Question

c.6 angle bisectors
what is hk?
diagram: points i, j, k, h with right angles at i (∠ihj) and k (∠khj), ∠ijh = 48°, ∠kjh = 48°, hi = 3s, hk = s + 8
hk = \boxed{}

Explanation:

Step1: Identify the angle bisector property

Since \( HJ \) is the angle bisector of \( \angle IJK \), and \( HI \perp IJ \), \( HK \perp JK \), by the Angle Bisector Theorem (the distance from a point on the angle bisector to both sides of the angle is equal), we have \( HI = HK \).
Given \( HI = 3s \) and \( HK = s + 8 \), so we set up the equation \( 3s = s + 8 \).

Step2: Solve for \( s \)

Subtract \( s \) from both sides of the equation \( 3s = s + 8 \):
\( 3s - s = s + 8 - s \)
\( 2s = 8 \)
Divide both sides by 2:
\( s=\frac{8}{2}=4 \)

Step3: Calculate \( HK \)

Substitute \( s = 4 \) into the expression for \( HK \), which is \( s + 8 \):
\( HK=4 + 8 = 12 \)

Answer:

\( 12 \)