Two congruent circles of radius 𝑟 centimeters intersect. Their centers are 56 centimeters apart and their common chord has a length of 42 centimeters. Determine 𝑟.
In order to answer this question, we’re going to need to draw a diagram. We’re told, first of all, that these two circles are congruent with a radius of 𝑟 centimeters, which means they are exactly the same size. We’re told that these two circles intersect, so they look a little something like this. The next piece of information we’re given is that their centers are 56 centimeters apart, so we can sketch in the line segment connecting their centers. We’re also told that their common chord has a length of 42 centimeters. So, this is a line connecting two points on the circumference of each circle. And it’s a line they have in common. So, it’s this chord here. We’re asked to determine 𝑟, the radius of these two congruent circles.
Now, we can sketch in the radius anywhere we like, but it makes most sense to sketch it in joining the center to a point on the circumference that we’re already interested in. So, we can use this radius here. We can also sketch in the radius in our other circle. And We now see that we have an isosceles triangle consisting of the two lines labelled 𝑟 and the line segment of 56 centimeters. The common chord, therefore, divides this isosceles triangle into two congruent right triangles.
We can sketch one of these triangles out separately to our main diagram if it helps. We now see that we have a right triangle and 𝑟 is the length of its hypotenuse. We actually know the length of the other two sides. The horizontal side will be half the distance between the centers of the circles. That’s half of 56 centimeters, which is 28 centimeters. And the vertical side will be half of the length of the common chord. That’s half of 42 centimeters, which is 21 centimeters.
As we know two sides in a right triangle and we wish to calculate the third, we can apply the Pythagorean theorem. The Pythagorean theorem tells us that in a right triangle, the square of the hypotenuse, that’s the longest side, is equal to the sum of the squares of the two shorter sides. You may sometimes see this written as 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 represents the hypotenuse and 𝑎 and 𝑏 represent the two shorter sides. In our triangle then, we can express the Pythagorean theorem using our three lengths. We have that 𝑟 squared is equal to 21 squared plus 28 squared. And now, we have an equation we can solve in order to find the value of 𝑟.
First, we evaluate 21 squared and 28 squared and then add them together, giving 𝑟 squared equals 1225. To find the value of 𝑟, we need to take the square root of each side of this equation, remembering that we only take the positive square root as 𝑟 represents a length. We have 𝑟 equals the square root of 1225, which is 35. So, by drawing a diagram to help visualize the situation and then applying the Pythagorean theorem, we’ve found that the radius of these two congruent circles is 35 centimeters.