Video Transcript
Find the area of the trapezoid 𝐴𝐵𝐶𝐷.
We can remember that a trapezoid is a quadrilateral with one pair of parallel sides. These parallel sides would be 𝐴𝐵 and 𝐶𝐷. We’re given that the length of 𝐶𝐷 is 74 centimeters. And we’re also given the lengths of the two nonparallel sides. 𝐶𝐵 is 50 centimeters and 𝐴𝐷 is 14 centimeters. We could find the area of this trapezoid either by considering it as an area of a triangle and an area of a rectangle and adding those together or by using the formula for the area of a trapezoid. The area of a trapezoid is equal to half ℎ times 𝑏 sub one plus 𝑏 sub two, where ℎ is the perpendicular height and 𝑏 sub one and 𝑏 sub two are the two parallel bases.
Even if we chose to use this formula or to find the area of the triangle and the rectangle separately, there’s still one important dimension that’s missing, the length of 𝐴𝐵, which we can define as 𝑥 centimeters. The length of 𝐸𝐷 will also be the same length of 𝑥 centimeters. And we know this because 𝐴𝐵𝐸𝐷 is a quadrilateral. And since it has two right angles, then it means that the angle at 𝐴 and the angle at 𝐵 will also be right angles. 𝐴𝐵𝐸𝐷 is therefore a rectangle, and the opposite sides are congruent. As we’re given that the whole length of 𝐶𝐷 is 74 centimeters, then the length of 𝐶𝐸, which we could call 𝑦, would be 𝑦 centimeters. And we know that 𝑥 plus 𝑦 must be equal to 74.
At this point, we can’t directly work out the length of 𝑥, but we can work out the length of 𝑦 by considering that we must have a right triangle at 𝐵𝐸𝐶. Since we have the angles on a straight line adding up to 180 degrees, then we know that the angle 𝐵𝐸𝐶 is equal to 90 degrees since it’s 180 degrees subtract this angle of 90 degrees at angle 𝐵𝐸𝐷. Looking at this right triangle 𝐵𝐸𝐶, we have a known hypotenuse of 50 centimeters, an unknown length of 𝑦 centimeters, and we know the length of 𝐵𝐸 will also be 14 centimeters as it’s congruent with the length of 𝐴𝐷.
As we have a right triangle, we can apply the Pythagorean theorem, which tells us that the square on the hypotenuse is equal to the sum of the squares on the other two sides. So we’ll have 50 squared is equal to 14 squared plus 𝑦 squared. Evaluating the squares, we have 2500 equals 196 plus 𝑦 squared. Subtracting 196 from both sides of this equation, we have 2304 equals 𝑦 squared. Taking the square root then of both sides, we have that 𝑦 is equal to 48 centimeters. So now we have found that the length of 𝐶𝐸 is 48 centimeters. Remember that this means we can calculate the length of 𝑥 centimeters as both 48 and 𝑥 must add to give 74 centimeters.
Subtracting 48 from 74, we find that 𝑥 is equal to 26 centimeters. Therefore, 𝐸𝐷 is 26 centimeters, and importantly 𝐴𝐵 is also 26 centimeters. So now we can apply the formula to find the area of this trapezoid. Remember that to find the area, we’re multiplying a half by the perpendicular height, which is 14 centimeters, and multiplying it by the sum of the two parallel bases. A half multiplied by 14 is seven, and 26 plus 74 is equal to 100. Seven multiplied by 100 of course gives us 700, and the units here will be square centimeters. And so the area of 𝐴𝐵𝐶𝐷 is 700 square centimeters.
Of course, we could’ve found this area by finding the area of the triangle 𝐵𝐸𝐶 and the rectangle 𝐴𝐵𝐸𝐷 and adding those values together. To find the area of the triangle, we’d apply the formula that the area of a triangle is half times the base times the perpendicular height. So 𝐵𝐸𝐶 has an area of 336 square centimeters. As 𝐴𝐵𝐸𝐷 is a rectangle, we multiply the length by the width to find its area. So 26 times 14 would give us 364 square centimeters. Adding together 336 square centimeters and 364 square centimeters would also give us an area of 700 square centimeters and so confirming our original answer.
