Question

The semicircle shown at left has center c and diameter \overline{WZ}
WZ
start overline, W, Z, end overline. The radius \overline{XY}
XY
start overline, X, Y, end overline of the semicircle has length 222. The chord \overline{YZ}
YZ
start overline, Y, Z, end overline has length 222. What is the area of the shaded sector formed by obtuse angle W, X, Y?

Answer 1

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The semicircle shown at left has center X and diameter W Z. The radius XY of the semicircle has length 2. The chord Y Z has length 2. What is the area of the shaded sector formed by obtuse angle WXY?

$\bold{ \red{\star{\blue{GIVEN }}}}$

RADIUS = 2

CHORD = 2

RADIUS --> XY , XZ , WX

( BEZ THEY TOUCH CIRCUMFERENCE OF THE CIRCLES AFTER STARTING FROM CENTRE OF THE CIRCLE)

$\bold{\blue{\star{\red{TO \: \: FIND}}}}$

THE AREA OF THE SHADED SECTOR FORMED BY OBTUSE ANGLE WXY.

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AREA COVERED BY THE ANGLE IN A SEMI SPHERE

$AREA = ANGLE \: \: IN \: \: RADIAN \times RADIUS$

$\huge\mathbb{\red A \pink{N}\purple{S} \blue{W} \orange{ER}}$

Total Area Of The Semi Sphere:-

$AREA = \pi \times radius \\ \\ AREA = \pi \times 2 = 2\pi$

Area Under Unshaded Part .

Given a triangle with each side 2 units.

This proves that it's is a equilateral triangle which means it's all angles r of 60° or π/3 Radian

So AREA :-

$AREA = \frac{\pi}{3} \times radius \\ \\ AREA = \frac{\pi}{3} \times 2 \\ \\ AREA = \frac{2\pi}{3}$

$\green{Now:- } \\ \green{ \: Area \: Under \: Unshaded \: Part }$

Total Area - Area Under Unshaded Part

$Area= 2\pi - \frac{2\pi}{3} \\ Area = \frac{6\pi - 2\pi}{3} \\ Area = \frac{4\pi}{3} \: \: ans$