ABC is a triangle.P,Q and R ARE THE MIDPOINTS OF BC , AC AND AB RESPECTIVELY.L IS ANY POINT ON BC DRAWN THROUGH VERTEX A.
PROVE THAT; RLPQ IS CYCLIC QUADRILATERAL IF AL IS PERPENDICULAR TO BC. AND QR IS JOINED
To prove that the quadrilateral RLPQ is cyclic, we need to show that the opposite angles are supplementary.
First, let's draw a diagram to better understand the situation.
```
A
/ \
/ \
/ \
/_______\
R P Q
```
Given: ABC is a triangle, P, Q, and R are the midpoints of BC, AC, and AB respectively, L is any point on BC drawn through vertex A, and AL is perpendicular to BC.
To prove that quadrilateral RLPQ is cyclic, we need to prove that ∠RLQ + ∠RPQ = 180°.
Now, let's break down the proof step by step:
Step 1: Since P and Q are the midpoints of AC and BC respectively, we know that PQ is parallel to AB and half its length. Therefore, ∠APQ = ∠ABC (alternate angles between parallel lines).
Step 2: Since R is the midpoint of AB, RL is parallel to QC and half its length. Therefore, ∠RLQ = ∠QCB (alternate angles between parallel lines).
Step 3: Since AL is perpendicular to BC, and RL is parallel to QC, we have ∠ALR = ∠QCB (corresponding angles between parallel lines and transversal).
Step 4: Combining Step 2 and Step 3, we have ∠RLQ = ∠ALR.
Step 5: Using the same reasoning, we can prove that ∠RPQ = ∠ALR.
Step 6: From Step 4 and Step 5, we have ∠RLQ = ∠RPQ.
Step 7: Since opposite angles in a cyclic quadrilateral sum up to 180°, we can conclude that RLPQ is a cyclic quadrilateral.
Therefore, we have proven that quadrilateral RLPQ is cyclic if AL is perpendicular to BC, and QR is joined.