### Video Transcript

In the figure, line segment π΅π· meets line segment π΄πΈ at πΆ, which is also the midpoint of line segment π΅π·. Find the length of line segment πΆπΈ.

The first thing we notice is that πΆ is the midpoint of line segment π΅π·. We know that the length of line segment π΅πΆ equals 27. And that means that the length of π·πΆ will also be equal to 27. Because the midpoint πΆ divides line π΅π· in half, each side equals 27. Weβre interested in line segment πΆπΈ. But at this point, it doesnβt seem like thereβs enough information to solve the question. We need to consider if we can say anything else about the angles or the side lengths.

We know that angle π΅πΆπ΄ and angle π·πΆπΈ are vertical angles. Theyβre the angles across from each other when two lines intersect. Therefore, the measure of angle π΅πΆπ΄ is equal to the measure of angle π·πΆπΈ. We can also say that two right-angled triangles are congruent if the leg and an acute angle of one triangle are congruent to the corresponding parts of another triangle.

In this image, itβs a little bit difficult to see which parts correspond. So we can rearrange the triangles to better see the corresponding parts. Hereβs triangle πΈπ·πΆ. We know the length of side π·πΆ equals 27. And weβve rotated triangle π΄π΅πΆ around so that their right angles are in the same position. We know that side length π΅πΆ is 27. And here we have two congruent sides, two congruent corresponding sides. We also know the measure of angle π·πΆπΈ equals the measure of π΅πΆπ΄. And that means we have corresponding congruent angles.

Based on this, we can say that triangle πΈπ·πΆ is congruent to triangle π΄π΅πΆ. And if these two triangles are congruent, all of their corresponding parts will be congruent. That means the length of πΈπ· will be equal to the length of π΄π΅. And the hypotenuse πΈπΆ will be equal to the length of the hypotenuse of the other triangle, line π΄πΆ. For triangle π΄π΅πΆ, we also know the length of side length π΄π΅ is equal to 36. We can find the length of π΄πΆ using the Pythagorean theorem. And once we find the length of π΄πΆ, that will also be the length of πΈπΆ.

The Pythagorean theorem tells us π squared plus π squared equals π squared, where π and π are the two smaller sides of a right-angled triangle. π will be the length of the hypotenuse. And for us, that means side length π΄π΅ squared plus side length π΅πΆ squared will be equal to the hypotenuse π΄πΆ squared. And the hypotenuse π΄πΆ is equal to the hypotenuse πΈπΆ.

Letβs plug in what we know. π΄π΅ equals 36. π΅πΆ equals 27. 36 squared plus 27 squared equals π΄πΆ squared. 36 squared plus 27 squared equals 2025, which is the hypotenuse squared. To find out what the hypotenuse is, we need to take the square root of both sides of the equation. The square root of 2025 equals 45. And in triangle π΄π΅πΆ, the hypotenuse equals 45. And by corresponding parts, we can say that triangle πΈπ·πΆ has a hypotenuse of 45 as well.