Question : In $\triangle $PQR, the bisector of $\angle $R meets side PQ at S, PR = 10 cm, RQ = 14 cm and PQ = 12 cm. What is the length of SQ?
Option 1: 5 cm
Option 2: 6 cm
Option 3: 7 cm
Option 4: 8 cm
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Correct Answer: 7 cm
Solution :
In $\triangle $PQR, the bisector of $\angle $R meets side PQ at S. According to the Angle Bisector Theorem, the ratio of the lengths of the two segments created by the angle bisector is equal to the ratio of the lengths of the other two sides of the triangle. $⇒\frac{PS}{SQ} = \frac{PR}{RQ}$ We know that PR = 10 cm, RQ = 14 cm, and PQ = 12 cm. We can express PS as PQ – SQ = 12 – SQ. $⇒\frac{12 - SQ}{SQ} = \frac{10}{14}$ $⇒SQ = \frac{12 \times 14}{10 + 14} = \frac{168}{24} = 7$ cm Hence, the correct answer is 7 cm.
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