Question

find the area of the shaded portion in the adjoining figure, it being given that ABCD is a square of 12 cm, CE = 4cm,FA=5cm and BG=5cm​

Answer 1

Answer:-

Area of the shaded region is 68.5 cm²

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Step-by-step explanation:-

Given:-

• ABCD is a square
• BC = 12 cm
• AG = 7 cm
• BG = 5 cm
• AF = 5 cm
• EC = 4 cm

To Find:-

• Area of the shaded region

Important Step:-

Let us first find the measure of each sides.

In the given figure,

AB || CD

AD || BC

It is given that BC = 12 cm

Now,

BC = AD = 12 cm

Also it is given that F is a point on AD

AF = 5 cm

FD = AD - AF = 12 - 5 = 7 cm

Now,

It is given that,

AG = 7 cm

BG = 5 cm

Hence, measure of,

AB = AG + BG = 7 + 5 = 12 cm

Now,

AB = CD = 12 cm

Also,

E is a point of CD

EC = 4 cm

DE = CD - EC = 12 - 4 = 8 cm

So we got measure of all the sides as:-

AD = 12 cm

FD = 7 cm

AB = 12 cm

DC = 12 cm

DE = 8 cm

Solution:-

In the given figure,

ABCD is a square with side 12 cm

So,

Let us first find the area of the square,

We know,

Area of square = (Side)²

Therefore,

$\sf{Area_{(square)} = (12)^2 = 144\:cm^2}$

Now,

Also from the given figure we can see that three triangles are there:-

1st triangle = EDF

2nd triangle = GCB

3rd triangle = GAF

Note:- All the triangles are right-angled

For ∆EDF

DE = 8 cm

DF = 7 cm

We know,

Area of triangle = $\sf{\dfrac{1}{2}\times base\times height}$

Therefore,

$\sf{Area_{(triangle\:1)} = \dfrac{1}{2}\times 8\times 7}$

= $\sf{Area_{(triangle\:1)} = 28\:cm^2}$

For ∆GCB

BC = 12 cm

BG = 5 cm

$\sf{Area_{(triangle\:2)} = \dfrac{1}{2}\times 5\times 12}$

= $\sf{Area_{(triangle\:2)} = 30\:cm^2}$

For ∆GAF

AF = 5 cm

AG = 7 cm

$\sf{Area_{(triangle\:3)} = \dfrac{1}{2}\times 7\times 5}$

= $\sf{Area_{(triangle\:3)} = \dfrac{35}{2}}$

= $\sf{Area_{(triangle\:3)} = 17.5\:cm^2}$

Now,

Total Area of all the three triangles:-

= $\sf{Area_{(triangle\:1)} + Area_{(triangle\:2)} + Area_{(triangle\:3)}}$

= $\sf{(28+30+17.5)\:cm^2}$

= $\sf{75.5\:cm^2}$

Therefore, total area of all the three triangle is 75.5 cm²

Now,

Area of shaded region = Area of square - Area of three triangles

= $\sf{(144-75.5)\:cm^2}$

= $\sf{68.5\:cm^2}$

Therefore area of the shaded region is 68.5 cm²

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