Question

u is the incenter of \\( \triangle ghy \\). find each measure.

1. \\( m \angle ugm \\)
2. \\( m \angle phu \\)
3. \\( hu \\)

Explanation:

Step1: Recall the property of in - center

The in - center of a triangle is the intersection of the angle bisectors of the triangle.

Step2: Find \(m\angle UGM\)

Since \(U\) is the in - center of \(\triangle GHY\) and \(GU\) is an angle bisector. Given \(\angle UGY = 28^{\circ}\), then \(m\angle UGM=m\angle UGY = 28^{\circ}\)

Step3: Find \(m\angle PHU\)

Since \(U\) is the in - center of \(\triangle GHY\) and \(HU\) is an angle bisector. Given \(\angle MGH=21^{\circ}\), then \(m\angle PHU=\frac{1}{2}\angle MGH\)

$$m\angle PHU = 21^{\circ}$$

Step4: Find \(HU\)

Since \(U\) is the in - center of \(\triangle GHY\), the in - center is equidistant from the sides of the triangle. \(UM = UP=UB\). Given \(UP = 5\), then \(HU=\sqrt{HP^{2}+UP^{2}}\) (using the Pythagorean theorem in right - triangle \(HPU\) where \(HP = 12\) and \(UP = 5\))

$$HU=\sqrt{12^{2}+5^{2}}=\sqrt{144 + 25}=\sqrt{169}=13$$

Answer:

1. \(28^{\circ}\)
2. \(21^{\circ}\)
3. \(13\)