Question

practice problems
7 problems
1 a triangle has sides with lengths 8, 15, and 17.
a. verify this is a right triangle.
b. approximate the measures of the acute angles in this triangle.

Explanation:

Step1: Verify right - triangle using Pythagorean theorem

The Pythagorean theorem states that for a right - triangle \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse (the longest side).
Let \(a = 8\), \(b=15\), \(c = 17\)

$$
LATEXBLOCK0
$$

And \(c^{2}=17^{2}=289\)
Since \(a^{2}+b^{2}=c^{2}\), the triangle is a right - triangle.

Step2: Find acute angles using trigonometric ratios

Let \(\theta_{1}\) be the angle opposite to the side of length \(8\) and \(\theta_{2}\) be the angle opposite to the side of length \(15\)

For \(\theta_{1}\):
\(\sin\theta_{1}=\frac{8}{17}\)
\(\theta_{1}=\sin^{- 1}(\frac{8}{17})\approx28.1^{\circ}\)

For \(\theta_{2}\):
\(\sin\theta_{2}=\frac{15}{17}\)
\(\theta_{2}=\sin^{-1}(\frac{15}{17})\approx61.9^{\circ}\)

Answer:

a. The triangle is a right - triangle because \(8^{2}+15^{2}=17^{2}\) (i.e., \(64 + 225=289\)).
b. The acute angles are approximately \(28.1^{\circ}\) and \(61.9^{\circ}\)