Question

S is any point on the side QR of a \(\Delta PQR\) such that \(\angle PSR = \angle QPR\). Prove that \(\frac{QR}{RP} = \frac{RP}{RS}\).

• A. Proof Completed
• B. Triangles Not Similar
• C. Insufficient Data
• D. Cannot be Determined

Hint:

To avoid mistakes in writing the ratios of similar triangles, always list the triangles with their corresponding vertices in order:
\[ \Delta QPR \sim \Delta PSR \]
The ratios can then be written directly from the order of letters:
\[ \frac{QP}{PS} = \frac{PR}{SR} = \frac{QR}{PR} \]
This eliminates any visual confusion from the diagram.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
In triangle \(PQR\), a point \(S\) lies on side \(QR\) such that the angle \(\angle PSR\) is equal to \(\angle QPR\). We need to prove the ratio relation \(\frac{QR}{RP} = \frac{RP}{RS}\).

Step 2: Key Formula or Approach:
We can prove relations of ratio of sides by establishing similarity between the appropriate triangles.
Specifically, we will look at the larger triangle \(\Delta QPR\) and the smaller triangle \(\Delta PSR\) and use the AA (Angle-Angle) similarity criterion.

Step 3: Detailed Explanation:
1. Consider \(\Delta QPR\) and \(\Delta PSR\):
- \(\angle PRQ = \angle PRS\) (This is the exact same angle, as \(S\) lies on the line \(QR\), making it a common angle).
- \(\angle QPR = \angle PSR\) (This is given in the problem statement).
2. Since two angles of \(\Delta QPR\) are equal to corresponding angles of \(\Delta PSR\), the two triangles are similar by the AA (Angle-Angle) similarity criterion:
\[ \Delta QPR \sim \Delta PSR \]
3. Corresponding sides of similar triangles are proportional. Let us write down the ratio of corresponding sides of these two similar triangles:
- The side opposite to \(\angle QPR\) in \(\Delta QPR\) is \(QR\).
- The side opposite to \(\angle PSR\) in \(\Delta PSR\) is \(PR\).
- The side opposite to the third angle \(\angle PQR\) in \(\Delta QPR\) is \(PR\).
- The side opposite to the third angle \(\angle SPR\) in \(\Delta PSR\) is \(RS\).
4. Therefore, the ratio of these corresponding sides must be equal:
\[ \frac{QR}{PR} = \frac{PR}{SR} \]
Which can be written as:
\[ \frac{QR}{RP} = \frac{RP}{RS} \]
This completed the proof.

Step 4: Final Answer:
The ratio has been proven using AA similarity between \(\Delta QPR\) and \(\Delta PSR\). Thus, the proof is completed.