Video Transcript
Given that the area of the parallelogram 𝐴𝐵𝐶𝐷 is equal to 268 square centimeters, find the area of triangle 𝑋𝐵𝐶.
In this example, we’re asked to find the area of the triangle 𝑋𝐵𝐶, which is inscribed in the parallelogram 𝐴𝐵𝐶𝐷. And we’re told that the area of parallelogram 𝐴𝐵𝐶𝐷 is 268 square centimeters.
The first thing we can note is that the parallelogram and triangle 𝑋𝐵𝐶 share a common base, 𝐵𝐶. So their bases are congruent. Also, since they lie between the same parallel lines, they must have the same perpendicular height, which we’ll call ℎ. We know that the area of a parallelogram with base 𝑏 and perpendicular height ℎ is given by 𝑏 multiplied by ℎ. So, applying this to our parallelogram 𝐴𝐵𝐶𝐷, we have that the area of 𝐴𝐵𝐶𝐷 is equal to the side length 𝐵𝐶 multiplied by the height ℎ. And we know that this is equal to 268 square centimeters.
Now, we also know that the area of any given triangle is one-half of its base times its perpendicular height, which in the case of triangle 𝑋𝐵𝐶 is one-half of 𝐵𝐶 multiplied by the perpendicular height ℎ. And we see in fact that this is one-half of the area of parallelogram 𝐴𝐵𝐶𝐷. So the area of triangle 𝑋𝐵𝐶 equals one-half of 268, and that’s 134 square centimeters.
We note finally that this is an illustration of the general property that if a triangle and a parallelogram share the same base and lie between the same pair of parallel lines, then the area of the parallelogram is twice the area of the triangle.
