### Video Transcript

Given that πΈπ΄ equals 5.2 centimeters, πΈπΆ equals six centimeters, πΈπ΅ equals 7.5 centimeters, and πΈπ· equals 6.5 centimeters, do the points π΄, π΅, πΆ, and π· lie on a circle?

Letβs consider the diagram weβve been given. It features two line segments, π΄π΅ and πΆπ·, which intersect at a point πΈ. Weβre given in the question the length of the line segment from point πΈ to each of the four points π΄, π΅, πΆ, and π·. πΈπ΄ is 5.2 centimeters, πΈπΆ is six centimeters, πΈπ΅ is 7.5 centimeters, and πΈπ· is 6.5 centimeters. Weβre asked to determine whether the points π΄, π΅, πΆ, and π· lie on a circle.

Now this sort of setup as we have in the diagram, two intersecting line segments, should remind us of the intersecting chords theorem. This tells us that if two chords π΄π΅ and πΆπ· intersect at a point πΈ, then π΄πΈ multiplied by πΈπ΅ is equal to πΆπΈ multiplied by πΈπ·. Essentially, what this is telling us is that the product of the lengths of the two segments that point πΈ divides each chord into is the same for both chords. Now the converse of this is also true, which means that if this relationship holds for two intersecting line segments, then those line segments are chords of a circle. So the endpoints of those line segments lie on its circumference.

To answer this question then, we need to test whether this relationship is satisfied by the lengths that weβve been given. First, for the line segment π΄π΅, π΄πΈ multiplied by πΈπ΅, or πΈπ΄ multiplied by πΈπ΅, is 5.2 multiplied by 7.5, which is equal to 39. Then for the line segment πΆπ·, πΆπΈ multiplied by πΈπ·, or πΈπΆ multiplied by πΈπ·, is six multiplied by 6.5, which is also equal to 39. As the product is the same for both line segments, the intersecting chords theorem is satisfied, and so the two line segments π΄π΅ and πΆπ· are chords of the same circle.

The points π΄, π΅, πΆ, and π· do lie on a circle, and so our answer to the question is yes.