P is the mid-point of side BC of ∆ABC. If Q is mid-point of AP and BQ when produced meets AC at L. Is LA = 1/3(CA)?
Type 1: if your answer is yes. Type 2: if your answer is no. Type 3: if answer cannot be determined.
Explanation:
Draw a line from P parallel to BL which meets AC at M.
Now, let’s consider ∆APM and ∆AQL
∠QAL is common in both triangles, and
∠AQL = ∠APM (Corresponding angles ∵ QL || PM)
∴ ∆APM ∼ ∆AQL ...(By A-A test of similarity)
⇒ AQ/AP = AL/AM
⇒ 1/2 = AL/AM
⇒ AL = LM …(1)
Similarly, ∆CPM ~ ∆CBL, and
⇒ CM = LM …(2)
From (1) and (2), we get
CM = ML = LA = 1/3 × AC
Hence, 1.
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