Question Video: Finding the Length of the Hypotenuse in a Right-Angled Triangle Using Right-Angled Triangle Trigonometry Mathematics • Second Year of Preparatory School

Video Transcript

Given that 𝐸𝐷 equals 7.5 centimeters, find the lengths of line segment 𝐴𝐶 and line segment 𝐵𝐸.

We can start by adding the length of the line segment 𝐸𝐷, which is 7.5 centimeters, to the diagram. The two lengths of 𝐴𝐶 and 𝐵𝐸 that we need to calculate are marked on the diagram.

Now, it doesn’t appear as though there is enough information to help us work out these lengths. However, the two congruent line segments will be very helpful. We can note that the triangle 𝐴𝐶𝐷 must be a right triangle, since we have the angle measure of angle 𝐴𝐷𝐶 marked as 90 degrees. We can observe that the line segment 𝐷𝐸 is from one of the vertices of this triangle to the midpoint of the opposite side.

Remember, we can say that 𝐸 is the midpoint of the side 𝐴𝐶 because it bisects this line segment into two congruent lengths. Therefore, we can describe the line segment 𝐷𝐸 as a median of triangle 𝐴𝐶𝐷. And there is a useful theorem regarding a median such as this. It is that in a right triangle, the length of the median drawn from the vertex of the right angle equals half the length of the triangle’s hypotenuse. So, in this triangle, the median 𝐸𝐷 is half the length of 𝐴𝐶, which is the hypotenuse of the right triangle. Alternatively, by multiplying both sides by two, we could write this equation as two times 𝐸𝐷 equals 𝐴𝐶. We know that 𝐸𝐷 is 7.5 centimeters. So two times this gives us that 𝐴𝐶 is 15 centimeters. And that is the first required side length found.

Next, we need to find the length of the line segment 𝐵𝐸. We can consider the triangle 𝐴𝐵𝐶, which is also a right triangle. Then, we have the same scenario as before. And in this triangle, we can say that 𝐵𝐸 is a median, as it’s a line from a vertex to the midpoint of the opposite side. Importantly, it’s also the median drawn from the vertex of the right angle. So this median 𝐵𝐸 is half the length of the hypotenuse 𝐴𝐶. And we’ve already calculated that 𝐴𝐶 is 15 centimeters. So half of this is 7.5 centimeters.

We can therefore give the answers for the lengths of both line segments. 𝐴𝐶 is 15 centimeters, and 𝐵𝐸 is 7.5 centimeters.