Question Video: Applying the Pythagorean Theorem to Solve Problems Mathematics • 8th Grade

Video Transcript

𝐴𝐵𝐶𝐷 is a trapezoid-shaped piece of land where the line segment 𝐴𝐷 is parallel to the line segment 𝐵𝐶 and the line segment 𝐴𝐵 is perpendicular to the line segment 𝐵𝐶. Find the length of the line segment 𝐴𝐵 given 𝐴𝐷 is equal to 24 meters, 𝐵𝐶 is equal to 48 meters, and 𝐷𝐶 is equal to 30 meters. Give the answer to the nearest meter.

We are told in the question that the line segments 𝐴𝐷 and 𝐵𝐶 are parallel. We want to calculate the length of 𝐴𝐵, which is perpendicular to both 𝐴𝐷 and 𝐵𝐶. We will let this length be equal to 𝑥 meters. Next, we can draw a line from the point 𝐷 to the line segment 𝐵𝐶. We will let the point of intersection of these two lines be point 𝐸. We draw this line 𝐷𝐸 such that it is perpendicular to the line 𝐵𝐶. This creates a right triangle 𝐷𝐸𝐶, where the length of 𝐷𝐸 will be equal to the length of 𝐴𝐵.

The hypotenuse of our right triangle is equal to 30 meters, and we see that one of the shorter sides 𝐷𝐸 is equal to 𝑥 meters. We can calculate the length of 𝐸𝐶 by subtracting 24 meters from 48 meters. This gives us an answer of 24 meters. In any right triangle, we can calculate the length of a missing side using the Pythagorean theorem. This states that 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side or hypotenuse.

Substituting in our values, we have 𝑥 squared plus 24 squared is equal to 30 squared. 24 squared is equal to 576 and 30 squared is equal to 900. We can then subtract 576 from both sides of this equation, giving us 𝑥 squared is equal to 324. We can then square root both sides of this equation. As 18 squared is equal to 324 and 𝑥 must be positive, we know that 𝑥 is equal to 18.

The length of 𝐴𝐵 must therefore be equal to 18 meters. In this question, we do not need to do any rounding to give our answer to the nearest meter. The length of 𝐴𝐵 is exactly 18 meters.