Question Video: Finding the Length of the Hypotenuse in a Right-Angled Triangle Using Right-Angled Triangle Trigonometry | Nagwa Question Video: Finding the Length of the Hypotenuse in a Right-Angled Triangle Using Right-Angled Triangle Trigonometry | Nagwa

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

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Given that πΈπ· = 7.5 cm, find the lengths of line segment π΄πΆ and line segment π΅πΈ.

03:04

### Video Transcript

Given that πΈπ· equals 7.5 centimeters, find the lengths of line segment π΄πΆ and line segment π΅πΈ.

We can start by adding the length of the line segment πΈπ·, which is 7.5 centimeters, to the diagram. The two lengths of π΄πΆ and π΅πΈ that we need to calculate are marked on the diagram.

Now, it doesnβ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 π΄πΆπ· must be a right triangle, since we have the angle measure of angle π΄π·πΆ marked as 90 degrees. We can observe that the line segment π·πΈ is from one of the vertices of this triangle to the midpoint of the opposite side.

Remember, we can say that πΈ is the midpoint of the side π΄πΆ because it bisects this line segment into two congruent lengths. Therefore, we can describe the line segment π·πΈ as a median of triangle π΄πΆπ·. 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 triangleβs hypotenuse. So, in this triangle, the median πΈπ· is half the length of π΄πΆ, which is the hypotenuse of the right triangle. Alternatively, by multiplying both sides by two, we could write this equation as two times πΈπ· equals π΄πΆ. We know that πΈπ· is 7.5 centimeters. So two times this gives us that π΄πΆ is 15 centimeters. And that is the first required side length found.

Next, we need to find the length of the line segment π΅πΈ. We can consider the triangle π΄π΅πΆ, which is also a right triangle. Then, we have the same scenario as before. And in this triangle, we can say that π΅πΈ is a median, as itβs a line from a vertex to the midpoint of the opposite side. Importantly, itβs also the median drawn from the vertex of the right angle. So this median π΅πΈ is half the length of the hypotenuse π΄πΆ. And weβve already calculated that π΄πΆ is 15 centimeters. So half of this is 7.5 centimeters.

We can therefore give the answers for the lengths of both line segments. π΄πΆ is 15 centimeters, and π΅πΈ is 7.5 centimeters.

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