Question

if (angle hikcongangle jik), (hk = p + 28), and (jk = 5p), what is the value of (p)?

Explanation:

Step1: Use the Angle - Bisector Theorem (distance from a point to sides of an angle)

Since \( \angle HIK\cong\angle JIK\), \(IK\) is the angle bisector of \( \angle JIH\). Also, \(KJ\perp JI\) and \(KH\perp HI\). By the Angle - Bisector Theorem, the distance from a point on the angle bisector to the two sides of the angle is equal. So \(HK = JK\).

Step2: Set up the equation

We know that \(HK=p + 28\) and \(JK = 5p\). Since \(HK=JK\), we have the equation \(p + 28=5p\).

Step3: Solve the equation for \(p\)

Subtract \(p\) from both sides of the equation:
\(p+28 - p=5p - p\)
\(28 = 4p\)
Divide both sides by \(4\): \(p=\frac{28}{4}=7\)

Answer:

\(7\)