Question

wer the questions about each triangle.
(a) one triangle has side lengths 20, 16, and 12.
compute the sum of the squares of the shorter lengths.
$12^2 + 16^2 = \square$
compute the square of the longest length.
$20^2 = \square$
what kind of triangle is it?
\\(\circ\\) acute triangle
\\(\circ\\) right triangle
\\(\circ\\) obtuse triangle
(b) the other triangle has side lengths 19, 15, and 16.
compute the sum of the squares of the shorter lengths.
$15^2 + 16^2 = \square$
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

Explanation:

Part (a)

Step 1: Calculate the sum of the squares of the shorter lengths

We need to compute \(12^2 + 16^2\). First, find \(12^2 = 144\) and \(16^2 = 256\). Then add them together: \(144 + 256 = 400\).

Step 2: Calculate the square of the longest length

The longest length is 20, so we compute \(20^2 = 400\).

Step 3: Determine the type of triangle

Since the sum of the squares of the two shorter sides (\(12^2 + 16^2 = 400\)) is equal to the square of the longest side (\(20^2 = 400\)), by the Pythagorean theorem, this is a right triangle.

Part (b)

Step 1: Calculate the sum of the squares of the shorter lengths

We need to compute \(15^2 + 16^2\). First, find \(15^2 = 225\) and \(16^2 = 256\). Then add them together: \(225 + 256 = 481\).

Step 2: Calculate the square of the longest length

The longest length is 19, so we compute \(19^2 = 361\). Wait, no, 19 squared is \(19\times19 = 361\)? Wait, no, 1919: 2019=380, minus 119=19, so 380 - 19 = 361? Wait, no, that's incorrect. Wait, 1919: 1019=190, 919=171, so 190 + 171 = 361? Wait, no, 15^2 + 16^2 is 225 + 256 = 481. Then 19^2: 1919. Let's calculate again: 2020=400, minus 20 + 19 = 39, so 400 - 39 = 361? Wait, no, that's wrong. Wait, 1919: 1910=190, 19*9=171, 190+171=361. But 481 is greater than 361? Wait, no, the longest side is 19? Wait, the side lengths are 19, 15, 16. So 19 is the longest. So \(19^2 = 361\). Then, since the sum of the squares of the two shorter sides (\(15^2 + 16^2 = 481\)) is greater than the square of the longest side (\(19^2 = 361\)), the triangle is acute.

Final Answers

Part (a)

• Sum of squares of shorter lengths: \(12^2 + 16^2 = \boldsymbol{400}\)
• Square of longest length: \(20^2 = \boldsymbol{400}\)
• Type of triangle: \(\boldsymbol{\text{Right triangle}}\)

Part (b)

• Sum of squares of shorter lengths: \(15^2 + 16^2 = \boldsymbol{481}\)
• Square of longest length: \(19^2 = \boldsymbol{361}\)
• Type of triangle: \(\boldsymbol{\text{Acute triangle}}\)

Answer:

Part (a)

Step 1: Calculate the sum of the squares of the shorter lengths

We need to compute \(12^2 + 16^2\). First, find \(12^2 = 144\) and \(16^2 = 256\). Then add them together: \(144 + 256 = 400\).

Step 2: Calculate the square of the longest length

The longest length is 20, so we compute \(20^2 = 400\).

Step 3: Determine the type of triangle

Since the sum of the squares of the two shorter sides (\(12^2 + 16^2 = 400\)) is equal to the square of the longest side (\(20^2 = 400\)), by the Pythagorean theorem, this is a right triangle.

Part (b)

Step 1: Calculate the sum of the squares of the shorter lengths

We need to compute \(15^2 + 16^2\). First, find \(15^2 = 225\) and \(16^2 = 256\). Then add them together: \(225 + 256 = 481\).

Step 2: Calculate the square of the longest length

The longest length is 19, so we compute \(19^2 = 361\). Wait, no, 19 squared is \(19\times19 = 361\)? Wait, no, 1919: 2019=380, minus 119=19, so 380 - 19 = 361? Wait, no, that's incorrect. Wait, 1919: 1019=190, 919=171, so 190 + 171 = 361? Wait, no, 15^2 + 16^2 is 225 + 256 = 481. Then 19^2: 1919. Let's calculate again: 2020=400, minus 20 + 19 = 39, so 400 - 39 = 361? Wait, no, that's wrong. Wait, 1919: 1910=190, 19*9=171, 190+171=361. But 481 is greater than 361? Wait, no, the longest side is 19? Wait, the side lengths are 19, 15, 16. So 19 is the longest. So \(19^2 = 361\). Then, since the sum of the squares of the two shorter sides (\(15^2 + 16^2 = 481\)) is greater than the square of the longest side (\(19^2 = 361\)), the triangle is acute.

Final Answers

Part (a)

• Sum of squares of shorter lengths: \(12^2 + 16^2 = \boldsymbol{400}\)
• Square of longest length: \(20^2 = \boldsymbol{400}\)
• Type of triangle: \(\boldsymbol{\text{Right triangle}}\)

Part (b)

• Sum of squares of shorter lengths: \(15^2 + 16^2 = \boldsymbol{481}\)
• Square of longest length: \(19^2 = \boldsymbol{361}\)
• Type of triangle: \(\boldsymbol{\text{Acute triangle}}\)