Video Transcript
In the trapezoid 𝐴𝐵𝐶𝐷 below, line segment 𝐴𝐷 is parallel to line segment 𝐵𝐶
and line segment 𝐴𝐸 is perpendicular to line segment 𝐵𝐶. Find its area.
Let’s begin by adding the information written in the question to the diagram. Line segment 𝐴𝐷 is parallel to line segment 𝐵𝐶 and line segment 𝐴𝐸 is
perpendicular to line segment 𝐵𝐶. This also means that line segments 𝐴𝐸 and 𝐷𝐶 are parallel, as both are
perpendicular to the line segment 𝐵𝐶. We’re asked to find the area of this trapezoid, so we can recall that this is
calculated by finding half the sum of the lengths of the parallel sides and
multiplying this by the trapezoid’s height. If we denote the lengths of the parallel sides as 𝑎 and 𝑏 and the height as ℎ, then
this is written as a half 𝑎 plus 𝑏 multiplied by ℎ.
We’re given the lengths of the parallel sides of this trapezoid. They’re six and 15 centimeters, but we don’t know its height. This is the length of the line segment 𝐴𝐸 on the figure. We can observe though that this line segment is one side in the right triangle
𝐴𝐵𝐸, in which we know the length of one other side. 𝐴𝐵 is 15 centimeters. In fact, we know the lengths of both of the other sides, because if 𝐴𝐸 and 𝐷𝐶 are
parallel, 𝐴𝐷 and 𝐸𝐶 must be of equal length. So 𝐸𝐶 is also six centimeters and 𝐸𝐵 is nine centimeters as this is the amount
left over from the full 15-centimeter length of 𝐵𝐶.
We then recall that if we know the lengths of two sides in a right triangle, we can
always calculate the length of the third side by applying the Pythagorean
theorem. This states that in a 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, this can be expressed as 𝑎 squared plus 𝑏 squared equals 𝑐
squared. Applying the Pythagorean theorem in triangle 𝐴𝐵𝐸, in which side 𝐴𝐵 is the
hypotenuse, gives ℎ squared plus nine squared equals 15 squared.
To solve this equation for ℎ, we first evaluate the squares and then subtract 81 from
each side of the equation to give ℎ squared equals 144. From here, we find the value of ℎ by square rooting, taking only the positive value
as ℎ represents a length. So we’ve found that ℎ is equal to 12.
Incidentally, we may recognize this triangle as a Pythagorean triple, that is, a
right triangle in which all three side lengths are integers, or indeed an
enlargement of the most commonly known Pythagorean triple with side lengths of
three, four, and five units.
We can now substitute the height of the trapezoid into the formula to calculate its
area, giving that the area of 𝐴𝐵𝐶𝐷 is equal to a half multiplied by six plus 15
multiplied by 12. Six plus 15 is 21, and a half of 12 is six, so we have 21 multiplied by six, which is
126. Including the units, we’ve found that the area of trapezoid 𝐴𝐵𝐶𝐷 is 126 square
centimeters.
