Pythagorean theorem with right triangle

We're asked to find the value of x in the isosceles triangle shown below. So that is the base of this triangle. So pause this video and see if you can figure that out.

Well, the key realization to solve this is to realize that this altitude that they dropped is going to form a right angle here and a right angle here. Notice both of these triangles, because this whole thing is an isosceles triangle, we are going to have two angles that are the same. This angle is the same as that angle.

Because it's an isosceles triangle, this 90° is the same as that 90°, and so the third angle needs to be the same. So that is going to be the same as that right over there. Since you have two angles that are the same, and you have a side between them that is the same, this side, this altitude of length 12, is on both triangles.

We know that both of these triangles are congruent. So they're both going to have 13. They're going to have one side that's 13, one side that is 12, and so this and this side are going to be the same. So this is going to be X over 2, and this is going to be X over 2.

Now we can use that information and the fact, and the Pythagorean theorem to solve for x. Let's use the Pythagorean theorem on this right triangle on the right-hand side. We can say that (X over 2)², that's the base right over here, this side right over here, we could write that (X over 2)² plus the other side, 12², is going to be equal to our hypotenuse squared, which is going to be equal to 13².

This is just the Pythagorean theorem now. So we can simplify this. This is going to be X² over 4, that's just (X² over 2)² plus 144 is equal to 169. Now I can subtract 144 from both sides. I'm going to try to solve for x; that's the whole goal here.

So subtracting 144 from both sides, and what do we get on the left-hand side? We have X² over 4 is equal to 169 - 144. Let's see, 169 - 144 is 25. So this is going to be equal to 25. We can multiply both sides by 4 to isolate the X². So we get X² is equal to 25 * 4, which is equal to 100.

Now, if we were just looking at this purely mathematically, you'd say, "Oh, X could be positive or negative 10." But since we're dealing with distances, we know that we want the positive value of it. So X is equal to the principal root of 100, which is equal to 10.

So there you have it; we have solved for X. This distance right here, the whole thing, the whole thing is going to be equal to 10. Half of that is going to be 5. So if we just looked at this length right over here—I'm doing that in the same color—let me see. So this length right over here, that's going to be 5.

Indeed, 5² + 12²—that's 25 + 144—is 169, which is 13². So the key realization here is isosceles triangle; the altitude splits it into two congruent right triangles. It also splits this base into two. So this is X over 2, and this is X over 2, and we use that information and the Pythagorean theorem to solve for x.