Question

Assertion: In ΔABC, AP ⊥ BC. E and F are midpoints of AB and AC, then AQ = QP.

Reason: Q is the midpoint of AP from midpoint theorem.

Options

• Both A and R are true and R is the correct reason for A.

• Both A and R are true but R is the incorrect reason for A.

• A is true but R is false.

• A is false but R is true.

Answer 1

Both A and R are true and R is the correct reason for A.

Explanation:

Assertion:

1. Midpoint Theorem Application: In ΔABC, E and F are the midpoints of AC and AB, respectively. The midpoint theorem states that the line segment FE is parallel to BC.
2. Altitude and Intersection: The altitude AP is perpendicular to BC and thus AP is parallel to FE. The point Q is the intersection of the altitude AP and the line segment FE.
3. Applying the Converse: Consider ΔABP. F is the midpoint of AB. Since FE || BP (because FE || BC), by the converse of the midpoint theorem, the line drawn through F parallel to BP must bisect AP. Therefore, Q, which is on FE and also on AP, is the midpoint of AP.
4. Conclusion: Since Q is the midpoint of AP, AQ = QP.

Reason:

The reason states that Q is the midpoint of AP from the midpoint theorem. This is correct because of the steps outlined above.

Assertion (A) is true: AQ = QP is a valid conclusion based on the midpoint theorem and its converse. 

Reason (R) is true: Q is indeed the midpoint of AP. 

Reason (R) is the correct explanation for (A): The application of the midpoint theorem and its converse to the triangle ABP is the direct reason why AQ = QP.

Therefore, Both A and R are true and R is the correct reason for A.