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81.
G is the centroid of the equilateral Δ ABC, if AB = 9 cm, then AG is equal to:

$3 square root of 3 space cm$

3 cm

^{$fraction numerator 3 square root of 3 over denominator 2 end fraction cm$}

6 cm

$3 square root of 3 space cm$

Median AD of equilateral triangle:
$equals space fraction numerator square root of 3 over denominator 2 end fraction straight a equals space fraction numerator square root of 3 over denominator 2 end fraction space cross times space 9$
As Centroid divides the median in the ratio of 2 : 1.
∴ AG : GD = 2 : 1
$therefore space space space space AG space equals space 2 over 3 space cross times space AD$
$AG space equals space 2 over 3 space cross times space fraction numerator square root of 3 over denominator 2 end fraction space cross times space 9 AG space equals space 3 square root of 3 space cm$

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82.

In the figure (not drawn to scale) given below, if AD = DC = BC and $angle BCE space equals space 96 degree$ , then $angle BCE space space equals space 96 degree comma$ then $angle DBC$ is

32°
84°
64°
96°

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83.

In Δ ABC, $angle straight B space equals space 5 angle straight C$ and $angle straight A space equals space 3 angle straight C$ , then the measure of $angle straight C$ is