Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
The formula for calculating the area of triangles comes from dividing a parallelogram in half, so the area is half of base times height. When finding the area of a triangle, the height is an altitude and the base must be the side intersected by the altitude. When given the area and asked for a base or height, a common mistake is to forget to multiply both sides of the equation by 2 before dividing.
In this problem, we're given one triangle and we have two heights that are drawn in. So that's a key part to this. Is that bc can be considered a height to base ac ad cd could be considered a height to base ba.
Well, let's write in what we know. We're told that ac is equal to 8 centimeters. So let's start by writing that in. ac is 8 centimeters. We also know that bc is 6 centimeters. So here's bc. I'm going to label that as 6 centimeters. We know that ab is 12 centimeters. I'm going to label this whole distance here as 12 centimeters. So how in the world are we going to find what cd is?
Well, what we could do is we could say this is the same triangle. So if I calculate the area using two of my dimensions, then I can use my other dimension to calculate my missing height.
So let's start off by saying, what's the area of a triangle? Well, it's equal to my base times the corresponding height divided by 2. So if I pick what I know, I know that one base is equal to 8 centimeters and the corresponding height is 6 centimeters, so h equals 6 centimeters. So I can plug that in here, so you an say that we have 8 times 6 divided by 2. So we see that 8 times 6 divided by 2 is going to give us 24 square centimeters.
So that's going to be the area of our triangle no matter how we describe it. Which means, to find our missing side dc, we can say that the area which is 24 square centimeters is equal to the base which we know which is 12, times our missing side dc or our missing height dc all divided by 2. So now we have one variable, one equation. 12 divided by 2 is 6, so you're going to say 24 square centimeters is equal to 6 times dc and I guess we have to remember here that this is going to be in centimeters. So now what I'm going to do is I'm going to divide by 6 centimeters, and 24 square centimeters divided by 6 centimetres is 4 centimeters and that's going to be equal to dc.
So remember, dc is a distance so we need to have to have centimeters just centimeters to the first power. If you had centimeters squared, you probably did something wrong.
Key thing here was remembering that we have 2 heights, to 2 bases and we know 3 of those 4 sides.