### Video Transcript

A circle has two chords, the line segments π΄πΆ and π΅π·, intersecting at πΈ. Given that the ratio π΄πΈ to π΅πΈ equals one to three and πΆπΈ equals six centimeters, find the length of π·πΈ.

Letβs add the information weβve been given to the diagram first. Weβre told that the length of πΆπΈ is six centimeters. And weβre also told that the ratio of the length of the line segments π΄πΈ to π΅πΈ is one to three. We can therefore express the lengths of these two line segments as π₯ centimeters and three π₯ centimeters for some nonzero value of π₯. Weβre asked to find the length of the line segment π·πΈ. And we see that the information weβve got concerns the lengths of line segments in two different chords of a circle.

We can therefore recall a basic case of the power of a point theorem, which relates the lengths of line segments in two different chords. Let πΈ be a point inside circle π. If π΄, π΅, πΆ, and π· are points on the circle such that the line segments π΄πΆ and π΅π· are two intersecting chords at πΈ, then π΄πΈ multiplied by πΆπΈ is equal to π΅πΈ multiplied by π·πΈ. This is exactly the setup we have here. We know the length of πΆπΈ. Itβs six centimeters. And we have expressions for the lengths of π΄πΈ and π΅πΈ. Itβs the length of π·πΈ that we want to find. So we can substitute the values or expressions we know and form an equation. π₯ multiplied by six is equal to three π₯ multiplied by π·πΈ.

To find the value of π·πΈ, we need to divide both sides of this equation by three π₯. And remember, we said π₯ was nonzero, so itβs fine to do this. We have six π₯ over three π₯ is equal to π·πΈ. And again, as π₯ is nonzero, we can cancel a factor of π₯ in the numerator and denominator. This leaves six over three is equal to π·πΈ. And of course six divided by three is equal to two. So by recalling the intersecting chords theorem, which is a special case of the power of a point theorem, we found that the length of π·πΈ is two centimeters.