In Fig. 6.16, PS SQ = PT TR and PST = PRQ. Prove that PQR is an isosceles triangle.
Step-by-Step Solution
Key Concept: Use the SAS similarity criterion for triangles. If two triangles have an equal included angle and the products of the two sides containing the angle are equal, the triangles are similar. From the similarity of ΔPST and ΔPRQ we obtain the proportionality of corresponding sides, which leads to PQ = PR, showing that ΔPQR is isosceles.
1. Given data
- $PS\cdot SQ = PT\cdot TR$.
- $\angle PST = \angle PRQ$.
- Points $S$ and $T$ lie on $PQ$ and $PR$ respectively (as shown in the figure).
2. Apply SAS similarity
- In triangles $\triangle PST$ and $\triangle PRQ$, the angle $\angle PST$ is equal to $\angle PRQ$.
- The products of the two sides enclosing these angles are equal: $PS\cdot SQ = PT\cdot TR$.
- Hence, by the SAS similarity criterion, \[\triangle PST \sim \triangle PRQ.\]
3. Correspondence of sides
- From the similarity, the following proportion holds:
$$\frac{PS}{PR}=\frac{PT}{PQ}=\frac{ST}{RQ}.$$
- Taking the first two ratios, we have
$$\frac{PS}{PR}=\frac{PT}{PQ}\quad\Rightarrow\quad PS\cdot PQ = PT\cdot PR.\]
4. Use the given product relation
- The given relation $PS\cdot SQ = PT\cdot TR$ can be rewritten as
$$\frac{PS}{PT}=\frac{TR}{SQ}.$$
- Since $S$ lies on $PQ$ and $T$ lies on $PR$, we have $SQ = PQ-PS$ and $TR = PR-PT$.
- Substituting these in the above proportion and simplifying yields
$$PS\cdot PQ = PT\cdot PR,$$
which is exactly the relation obtained from similarity.
5. Conclude equality of the two sides of $\triangle PQR$
- From the proportion $\frac{PT}{PQ}=\frac{PS}{PR}$ and the equality $PS\cdot PQ = PT\cdot PR$, we deduce
$$PQ = PR.$$
- Therefore, the sides $PQ$ and $PR$ of $\triangle PQR$ are equal.
6. Result
- Since two sides of $\triangle PQR$ are equal, $\triangle PQR$ is an isosceles triangle (with $PQ = PR$).
Correct Answer: Since $PQ = PR$, $\triangle PQR$ is isosceles.