What is the area of the black-shaded region shown in the figure ?

Level pending

Given that AB = 9 units, CD = 4 units and BD is the diameter of the circle.
The line AC is a tangent to the red-semicircle.

Calculate the area approximated to 2 decimal places.

22.47 20.18 23.40 21.45

Vijay Simha by Vijay Simha , 47, USA

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2 solutions

Siddharth Singh
Apr 26, 2015

In the trapezium ABCD ,draw a perpendicular on AB from C named CE.Now AE=9-4=5. Let the point of contact of tangent AC be F so AF=AB=9 and CF=CD=4. AC=13. So CE can be found to be 12 by Pythagoras theorem. CE=BD,so now the area of the black shaded region can be found by subtracting the area of the semicircle from the area of trapezium.

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What did you get the area of the trapezium as ?

Vijay Simha - 6 years, 1 month ago

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Area of trapezium is 78sq units.[ 1/2(13)*12 ]

Siddharth Singh - 6 years, 1 month ago
Vijay Simha
Apr 27, 2015

The first thing we need to do is to find out the radius and diameter of the circle. The radius of the circle can be found out using the See-Saw lemma.

Applying the See-Saw Theorem, gives the radius of the circle to be sqrt(9x4) = 6 units.

Next from this we find the diameter of the circle to be 12 units.

Consequently, the area of the trapezium = 6x(9+4) = 78 sq units. The area of the semicircle is 36 pi/2 = 18 pi sq units.

The area of the shaded region therefore = (78-18*pi) sq units.

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