In the figure, line segment 𝐵𝐷 meets line segment 𝐴𝐸 at 𝐶, which is also the midpoint of line segment 𝐵𝐷. Find the length of line segment 𝐶𝐸.
The first thing we notice is that 𝐶 is the midpoint of line segment 𝐵𝐷. We know that the length of line segment 𝐵𝐶 equals 27. And that means that the length of 𝐷𝐶 will also be equal to 27. Because the midpoint 𝐶 divides line 𝐵𝐷 in half, each side equals 27. We’re interested in line segment 𝐶𝐸. But at this point, it doesn’t seem like there’s 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 𝐵𝐶𝐴 and angle 𝐷𝐶𝐸 are vertical angles. They’re the angles across from each other when two lines intersect. Therefore, the measure of angle 𝐵𝐶𝐴 is equal to the measure of angle 𝐷𝐶𝐸. 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, it’s a little bit difficult to see which parts correspond. So we can rearrange the triangles to better see the corresponding parts. Here’s triangle 𝐸𝐷𝐶. We know the length of side 𝐷𝐶 equals 27. And we’ve rotated triangle 𝐴𝐵𝐶 around so that their right angles are in the same position. We know that side length 𝐵𝐶 is 27. And here we have two congruent sides, two congruent corresponding sides. We also know the measure of angle 𝐷𝐶𝐸 equals the measure of 𝐵𝐶𝐴. And that means we have corresponding congruent angles.
Based on this, we can say that triangle 𝐸𝐷𝐶 is congruent to triangle 𝐴𝐵𝐶. And if these two triangles are congruent, all of their corresponding parts will be congruent. That means the length of 𝐸𝐷 will be equal to the length of 𝐴𝐵. And the hypotenuse 𝐸𝐶 will be equal to the length of the hypotenuse of the other triangle, line 𝐴𝐶. For triangle 𝐴𝐵𝐶, we also know the length of side length 𝐴𝐵 is equal to 36. We can find the length of 𝐴𝐶 using the Pythagorean theorem. And once we find the length of 𝐴𝐶, that will also be the length of 𝐸𝐶.
The Pythagorean theorem tells us 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑎 and 𝑏 are the two smaller sides of a right-angled triangle. 𝑐 will be the length of the hypotenuse. And for us, that means side length 𝐴𝐵 squared plus side length 𝐵𝐶 squared will be equal to the hypotenuse 𝐴𝐶 squared. And the hypotenuse 𝐴𝐶 is equal to the hypotenuse 𝐸𝐶.
Let’s plug in what we know. 𝐴𝐵 equals 36. 𝐵𝐶 equals 27. 36 squared plus 27 squared equals 𝐴𝐶 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 𝐴𝐵𝐶, the hypotenuse equals 45. And by corresponding parts, we can say that triangle 𝐸𝐷𝐶 has a hypotenuse of 45 as well.