Video: CBSE Class X • Pack 5 • 2014 • Question 14

CBSE Class X • Pack 5 • 2014 • Question 14


Video Transcript

In the figure, a square 𝑂𝐴𝐵𝐶 is inscribed in a quadrant 𝑂𝑃𝐵𝑄 of a circle. If 𝑂𝐴 equals 20 centimeters, find the area of the shaded region. Use 𝜋 equals 3.14.

The area of the shaded region is equal to the area of quadrant 𝑂𝑃𝐵𝑄 minus the area of the square 𝑂𝐴𝐵𝐶. A quadrant is one-fourth of a circle. If we find the area of a circle by multiplying 𝜋 times the radius squared, the area of the quadrant is one-fourth of that. So we would say 𝜋𝑟 squared divided by four. And of course, the area of a square equals one of its sides squared.

Our question tells us that 𝑂𝐴, the side of the square, measures 20 centimeters. We can plug in 20 for the side of the square. We’re missing one piece of information we need to find the area of the quadrant. We need to know what the radius is of this circle. We remember that a radius is a straight line from the center to the circumference of a circle. Our circle is centered at point 𝑂. It has a radius at 𝑂𝑄 and 𝑂𝑃. We also know a third radius for this quadrant.

The line segment 𝑂𝐵 is a straight line from the center of this circle to its outside. This diagonal across the square turns it into two congruent triangles. Because we have a square, we know that side length 𝐴𝐵 is also equal to 20 centimeters. If we find the length of 𝑂𝐵, we’ll find the radius of this quadrant. We could of course use the Pythagorean theorem and say 20 squared plus 20 squared equals the hypotenuse squared.

But a simpler way is to recognize that this is a 45-45-90 degree triangle. And that means its side lengths will be in the ratio one to one to the square root of two. Our two smaller sides measure 20 centimeters. And that means our larger side will be 20 centimeters times the square root of two. The radius of this triangle is 20 times the square root of two.

To solve, we plug in 20 times the square root of two in for our radius. 20 times the square root of two squared equals 800. 800𝜋 over four minus 20 squared which equals 400. In the next step, we can divide 800 by four. And we’ll be left with 200𝜋 minus 400. This is really the furthest we can go without using our approximation 3.14 for 𝜋. We substitute 3.14 for 𝜋.

At this point, I would probably do something like this. Instead of leaving 200, I would say 100 times two times 3.14. Two times 3.14 equals 6.28. Multiply that by 100. Move the decimal point two places to the right. And we get 628. Don’t forget our subtract 400 that’s there.

628 minus 400 is the area of our shaded region. And it equals 228. We’re dealing with area. And all of our units were measured in centimeters. So the final answer should be written as 228 centimeters squared.

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