Question

given the following right triangle; several students were trying to find the length of the missing leg. which of the students is approaching this problem correctly? image of right triangle with one leg 5, hypotenuse 13, and the other leg x choose the correct answer below. a. joel knows that the pythagorean theorem involves squares, so he subtracts 13 - 5, which gives him 8, and then squares the result. b. mario squares each of the given sides, subtracts these numbers, and then takes the square root of the difference. c. alessandra squares each of the given sides, adds these numbers, and then takes the square root of the sum. d. fiona multiplies the two given sides. she then takes the square root of that product.

Explanation:

To find the missing leg \( x \) of a right triangle with hypotenuse \( 13 \) and one leg \( 5 \), we use the Pythagorean theorem: \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse and \( a, b \) are the legs. Rearranging for the missing leg \( x \) (let \( a = 5 \), \( c = 13 \)): \( x^2 = c^2 - a^2 \), so \( x=\sqrt{c^2 - a^2} \). This means we square the hypotenuse and the known leg, subtract the two squares, then take the square root of the difference.

• Option A: Joel subtracts \( 13 - 5 \) first, which is incorrect. The Pythagorean theorem involves squaring first, not subtracting the lengths directly.
• Option B: Mario squares each side (\( 13^2 \) and \( 5^2 \)), subtracts the results (\( 13^2 - 5^2 \)), then takes the square root. This matches the rearranged Pythagorean theorem for a missing leg.
• Option C: Alessandra adds the squares, which is for finding the hypotenuse (when both legs are known), not a missing leg.
• Option D: Fiona multiplies the sides and takes the square root, which is not part of the Pythagorean theorem.

Answer:

B. Mario squares each of the given sides, subtracts these numbers, and then takes the square root of the difference.