Question

the side lengths of two triangles are given below
answer the questions about each triangle.
(a) one triangle has side lengths 15, 17, and 8.
compute the sum of the squares of the shorter lengths.
8² + 15² =
compute the square of the longest length.
17² =
what kind of triangle is it?
acute triangle
right triangle
obtuse triangle
(b) the other triangle has side lengths 11, 14, and 12.
compute the sum of the squares of the shorter lengths.
11² + 12² =
compute the square of the longest length.
14² =
what kind of triangle is it?
acute triangle
right triangle
obtuse triangle

Explanation:

Step1: Compute sum of squares of shorter lengths for (a)

For side lengths \(8\), \(15\), and \(17\) (where \(8\) and \(15\) are shorter), use the formula \(a^{2}+b^{2}\).
\(8^{2}+15^{2}=64 + 225=289\)

Step2: Compute square of the longest length for (a)

For longest side \(c = 17\), use the formula \(c^{2}\).
\(17^{2}=289\)
Since \(8^{2}+15^{2}=17^{2}\), by Pythagorean theorem, it is a right - triangle.

Step3: Compute sum of squares of shorter lengths for (b)

For side lengths \(11\), \(12\), and \(14\) (where \(11\) and \(12\) are shorter), use the formula \(a^{2}+b^{2}\).
\(11^{2}+12^{2}=121+144 = 265\)

Step4: Compute square of the longest length for (b)

For longest side \(c = 14\), use the formula \(c^{2}\).
\(14^{2}=196\)
Since \(11^{2}+12^{2}=265>196 = 14^{2}\), by the rule \(a^{2}+b^{2}>c^{2}\) (where \(c\) is the longest side), it is an acute - triangle.

Answer:

(a) \(8^{2}+15^{2}=289\), \(17^{2}=289\), Right triangle
(b) \(11^{2}+12^{2}=265\), \(14^{2}=196\), Acute triangle