Question

in the rectangular prism below, $ru = 16$, $ut = 35$, and $st = 37$. find the length $rs$. if applicable, round your answer to the nearest whole number. the lengths on the figure are not drawn accurately.

Explanation:

Step1: Identify right triangle UTS

In a rectangular prism, angle at S (∠UST) is a right angle (since it's a rectangular prism, edges are perpendicular). So triangle UTS is a right triangle with UT = 35, ST = 37, and we can find US using the Pythagorean theorem.
The Pythagorean theorem states that for a right triangle with legs \( a, b \) and hypotenuse \( c \), \( a^{2}+b^{2}=c^{2} \). Here, \( ST \) is the hypotenuse, \( UT \) is one leg, and \( US \) is the other leg. So \( US^{2}+UT^{2}=ST^{2} \), which rearranges to \( US = \sqrt{ST^{2}-UT^{2}} \).
Substitute \( ST = 37 \) and \( UT = 35 \):
\( US=\sqrt{37^{2}-35^{2}}=\sqrt{(37 + 35)(37 - 35)} \) (using the difference of squares \( a^{2}-b^{2}=(a + b)(a - b) \))
\(=\sqrt{72\times2}=\sqrt{144}=12 \)

Step2: Identify right triangle RUS

Now, triangle RUS is also a right triangle (since in the rectangular prism, RU and US are perpendicular edges). RU = 16, US = 12, and RS is the hypotenuse. Again, use the Pythagorean theorem: \( RS=\sqrt{RU^{2}+US^{2}} \)
Substitute \( RU = 16 \) and \( US = 12 \):
\( RS=\sqrt{16^{2}+12^{2}}=\sqrt{256 + 144}=\sqrt{400}=20 \)

Answer:

20