Question

(a) one triangle has side lengths 18, 19, and 10. compute the sum of the squares of the shorter lengths. $10^2 + 18^2 = \square$ compute the square of the longest length. $19^2 = \square$ what kind of triangle is it? $\circ$ acute triangle $\circ$ right triangle $\circ$ obtuse triangle (b) the other triangle has side lengths 17, 8, and 15. compute the sum of the squares of the shorter lengths. $8^2 + 15^2 = \square$ compute the square of the longest length. $17^2 = \square$ what kind of triangle is it? $\circ$ acute triangle $\circ$ right triangle $\circ$ obtuse triangle

Explanation:

Part (a)

Step1: Calculate sum of squares of shorter sides

The shorter sides are 10 and 18. So we compute \(10^{2}+18^{2}\).
\(10^{2} = 100\) and \(18^{2}=324\). Then \(100 + 324=424\).

Step2: Calculate square of longest side

The longest side is 19. So we compute \(19^{2}\).
\(19\times19 = 361\).

Step3: Compare the two results

We have the sum of squares of shorter sides as 424 and the square of the longest side as 361. Since \(424>361\) (i.e., \(a^{2}+b^{2}>c^{2}\) where \(c\) is the longest side), the triangle is acute.

Step1: Calculate sum of squares of shorter sides

The shorter sides are 8 and 15. So we compute \(8^{2}+15^{2}\).
\(8^{2}=64\) and \(15^{2} = 225\). Then \(64+225 = 289\).

Step2: Calculate square of longest side

The longest side is 17. So we compute \(17^{2}\).
\(17\times17=289\).

Step3: Compare the two results

We have the sum of squares of shorter sides as 289 and the square of the longest side as 289. Since \(a^{2}+b^{2}=c^{2}\) (where \(c\) is the longest side), the triangle is right.

Answer:

• Sum of squares of shorter sides: \(424\)
• Square of longest side: \(361\)
• Type of triangle: Acute triangle

Part (b)