Question : G is the centroid of $\triangle$ABC. If AG = BC, then measure of $\angle$BGC is:
Option 1: 45°
Option 2: 60°
Option 3: 90°
Option 4: 120°
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Correct Answer: 90°
Solution :
Here D is the midpoint of BC and AG = BC
So, BD = GD ⇒ $\angle$BGD = $\angle$GBD and CD = GD ⇒ $\angle$GCD = $\angle$CGD
In $\triangle$BGC, $\angle$GBC + $\angle$GCB + $\angle$BGC = 180°
⇒ $\angle$GBC + $\angle$GCB + ($\angle$BGD + $\angle$CGD) = 180° [as $\angle$BGC = $\angle$BGD + $\angle$CGD]
⇒ $(\angle$GBC + $\angle$GBD) + ($\angle$GCB + $\angle$GCD) = 180°
⇒ 2$\angle$GBC + 2$\angle$GCB = 180°
⇒ $\angle$GBC + $\angle$GCB = 90°
$\therefore$ $\angle$BGC = 180° – ($\angle$GBC + $\angle$GCB) = 180° – 90° = 90°
Hence, the correct answer is 90°.
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