Question : A triangle $\text{PQR}$ has three sides equal in measurement and if $\text{PM}$ is perpendicular to $\text{QR}$, then which of the following equality holds?
Option 1: $3 \text{PM}^2=2 \text{PQ}^2$
Option 2: $3 \text{PQ}^2=4 \text{PM}^2$
Option 3: $3 \text{PM}^2=4 \text{PQ}^2$
Option 4: $3 \text{PQ}^2=2 \text{PM}^2$
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Correct Answer: $3 \text{PQ}^2=4 \text{PM}^2$
Solution :
In an equilateral triangle $\text{PQR}$, all three sides are equal, so let's denote the length of each side as $a$.
Given: $\text{PM}\perp \text{QR}$
$\therefore$ $PQ^2 = PM^2 + MQ^2$ [By Pythagoras theorem]
$\text{MQ}$ is half of $\text{QR}$ (as it bisects $\text{QR}$ when $\text{PM}$ is perpendicular to $\text{QR}$)
⇒ $MQ = \frac{1}{2} QR = \frac{1}{2}a$ and $PQ=a$
⇒ $a^2 = PM^2 + \left(\frac{1}{2}a\right)^2$
⇒ $a^2 = PM^2 + \frac{1}{4}a^2$
⇒ $a^2-\frac{1}{4}a^2=PM^2$
⇒ $\frac{3}{4}a^2 = PM^2$
Replacing $a$ by $PQ$
⇒ $\frac{3}{4}PQ^2 = PM^2$
$\therefore3PQ^2 = 4 PM ^2$
Hence, the correct answer is $3PQ^2 = 4 PM ^2$.
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