Question

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Answer 1

Let the side length of an equilateral triangle ∆PQR is a and PT be the altitude of ∆PQR as shown in figure.
Now, PQ = QR = PR = a
We know, in case of equilateral triangle, altitude divide the base in two equal parts. e.g., QT = TR = QR/2 = a/2
Given, QS = QR/3 = a/3
∴ ST = QT - QS = a/2 - a/3 = a/6
Also PT = √(PQ² - QT²) = √(a² -a²/4) = √3a/2

Now, use Pythagoras theorem for ∆PST
PS² = ST² + PT²
⇒PS² = (a/6)² + (√3a/2)² = a²/36 + 3a²/4
= (a² + 27a²)/36 = 28a²/36 = 7a²/9
⇒9PS² = 7PQ²
Hence proved