Triangle missing side example

The triangle shown below has an area of 75 square units. Find the missing side, so pause the video and see if you can find the length of this missing side.

All right, now let's work through this together. They give us the area; they give us this side right over here, this 11. They give us this length, 10, which if we rotate this triangle, you could view it as an altitude.

In fact, let me do that. Let me rotate this triangle, because then I think it might jump out at you how we can tackle this. So let me copy and let me paste it. If I move it here, but I'm going to rotate it.

So if I rotate the... oh, that is our rotated triangle. Now it might be a little bit clearer what we're talking about. This length x that we want to figure out, this is our base, and they give us our height, and they give us our area. We know how base, height, and area relate for a triangle.

We know that area is equal to one-half times the base times the height. They tell us, they tell us that our area is 75 units squared. So this is 75 is equal to one-half. What is our base? Our base is the variable x, so let's just write that down: one-half times x.

And then what is our height? Well, our height is actually the 10. If x is the length of our base, then the height of our triangle is going to be 10. We actually don't even need to use this 11; they're putting that there just to distract you.

So this is going to be our height times 10. So 75 is equal to one-half times x times 10. Or let me just rewrite it this way. We could say 75 is equal to one-half times 10 is equal to 5 times x, is equal to 5.

Let me do the x in that same color: is equal to 5 times x. So what is x going to be? There's a couple of ways you could think about it. You could say 5 times what is equal to 75? And you might be able to figure that out. You might say, okay, let's see, 5 times 10 is 50, and then let's say I need another 25, so I'd put another 5 there.

So it's really 5 times 15, or you could do it a little bit more systematically. You can divide both sides by what you're multiplying by: x. So if you divide this side by 5, 5 times x divided by 5, well, you're just going to have an x left over.

But you can't... these two things were equal, so you can't just do it to one side; you have to do it to both sides. So you have to divide both sides by 5. And what's 75 divided by 5? Well, that is 15.

So you get x is equal to... x is equal to 15. And you can verify that if x is equal to 15, base times height times one-half, well, it's 15 times 10 times one-half, or 15 times five, which is going to be 75 square units.