Suppose a circle of diameter 15 centimeters contains a chord of length 11.8 centimeters. What is the shortest distance between the chord and the center of the circle? Give your answer in centimeters to the nearest hundredth.
We could often find that questions like this are much easier if we begin with a sketch of the information. We know that the circle has a diameter of 15 centimeters, and let’s draw in the chord of 11.8 centimeters. A chord is defined as a line segment which joins two distinct points on the circumference of a circle. We are asked to find the shortest distance between the chord and the center of the circle.
We could calculate the distance of any point on this chord to the center of the circle, but there’s only one place which would be the shortest distance. And that is the point where there is a perpendicular line from the center to the chord. This is the distance that we will need to calculate, so let’s define it as 𝑥 centimeters. We still don’t have enough information to help us work out the value of 𝑥. And so, we should apply one of the properties that we know about chords and a line through the center of a circle.
This property tells us that if we have a circle with center 𝐴, containing a chord of the segment 𝐵𝐶, then the straight line that passes through 𝐴 and is perpendicular to line segment 𝐵𝐶 also bisects line segment 𝐵𝐶. What this means here is that this line coming from the center, which is perpendicular to the chord, also bisects the chord. And so, the chord is split into two line segments which are congruent. And so, if we wanted to work out the length of one of these line segments, it’s going to be half the length of the chord. 11.8 divided by two is 5.9 centimeters. And so, we know that this half chord length is 5.9 centimeters.
The final piece of this problem is really recognizing that we can create a right triangle. This line segment, which forms the hypotenuse of the right triangle, is in fact the radius of the circle because it’s a line segment from the center to a point on the circumference. So, here, we can apply the information that the diameter is 15 centimeters. The radius can be found by calculating half of the diameter. So, 15 divided by two will give us 7.5, and the units, of course, will still be centimeters.
Now, we have a right triangle with two lengths that we know and one that we don’t know. And that means we can apply the Pythagorean theorem. The Pythagorean theorem gives us that for any right triangle the square of the hypotenuse is equal to the sum of the squares on the other two sides. And so, we can apply this theorem by taking the hypotenuse, which is 7.5, and squaring it and setting it equal to 5.9 squared plus 𝑥 squared.
Working out 7.5 squared, we get 56.25. And working out 5.9 squared, we get 34.81. We rearrange this in order to get 𝑥 squared by itself on one side of the equation. And so, subtracting 34.81 from both sides, we have 21.44 is equal to 𝑥 squared. Next, we take the square root of both sides of the equation, remembering that because 𝑥 is a length, we need only consider the positive value of the root. We are asked to round our answer to the nearest hundredth. And 4.6303 will round down to 4.63. And, of course, the units will be in centimeters.
And so, we can give the answer, then, that the shortest distance between the chord and the center of the circle is 4.63 centimeters.