Question

If the angle bisector of \( \angle P \) meets \( QR \) at point \( M \), find the length of \( PM \).

• A. \( \dfrac{28\sqrt{5}}{9} \) cm
• B. \( \dfrac{42\sqrt{5}}{11} \) cm
• C. \( \dfrac{36\sqrt{3}}{7} \) cm
• D. \( 4\sqrt{3} \) cm

Hint:

When an angle bisector meets the opposite side, use the Angle Bisector Theorem to find segment ratios and apply the length formula.

Answer 1

The Correct Option is C

From the previous question: - \( PQ = 12 \), \( PR = 9 \), \( \angle P = 60^\circ \), \( QR = 3\sqrt{13} \) Let angle bisector of \( \angle P \) meet QR at M. 
By Angle Bisector Theorem: \[ \frac{QM}{MR} = \frac{PQ}{PR} = \frac{12}{9} = \frac{4}{3} \Rightarrow QM = \frac{4}{7} QR, MR = \frac{3}{7} QR \] So: \[ QM = \frac{4}{7} 3\sqrt{13} = \frac{12\sqrt{13}}{7}, MR = \frac{9\sqrt{13}}{7} \] Use Angle Bisector Length Formula: \[ PM^2 = PQ PR \left[ 1 - \left( \frac{QR^2}{(PQ + PR)^2} \right) \right] \] Substitute: \[ PM^2 = 12 9 \left[ 1 - \left( \frac{(3\sqrt{13})^2}{(12 + 9)^2} \right) \right] = 108 \left[ 1 - \left( \frac{117}{441} \right) \right] = 108 \frac{324}{441} = \frac{34992}{441} \] Simplify: \[ PM = \sqrt{\frac{34992}{441}} = \sqrt{79.333...} \approx 8.9 \] Try options: (A) \( \dfrac{28\sqrt{5}}{9} \approx 8.72 \) 

(b) \( \dfrac{42\sqrt{5}}{11} \approx 8.62 \) 

(c) \( \dfrac{36\sqrt{3}}{7} \approx 8.86 \) 

(d) \( 4\sqrt{3} \approx 6.93 \)
Only option 
(c) matches approx. So, Final Answer: \( \boxed{\dfrac{36\sqrt{3}}{7}} \)