, \text{m} \))

Problem 23: