Question

In the given figure, two triangles ABC and PQR are shown such that \(\angle A = \angle P\) and \(\angle C = \angle R\). If \(AD \perp BC\) and \(PS \perp QR\), then prove that (i) \(\Delta ADB \sim \Delta PSQ\) (ii) \(AD \times QS = BD \times PS\).

Hint:

Whenever you are asked to prove a product relation of the form \(W \times X = Y \times Z\), rearrange it as a ratio \(\frac{W}{Z} = \frac{Y}{X}\).
This immediately guides you to identify which pair of triangles you need to prove similar to get the required side ratios.

Answer 1

Correct Answer: 4

Step 1: Understanding the Question:
We are given two triangles \(\Delta ABC\) and \(\Delta PQR\) where \(\angle A = \angle P\) and \(\angle C = \angle R\).
Altitudes \(AD\) and \(PS\) are drawn such that \(AD \perp BC\) and \(PS \perp QR\).
We need to prove two parts:
- Part (i): Show that \(\Delta ADB\) is similar to \(\Delta PSQ\).
- Part (ii): Prove the geometric relation \(AD \times QS = BD \times PS\).

Step 2: Key Formula or Approach:
We will use similarity of triangles:
1. AA Similarity Criterion: If two angles of one triangle are equal to two angles of another, the triangles are similar.
2. If two triangles are similar, the ratios of their corresponding sides are equal.

Step 3: Detailed Explanation:

• Let us first establish the similarity between the main triangles \(\Delta ABC\) and \(\Delta PQR\):
- In \(\Delta ABC\) and \(\Delta PQR\):
\[ \angle BAC = \angle QPR \quad \text{(Given)} \] \[ \angle ACB = \angle PRQ \quad \text{(Given)} \] - By the AA similarity criterion:
\[ \Delta ABC \sim \Delta PQR \]

• Since \(\Delta ABC \sim \Delta PQR\), their corresponding interior angles must be equal:
\[ \angle B = \angle Q \]

• Part (i): Prove \(\Delta ADB \sim \Delta PSQ\)
- Now, compare the smaller triangles \(\Delta ADB\) and \(\Delta PSQ\):
\[ \angle ADB = \angle PSQ = 90^\circ \quad \text{(Since } AD \perp BC \text{ and } PS \perp QR\text{)} \] \[ \angle B = \angle Q \quad \text{(Proved above)} \] - Since two corresponding angles are equal, by AA similarity criterion:
\[ \Delta ADB \sim \Delta PSQ \] - This completes the proof for part (i).

• Part (ii): Prove \(AD \times QS = BD \times PS\)
- Since \(\Delta ADB \sim \Delta PSQ\), their corresponding sides are proportional:
\[ \frac{AD}{PS} = \frac{BD}{QS} \] - Cross-multiply the terms of the proportion:
\[ AD \times QS = BD \times PS \] - This completes the proof for part (ii).


Step 4: Final Answer:
Both parts of the theorem are proved successfully.