Video: CBSE Class X • Pack 5 • 2014 • Question 21

CBSE Class X • Pack 5 • 2014 • Question 21

06:39

Video Transcript

In the figure, a circle is inscribed in an equilateral triangle 𝐴𝐵𝐶 of side length 12 centimetres. Find the radius of the circle and the area of the shaded region. Take 𝜋 is equal to 3.14 and root three is equal to 1.73.

Since the circle is inscribed within the triangle, we know that the circumference of the circle meets each side of the triangle exactly once. Let’s call these points of intersection 𝑥, 𝑦, and 𝑧. We can add lines joining the centre of the circle to each of these points. They are the radii of the circle. Let’s call them 𝑟. We know that the radius and the tangent meet at angle of 90 degrees. So the radii meet the sides of the triangle at 90 degrees.

Next, we can split the triangle up into three equal triangles. These are triangle 𝐴𝑂𝐵, triangle 𝐵𝑂𝐶, and triangle 𝐴𝑂𝐶. The area of these three triangles makes up the area of the larger triangle 𝐴𝐵𝐶. The area of 𝐴𝑂𝐵 plus 𝐵𝑂𝐶 plus 𝐴𝑂𝐶 equals the area of 𝐴𝐵𝐶. Since each of these triangles are equal, we can say that the area of 𝐴𝑂𝐵 is equal to one-third of the area of 𝐴𝐵𝐶.

Now, one of the formulae we know for area of a triangle is a half multiplied by base multiplied by height. If we treat the base of triangle 𝐴𝑂𝐵 as the side 𝐴𝐵, we can write the area of 𝐴𝑂𝐵 as being a half multiplied by 12 multiplied by 𝑟. Remember, the radius is perpendicular to the tangent. So the radius is the height of this triangle. That gives us that the area of 𝐴𝑂𝐵 is six 𝑟.

Now, we don’t know the vertical height of triangle 𝐴𝐵𝐶. So we’ll use an alternative formula. We’ll use the formula half multiplied by 𝑎𝑏 multiplied by sin 𝐶. Since this is an equilateral triangle, each of its interior angles are equal. They’re all 60 degrees. Each of the sides of this triangle are 12 centimetres. So the area is a half multiplied by 12 multiplied by 12 multiplied by sin 60. Sin 60 is root three over two. So this becomes a half multiplied by 12 multiplied by 12 multiplied by root three over two. We can cross cancel by two and by two again. And that gives us that the area of triangle 𝐴𝐵𝐶 is 36 root three.

Now remember, we said that the area of 𝐴𝑂𝐵 is equal to a third of the area of 𝐴𝐵𝐶. So let’s substitute in the areas we have. That gives us six 𝑟 is equal to a third multiplied by 36 root three. A third of 36 is 12. So we get six 𝑟 equals 12 root three. And if we divide through by six, we get that 𝑟 is equal to two root three. We’re told to use root three as 1.73. So the radius is two multiplied by 1.73, which is 3.46. We’ve calculated the radius of this circle to be 3.46 centimetres.

Now that we have the radius, we need to find the area of the shaded region. The shaded area is given by the area of the triangle minus the area of the circle. Remember, we said that the area of triangle 𝐴𝐵𝐶 is given by a half 𝑎𝑏 sin 𝐶. And the area of the circle is 𝜋𝑟 squared. We already worked out that a half 𝑎𝑏 sin 𝐶 was 36 root three. 𝜋𝑟 squared is 𝜋 multiplied by two root three squared. It’s actually easier here to use two root three rather than 3.46 at this stage. Two root three all squared is the same as having two squared multiplied by root three squared. Two squared is four. And root three squared is three. So we have four multiplied by three which is 12.

Using the values of 𝜋 and root three that we were given in the question, and we get that the shaded area is 36 multiplied by 1.73 minus 3.14 multiplied by 12. We we’ll need to use some formal written methods for multiplication to work out the value of 36 multiplied by 1.73 and 3.14 multiplied by 12. To work out 36 multiplied by 1.73, let’s first calculate 36 multiplied by 173. Three multiplied by six is 18. So we put an eight in this first column and carry the one. Seven multiplied by six is 42. And when we add the one, we get 43. One multiplied by six is six and six plus four is 10.

Next, we’re going to multiply the digits of 173 by three. But this three isn’t really a three; it’s actually a 30. So we put a zero in this column to show that when we multiply, our answer is going to be 10 times bigger than we expect. Three multiplied by three is nine. Seven multiplied by three is 21. And one multiplied by three is three. Three plus two is five. Finally, we add these two numbers. And we get 173 multiplied by 36 is 6228. 1.73 is 100 times smaller than 173. So 36 multiplied by 1.73 is going to be 100 times smaller than 173 multiplied by 36. It’s 62.28.

We can use a similar method to calculate 3.14 multiplied by 12. 314 multiplied by 12 is 3768. 3.14 is 100 times smaller than 314. So our answer is also going to be 100 times smaller. 3.14 multiplied by 12 is 37.68. 62.28 minus 37.68 is 24.6. We have that the radius is 3.46 centimetres. And the shaded area is 24.6 centimetres squared.

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