### Video Transcript

A circle has two secants, line segment π΄π΅ and line segment π΄π·, intersecting at π΄. Given that π΄πΈ equals three centimeters, πΈπ· equals five centimeters, and π΄π΅ equals nine centimeters, find the length of π΅πΆ, giving your answer to the nearest tenth.

Letβs begin by adding the information weβve been given to the diagram. First, weβre told that the length of π΄πΈ is three centimeters and, secondly, that the length of πΈπ· is five centimeters. The final piece of information weβre given is that the length of π΄π΅ is nine centimeters. We want to find the length of π΅πΆ, which is a segment of the secant π΄π΅. To do this, we can recall the intersecting secants theorem, which is a special case of the power of a point theorem. This states that if π΄ is a point outside a circle and π΅, πΆ, π·, and πΈ are points on the circle such that the line segment π΄π΅ is a secant to the circle at π΅ and πΆ and the line segment π΄π· is a secant to the circle at π· and πΈ, then π΄πΆ multiplied by π΄π΅ is equal to π΄πΈ multiplied by π΄π·.

Letβs have a look at what we know. We were given that the length of π΄π΅ is nine centimeters. We were also given that the length of π΄πΈ is three centimeters. And we can work out the length of π΄π· as the sum of three centimeters and five centimeters, so itβs eight centimeters. We therefore have the equation π΄πΆ multiplied by nine is equal to three multiplied by eight. Now we can use this to calculate the length of π΄πΆ although it isnβt π΄πΆ we want to find; itβs π΅πΆ. But π΄πΆ plus π΅πΆ will give the length of the entire secant segment π΄π΅, which we know to be nine. So if we can find the length of π΄πΆ, we can then use the second equation to determine the length of π΅πΆ. Simplifying the first equation, we have nine π΄πΆ is equal to 24.

To solve for π΄πΆ, we need to divide both sides by nine. That gives π΄πΆ equals 24 over nine. And dividing both the numerator and denominator by three, this simplifies to eight over three. We can then substitute this value for the length of π΄πΆ into our second equation, giving eight over three plus π΅πΆ equals nine. π΅πΆ is then equal to nine minus eight over three. And we can perform this subtraction more simply if we express nine as 27 over three. 27 over three minus eight over three is equal to 19 over three, which, as a decimal, is equal to 6.3 recurring. The question specifies that we should give our answer to the nearest tenth. So, rounding to one decimal place, we found that the length of π΅πΆ to the nearest tenth is 6.3 centimeters.