Question

solve the right triangle.
write your answers in simplified, rationalized form. do not round.
hi =
m∠i =
m∠h =

Explanation:

Step1: Find the length of \(HI\) using the Pythagorean theorem

In a right - triangle \(IGH\) with \(IG = 8\sqrt{11}\), \(GH = 8\sqrt{11}\), and by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(a = IG\), \(b = GH\), \(c = HI\)).

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Step2: Find the measure of \(\angle I\)

Since \(\tan I=\frac{GH}{IG}\), and \(IG = GH = 8\sqrt{11}\), then \(\tan I=\frac{8\sqrt{11}}{8\sqrt{11}} = 1\).
We know that if \(\tan\theta=1\), then \(\theta = 45^{\circ}\) (because \(\tan45^{\circ}=1\) in the range \(0^{\circ}<\theta<90^{\circ}\) for a right - triangle).

Step3: Find the measure of \(\angle H\)

In a right - triangle \(IGH\), \(\angle G = 90^{\circ}\), and using the angle - sum property of a triangle \(\angle I+\angle H+\angle G=180^{\circ}\).
Since \(\angle I = 45^{\circ}\) and \(\angle G = 90^{\circ}\), then \(\angle H=180^{\circ}-\angle I - \angle G\).

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Answer:

\(HI = 8\sqrt{22}\), \(m\angle I=45^{\circ}\), \(m\angle H = 45^{\circ}\)