Ex 6.6, 1
In Fig. 6.56, PS is the bisector of ∠QPR of △PQR. Prove that SRQS=PRPQ.
Given: △PQR
and PS is the bisector of ∠QPR
i.e. ∠QPS=∠RPS
To Prove: SRQS=PRPQ
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Step-by-Step Solution
Step 1: Construction
To prove the Angle Bisector Theorem, we need to construct a parallel line. We draw a line through point R parallel to PS, and extend QP to meet this line at point T. This construction creates a transversal line QR intersecting parallel lines PS and $RT.
Step 2: Identify Equal Angles
Since PS∥RT and QT is a transversal, the corresponding angles ∠QPS and ∠QTR are equal. Also, since PS∥RT and PR is a transversal, the alternate interior angles ∠SPR and ∠PRT are equal. We are given that PS is the angle bisector of ∠QPR, so ∠ QPS = ∠ SPR.
Step 3: Equate Angles in Triangle PRT
From the previous step, we have ∠QPS=∠QTR and ∠SPR=∠PRT. Since ∠QPS=∠SPR (given), it follows that ∠ QTR = ∠ PRT.
Step 4: Identify Equal Sides in Triangle PRT
In △PRT, since ∠QTR=∠PRT, the sides opposite to these angles must be equal. Therefore, $PR = PT.
Step 5: Apply Basic Proportionality Theorem
Now consider △QRT. We have constructed PS∥RT. By the Basic Proportionality Theorem (also known as Thales Theorem or Intercept Theorem), if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides proportionally. Therefore, SRQS=PTQP.
Step 6: Substitute and Conclude
From Step 4, we established that PT=PR. Substituting PR for PT in the equation from Step 5, we get SRQS=PRPQ. This proves the Angle Bisector Theorem.