Video: Finding the Height of a Triangle by Dividing Polynomials

The area of a triangle is (12𝑥² + 4𝑥) cm², and its base is 4𝑥 cm. Write an expression for its height.

02:09

Video Transcript

The area of a triangle is 12𝑥 squared plus four 𝑥 square centimeters, and its base is four 𝑥 centimeters. Write an expression for its height.

Let’s begin by recalling how we find the area of a triangle. For a triangle whose base is 𝑏 units and whose perpendicular height is ℎ units, its area is a half base times height. And the area will be given in square units. Now, it doesn’t matter that we’re working with algebraic expressions. We can still substitute these into this formula.

Let’s let the height be equal to ℎ or ℎ centimeters. We’re told the area is 12𝑥 squared plus four 𝑥 and its base is four 𝑥. So we can write 12𝑥 squared plus four 𝑥 equals a half times four 𝑥 times ℎ. Now, because we can find a half of four 𝑥 quite easily, we should. This will make the next step a little bit easier. But if we couldn’t, what we could do is multiply both sides by two.

We’re not going to do that though. We’re going to write the right-hand side as two 𝑥 times ℎ. And then since we’re looking to find the value of ℎ or certainly an expression for ℎ, we solve for ℎ by dividing through by two 𝑥. So ℎ is 12𝑥 squared plus four 𝑥 all over two 𝑥.

There are a number of ways we can simplify this fraction or divide the numerator by the denominator. One way is to use the bus stop method. So let’s see what that looks like. Since we’re dividing by a monomial, we don’t need to use long division. We’re simply going to divide each term in our dividend, that’s the quadratic here, by the divisor, that’s two 𝑥. So 12 divided by two is six, and 𝑥 squared divided by 𝑥 is 𝑥. So we see that 12𝑥 squared divided by two 𝑥 is six 𝑥. Then we divide four 𝑥 by two 𝑥. Well, four divided by two is two, and 𝑥 divided by 𝑥 is one. So when we divide 12𝑥 squared plus four 𝑥 by two 𝑥, we get six 𝑥 plus two. And so that’s our expression for ℎ.

It is, of course, worth noting that we’re working in centimeters and square centimeters. So the units for the height ℎ are in centimeters also. The height is six 𝑥 plus two centimeters.

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