In the figure shown, 𝐴𝐵 equals 12 and 𝐵𝐶 equals five. If the area of the circle is 𝑥𝜋, find the value of 𝑥.
We’re told in the question that the length of 𝐴𝐵 is equal to 12 and the length of 𝐵𝐶 is equal to five. Joining the points 𝐴𝐶, as shown, creates a right-angled triangle as the angle 𝐴𝐵𝐶 is equal to 90 degrees. One of our circle theorems states that the angle in a semicircle is equal to 90 degrees. This means that the line 𝐴𝐶 is the diameter of the circle. If we let 𝑂 be the center of the circle, then 𝑂 is the midpoint of 𝐴𝐶. Therefore, 𝐴𝑂 is the radius of the circle.
We might remember at this point that there are some special right triangles that we need to know. For example, a three, four, five and five, 12, 13 right triangle. In our case, as the two shorter sides are lengths five and 12, then the longest side or hypotenuse will have length 13. The diameter of the circle 𝐴𝐶 is length 13. As the radius is half the length of the diameter, this will have length 13 over two or 13 divided by two. We could write this as the decimal 6.5 as a half of 13 is 6.5.
If we hadn’t recognized that we have a special triangle, we could use Pythagoras’ theorem to calculate the length of the diameter. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the longest side of the right triangle known as the hypotenuse. As the length 𝐴𝐶, the diameter of the circle, is the hypotenuse, we have 𝐴𝐶 squared is equal to 12 squared plus five squared. 12 squared is equal to 144 as 12 multiplied by 12 is 144. Five squared is equal to 25. This means that 𝐴𝐶 squared is equal to 144 plus 25. These two numbers sum to 169. The opposite or inverse of squaring is square rooting. Therefore, we need to square root both sides of this equation. The square root of 169 is equal to 13 as 13 multiplied by 13 is 169. We have once again proved that the length of the diameter 𝐴𝐶 is equal to 13.
We were told that the area of our circle is 𝑥𝜋. And the formula to calculate the area of a circle is 𝜋𝑟 squared. As the radius is equal to 13 over two, the area will be equal to 𝜋 multiplied by 13 over two squared. If we want to square any fraction, we can square the numerator and denominator separately. 13 squared is equal to 169, as we have already seen. And two squared is equal to four. Therefore, the area is equal to 𝜋 multiplied by 169 over four. This can be rewritten as 169 over four 𝜋.
Our answer is now written in the same form as the question, where 𝑥 is equal to 169 over four.