Question Video: Finding the Unknown Lengths in a Right Triangle Using the Pythagorean Theorem Mathematics • First Year of Preparatory School

Video Transcript

Given that 𝐴𝐵𝐶𝐷 is a trapezoid, determine the length of line segment 𝐷𝐶.

We’re told that the quadrilateral 𝐴𝐵𝐶𝐷 is a trapezoid, which means it has one pair of parallel sides. We can see these parallel sides marked on the figure. They’re the sides 𝐴𝐷 and 𝐵𝐶. We’re asked to determine the length of the line segment 𝐷𝐶, which is one of the nonparallel sides of the trapezoid, also known as its legs.

Now, it may appear at first that we don’t have enough information to be able to do this. However, if we sketch in the perpendicular from point 𝐷 to the side 𝐵𝐶, meeting this side at a point we’ll call 𝐸, then we create a right triangle in which 𝐷𝐶 is the hypotenuse. The length of this new line segment is the same as the length of line segment 𝐴𝐵, as both are perpendicular to the two parallel sides of the trapezoid. Hence, 𝐷𝐸 equals 12 centimeters.

We can also determine the length of line segment 𝐸𝐶. As 𝐵 is vertically below 𝐴 and 𝐸 is vertically below 𝐷, line segments 𝐴𝐷 and 𝐵𝐸 are of equal length. Hence, 𝐵𝐸 is also 12.1 centimeters. And the length of 𝐸𝐶 is the difference between the lengths of 𝐵𝐶 and 𝐵𝐸. That’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 𝑎 and 𝑏 and the length of the hypotenuse as 𝑐, then this can be expressed as 𝑎 squared plus 𝑏 squared equals 𝑐 squared.

In our triangle, 𝐷𝐶 is the hypotenuse as it is directly opposite the right angle. So we can form the equation 𝐷𝐶 squared equals nine squared plus 12 squared. To solve for 𝐷𝐶, we evaluate the squares and find their sum, giving 𝐷𝐶 squared equals 225. Finally, we take the square root of both sides of the equation, giving 𝐷𝐶 equals 15. Note that we’re only interested in the positive solution here as 𝐷𝐶 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, we’ve found that the length of line segment 𝐷𝐶 is 15 centimeters.