The area of a triangle is 12𝑥 squared plus 38𝑥 plus 28 centimeters squared, and its base is two 𝑥 plus four centimeters. Find an expression for its height.
We now know that we can find the area of a triangle by multiplying the height times the base and then dividing by two, but we wanna rewrite this expression so that it says ℎ equals something. How do we find the height if we know the area and the base? To find that, we’ll need to isolate the ℎ to get it by itself. Our first step would be to multiply by two on both sides.
Two divided by two is one; it cancels out. On the right side, this leaves us with height times the base equals two times the area. From there we can divide both sides of our equation by the base. 𝑏 divided by 𝑏 equals one, so two times the area divided by the base equals the height. Let’s take this formula and plug in what we know.
We know that the area equals 12𝑥 squared plus 38𝑥 plus 28 centimeters squared, and our base equals two 𝑥 plus four centimeters. We notice that we have centimeters squared in the numerator and centimeters in the denominator. One of the sets of centimeters cancel out, and that reminds us that our height will be given in centimeters not in centimeters squared. We won’t change anything about our numerator.
But for our denominator, I see a common factor of two. If I take out the factor of two, we’ll reduce the expression to two times 𝑥 plus two. And then we notice that we have a two in our numerator and in our denominator. Two divided by two equals one; these cancel each other out. But how would we divide 12𝑥 squared plus 38𝑥 plus 28 by 𝑥 plus two? We need to use something called polynomial long division.
From here we ask the question what can we multiply 𝑥 by two to equal 12𝑥 squared. If I multiply 12𝑥 by 𝑥, we’ll get 𝑥 squared. But we have to multiply our 12𝑥 by that second term as well, so we’ll have to multiply 12𝑥 by two. From here, we’ll treat the problem like any other long division problem, and we’ll subtract. 12𝑥 squared minus 12𝑥 squared equals zero; 38𝑥 minus 24𝑥 equals 14𝑥.
We bring down our next term. Now we ask that same question again: what can we multiply 𝑥 by to equal 14𝑥? It’s 14; 14 times 𝑥. We can’t forget to multiply by that second term; 14 times two equals 28. We do some more subtraction here; 14𝑥 minus 14𝑥 equals zero; 28 minus 28 equals zero. Now we have nothing remaining. 12𝑥 plus 14 is what we get when we divide two times the area by the base, and we can say that the height of this triangle equals 12𝑥 plus 14 centimeters.