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If Two Sides of a Triangle Are Unequal, Then the Angle Opposite to the Greater Side Is Greater Than Angle Opposite to the Smaller Side

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Question

From the information given in the figure, prove that PM=PN= √3×a

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Solution

PQ=QR=PR=a

=ΔPQR is equilateral
QS=SR=½ QR
QS=a/2
PS is altitude of Δ
=PS=√3/2 a (By Pythagoras Theorem)
Now ,MS=MQ+QS
=a + a/2
MS = 3a/2

Now, ΔPMS is right angled
By Pythagoras theorem,
PM²=MS²+RS²
=(3a/2)²+(√3/2a)²
=(9a²+3a²)/4
PM²=12a^2/4=3a²
so PM=√3a
Since figure is symmetrical, we can prove in the same way for PN
=PM=PN=√3a
Hence Proved

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