Question

∠ABC = 90°. ABP is an equilateral triangle. PQ || BC. Prove that

1. PQ ⊥ AB
2. AQ = QB
3. R is the mid-point of AC.

Answer 1

Given:

• ∠ABC = 90°
• Triangle ABP is equilateral.
• PQ is parallel to BC (PQ || BC).

i. PQ ⊥ AB

Step 1: ABP is an equilateral triangle.
=> ∠PAB = 60° (All angles in equilateral triangle)

Step 2: Given ∠ABC = 90°.

Step 3: Since PQ || BC and ∠ABC = 90°, line AB is perpendicular to BC (by definition of right angle).

Step 4: If PQ is parallel to BC and BC ⊥ AB, then PQ ⊥ AB (parallel lines preserve perpendicularity).

PQ ⊥ AB.

ii. Prove AQ = QB

Step 1: Since ABP is equilateral, AB = AP.

Step 2: Join point Q on AB such that PQ || BC.

Step 3: In triangle ABP, PQ || BC, by midpoint theorem, Q is the midpoint of AB.

⇒ AQ = QB.

iii. Prove R is the midpoint of AC

Step 1: Since PQ || BC, and Q is the midpoint of AB (from above),

Step 2: If a line through the midpoint of one side of a triangle is drawn parallel to another side, it bisects the third side (Midpoint theorem).

So, R is the midpoint of AC.