Question Video: Writing Algebraic Expressions to Describe the Areas of Composite Figures Involving Rectangles and triangles

Video Transcript

Find the area of the shaded region.

So essentially, we’ll want the area of the two pink triangles. And the area of a triangle is one-half times the base times the height. Now we know the height of the triangles. It’s six 𝑥. However, we know the basis completely together to be 15𝑥 plus five. So let’s maybe find a different way, because the 15𝑥 plus five we’re not sure how to split them up.

Instead, let’s think of this as an entire rectangle area and then taking away that white triangle, this one. And this will give us the area of the shaded region. The area of a rectangle is length times width, and the area of a triangle is one-half times the base times the height. So we know the length and the width of the rectangle. So we need to take 15𝑥 plus five times six 𝑥.

And then the triangle, if we will look at this upside down, it might be a little easier. But we will use this as the base, the 15𝑥 plus five, and the height of that triangle will be six 𝑥. So one-half base times height would be one-half times 15𝑥 plus five times six 𝑥.

Let’s go ahead and rewrite it down here so we have more room, because we need to distribute. 15𝑥 times six 𝑥 is 90𝑥 squared, and five times six 𝑥 would be 30𝑥. Bring down our subtraction sign. We can go ahead and take one-half times six, which is three. So we can think of this is as 15𝑥 plus five times three 𝑥. 15𝑥 times three 𝑥 is 45𝑥 squared.

Now it is important to put a bracket out front. That way, we recognise the minus sign — that’s pink — will need to be distributed to everything in the blue. And five times three 𝑥 would be 15𝑥. So now we need to distribute this negative sign. So we actually have minus 45𝑥 squared minus 15𝑥.

Now we can combine like terms. 90𝑥 squared minus 45𝑥 squared would be 45𝑥 squared. 30𝑥 minus 15𝑥 would be 15𝑥. Therefore, 45𝑥 squared plus 15𝑥 will be the area of the shaded region. It may also be written as 15𝑥 times three 𝑥 plus one, because we could take out a greatest common factor of 15𝑥 from both terms. Either answer will probably be fine.