# Question Video: Finding the Unknown Lengths in a Right Triangle Using the Pythagorean Theorem Mathematics • 8th Grade

Given that π΄π΅πΆπ· is a trapezoid, determine the length of line segment π·πΆ.

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### Video Transcript

Given that π΄π΅πΆπ· is a trapezoid, determine the length of line segment π·πΆ.

Weβre told that the quadrilateral π΄π΅πΆπ· is a trapezoid, which means it has one pair of parallel sides. We can see these parallel sides marked on the figure. Theyβre the sides π΄π· and π΅πΆ. Weβre asked to determine the length of the line segment π·πΆ, which is one of the nonparallel sides of the trapezoid, also known as its legs.

Now, it may appear at first that we donβt have enough information to be able to do this. However, if we sketch in the perpendicular from point π· to the side π΅πΆ, meeting this side at a point weβll call πΈ, then we create a right triangle in which π·πΆ is the hypotenuse. The length of this new line segment is the same as the length of line segment π΄π΅, as both are perpendicular to the two parallel sides of the trapezoid. Hence, π·πΈ equals 12 centimeters.

We can also determine the length of line segment πΈπΆ. As π΅ is vertically below π΄ and πΈ is vertically below π·, line segments π΄π· and π΅πΈ are of equal length. Hence, π΅πΈ is also 12.1 centimeters. And the length of πΈπΆ is the difference between the lengths of π΅πΆ and π΅πΈ. Thatβs 21.1 minus 12.1, which is nine centimeters.

We now have a right triangle in which we know the lengths of two of the sides and wish to calculate the length of the third side. We can do this by applying the Pythagorean theorem. This states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the two shorter sides. If we denote the lengths of the two shorter sides as π and π and the length of the hypotenuse as π, then this can be expressed as π squared plus π squared equals π squared.

In our triangle, π·πΆ is the hypotenuse as it is directly opposite the right angle. So we can form the equation π·πΆ squared equals nine squared plus 12 squared. To solve for π·πΆ, we evaluate the squares and find their sum, giving π·πΆ squared equals 225. Finally, we take the square root of both sides of the equation, giving π·πΆ equals 15. Note that weβre only interested in the positive solution here as π·πΆ represents a length.

We may notice that as all three side lengths of this triangle are integer values, it is a Pythagorean triple. In fact, it is an enlargement of perhaps the most commonly known Pythagorean triple, the right triangle with side lengths of three, four, and five units. Including the length units, weβve found that the length of line segment π·πΆ is 15 centimeters.