A circle has a tangent 𝐴𝐵 and a secant 𝐴𝐷 that cuts the circle at 𝐶. Given that 𝐴𝐵 equals seven centimeters and 𝐴𝐶 equals five centimeters, find the length of 𝐶𝐷. Give your answer to the nearest hundredth.
Let’s begin by adding the information we’re given to the diagram. We’re told that the length of 𝐴𝐵 is seven centimeters and the length of 𝐴𝐶 is five centimeters. The length 𝐶𝐷 is what we wish to find. Now, the information we’re working with consists of the length of a tangent and the lengths of different segments of a secant to a circle. We can approach this problem by recalling the intersecting tangents and secants theorem, which is a special case of the power of a point theorem. This states the following. Let 𝐴 be a point outside a circle and let 𝐵, 𝐶, and 𝐷 be points on the circle such that the line segment 𝐴𝐵 is a tangent segment and the line segment 𝐴𝐷 is a secant segment. Then, 𝐴𝐵 squared is equal to 𝐴𝐶 multiplied by 𝐴𝐷.
Now, this is exactly the setup we have here. So we can substitute some of the values we know. 𝐴𝐵 is equal to seven. So on the left-hand side of the equation, we have seven squared. 𝐴𝐶 is equal to five, and we don’t know the length of 𝐴𝐷. So we have the equation seven squared is equal to five multiplied by 𝐴𝐷. Now, we can solve this equation to determine the value of 𝐴𝐷. And this will be really useful because 𝐶𝐷 is a segment of 𝐴𝐷.
The length of 𝐴𝐷 is equal to the length of 𝐴𝐶 plus the length of 𝐶𝐷. And we already know the length of 𝐴𝐶 to be five. So if we can determine the length of 𝐴𝐷, we can then use this equation to find the length of 𝐶𝐷. Returning to our first equation and evaluating seven squared, we have that 49 is equal to five multiplied by 𝐴𝐷. We can then divide both sides of this equation by five, and we have that 𝐴𝐷 is equal to 49 over five. Or as a decimal, this is 9.8.
We can now substitute this value for the length of 𝐴𝐷 into our second equation, which gives 9.8 is equal to five plus 𝐶𝐷. To solve this equation for 𝐶𝐷, we simply need to subtract five from each side. And we find that 𝐶𝐷 is equal to 4.8. The question though specifies that we should give our answer to the nearest hundredth. So we need to include a zero in the second decimal place. So by recalling the intersecting tangents and secants theorem, which is a special case of the power of a point theorem, we found that the length of 𝐶𝐷 correct to the nearest hundredth is 4.80 centimeters.