Question

the side lengths of two triangles are given below.
answer the questions about each triangle.
(a) one triangle has side lengths 19, 14, and 16.
compute the sum of the squares of the shorter lengths.
$14^{2}+16^{2}=\square$
compute the square of the longest length.
$19^{2}=\square$
what kind of triangle is it?
○acute triangle
○right triangle
○obtuse triangle
(b) the other triangle has side lengths 18, 16, and 5.
compute the sum of the squares of the shorter lengths.
$5^{2}+16^{2}=\square$
compute the square of the longest length.
$18^{2}=\square$
what kind of triangle is it?
○acute triangle
○right triangle
○obtuse triangle

Explanation:

Step1: Calculate \(14^{2}+16^{2}\) for part (a)

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$$

Step2: Calculate \(19^{2}\) for part (a)

$$19^{2}=361$$

Since \(452>361\) (sum of squares of shorter sides \(>\) square of the longest side), it is an acute triangle.

Step3: Calculate \(5^{2}+16^{2}\) for part (b)

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$$

Step4: Calculate \(18^{2}\) for part (b)

$$18^{2}=324$$

Since \(281<324\) (sum of squares of shorter sides \(<\) square of the longest side), it is an obtuse triangle.

Answer:

(a) \(14^{2}+16^{2}=452\), \(19^{2}=361\), Acute triangle.
(b) \(5^{2}+16^{2}=281\), \(18^{2}=324\), Obtuse triangle.