A circle with centre 𝑀 and a point 𝐴 satisfy 𝑀𝐴 is equal to 28 centimetres and 𝑃 𝑀 of 𝐴 is equal to four. Using 𝜋 is equal to 22 over seven, find the area and the circumference of the circle to the nearest integer.
The notation 𝑃 𝑀 of 𝐴 refers to the power of a point theorem. Let’s firstly consider our circle and show how this theorem relates to it. We’re told in the question that we have a circle with centre 𝑀 and a point 𝐴. We’re also told that the length of 𝑀𝐴 is 28 centimetres. If we extend the line 𝐴𝑀 we have two points on the outside of the circle 𝐵 and 𝐶. For any line of this type, the power of a point theorem states that the power of a point is equal to 𝐴𝐵 multiplied by 𝐴𝐶.
The lengths 𝑀𝐵 and 𝑀𝐶 are both the radius of the circle. We will call this 𝑟. The length 𝐴𝐵 is, therefore, equal to 𝐴𝑀 minus 𝑀𝐵. 𝐴𝑀 is equal to 28 centimetres and 𝑀𝐵 is equal to 𝑟. Therefore, 𝐴𝐵 is equal to 28 minus 𝑟. We can use the same methods to calculate the length of 𝐴𝐶. This is equal to 𝐴𝑀 plus 𝑀𝐶. 𝐴𝑀 is once again equal to 28 and the length 𝑀𝐶 is 𝑟. Therefore, 𝐴𝐶 is equal to 28 plus 𝑟. Substituting these into our formula gives us 28 minus 𝑟 multiplied by 28 plus 𝑟. We know that the power of a point is equal to four. Therefore, this expression equals four.
At this point, we might notice that the right-hand side is the difference of two squares. If we don’t notice this, we can just expand it using the FOIL method. Multiplying the First terms 28 and 28 gives us 784. Multiplying the Outside terms gives us 28𝑟. Multiplying the Inside terms gives us negative 28𝑟. Finally, multiplying the last terms gives us negative 𝑟 squared.
The middle two terms cancel as 28𝑟 minus 28𝑟 is equal to zero. Our equation becomes four is equal to 784 minus 𝑟 squared. Subtracting four from both sides of this equation gives us zero is equal to 780 minus 𝑟 squared. We can then add 𝑟 squared to both sides of this equation to give us 𝑟 squared is equal to 780. Square rooting both sides gives us 𝑟 is the square root of 780. This would usually be positive or negative. However, as we’re dealing with a length, it must be positive: the square root of 780.
The question asked us to calculate the area and circumference of the circle. The area of any circle can be calculated by multiplying 𝜋 by the radius squared. The circumference can be calculated by multiplying 𝜋 by the diameter. This is the same as multiplying two 𝜋 by the radius as the diameter is double the radius. If we clear some room, we can now calculate these two values.
As the radius squared is equal to 780, the area is equal to 𝜋 multiplied by 780. In this question, we’re told to use 22 over seven as 𝜋. Multiplying this by 780 gives us 2451.428 et cetera. We need to round this to the nearest integer. We look at the first digit after the decimal point. As this is four, we will round down. The area of the circle to the nearest integer is 2451 centimetres squared.
The circumference can be calculated by multiplying two by 22 over seven by the root of 780. Typing this into the calculator gives us 175.550 et cetera. This time, as the number after the decimal point is five, we will round up. The circumference of the circle to the nearest integer is 176 centimetres.
This means that our two answers are 2451 centimetre squared and 176 centimetres.