### Video Transcript

𝐴𝐵𝐶𝐷 is a trapezoid, where line
segment 𝐴𝐷 is parallel to line segment 𝐵𝐶, 𝐴𝐷 equals 22 centimeters, and 𝐵𝐶
equals 13 centimeters. If the area of triangle 𝐴𝐵𝐶 is
65 square centimeters, what is the area of the trapezoid?

We’ve been given various
information about this trapezoid and asked to calculate its area. The area of a trapezoid is equal to
half the sum of the lengths of its parallel sides, usually denoted by 𝑎 and 𝑏,
multiplied by the height, denoted by ℎ.

We’ve been given the lengths of the
parallel sides, the line segments 𝐴𝐷 and 𝐵𝐶, which are 22 and 13 centimeters,
respectively. But we haven’t been given the
trapezoid’s height. We’ll need to use the other
information in the question to calculate this before we can find the trapezoid’s
area.

The other information given is that
the area of triangle 𝐴𝐵𝐶 is 65 square centimeters. That’s this triangle here. Now, the area of a triangle is
found using the formula a half multiplied by its base multiplied by its
perpendicular height. The base of triangle 𝐴𝐵𝐶 is
common with a side of the trapezoid, side 𝐵𝐶, which is of length 13
centimeters. The height of triangle 𝐴𝐵𝐶 is
the same as the height of the trapezoid. And so we can form an equation. 65 is equal to a half multiplied by
13ℎ.

We can solve this equation to find
the height of the triangle and therefore the height of the trapezoid. Multiplying both sides of the
equation by two gives 130 equals 13ℎ. And then dividing both sides of the
equation by 13 gives ℎ equals 10.

We’ve now found that the height of
the trapezoid is 10 centimeters. And so we’re ready to calculate its
area. Substituting 13 and 22 for the
lengths of the parallel sides and 10 for the height into the formula for the area of
a trapezoid gives that the area of the trapezoid 𝐴𝐵𝐶𝐷 is equal to a half
multiplied by 13 plus 22 multiplied by 10. 13 plus 22 is 35, and a half
multiplied by 10 is five. So the calculation simplifies to
five multiplied by 35, which is 175.

By first using the area of triangle
𝐴𝐵𝐶 to calculate the trapezoid’s height, we’ve found that the area of the
trapezoid is 175 square centimeters.