# Question Video: Finding the Length of a Right-Angled Triangleβs Side given the Others Two Sidesβ Lengths Using the Tangentsβ Properties Mathematics

Line π΄πΆ is tangent to circle π at π΄. Given that π΅π = 55 cm, π΄πΆ = 96 cm, what is π΅πΆ?

02:51

### Video Transcript

Line π΄πΆ is tangent to circle π at π΄. Given that π΅π equals 55 centimeters, π΄πΆ equals 96 centimeters, what is π΅πΆ?

Letβs begin by adding what we know about our circle and the lines within it to the diagram itself. First, weβre given the length of the line segment between π΅ and π. Itβs 55 centimeters. Weβre also told that the length of π΄πΆ is 96 centimeters, and weβre asked to calculate π΅πΆ. Thatβs the length of the line segment that joins π΅ to πΆ. So letβs define that to be equal to π₯ centimeters. Now, we also know that line π΄πΆ is a tangent to the circle at point π΄. And we also see that line segment π΅π΄ passes through point π. This means that line segment π΅π΄ must be the diameter of the circle.

So what do we know about the relationship between a diameter and tangent of a circle? In fact, they are perpendicular to one another. In other words, angle π΅π΄πΆ is equal to 90 degrees. So triangle π΄π΅πΆ is in fact a right triangle. And of course, we know that if we know two of the dimensions of a right triangle, we can calculate the third by using the Pythagorean theorem. And whilst it might not look like it, we do actually know two of the dimensions in this triangle. Remember, we were told π΅π is equal to 55 centimeters. π΅π is in fact the radius of the circle. This means line segment π΅π΄ must be double the length of line segment π΅π since π΅π΄ is the diameter. Well, twice 55 centimeters is 110 centimeters, so line segment π΅π΄ is 110 centimeters in length.

We can therefore use the Pythagorean theorem to find the length that weβve called π₯ centimeters. The Pythagorean theorem says that, for a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. In this case, the hypotenuse is equal to π₯ centimeters. So 110 squared plus 96 squared is equal to π₯ squared. 110 squared is 12100, whilst 96 squared is 9216. Finding their sum and we see that π₯ squared is equal to 21316.

To solve this equation for π₯, we find the square root of both sides. And remember, since π₯ represents a dimension, weβre only interested in the positive square root of 21316. The positive square root of this number is in fact 146. So we found that π₯ is equal to 146. We can therefore say that π΅πΆ equals 146 centimeters.