 So in this video, we're going to do a little bit of trigonometry, right triangle trigonometry, but with an Asosceles triangle. It turns out that Asosceles triangles are deeply connected to right triangles as we're going to see in just a second. We'll recall what is an Asosceles triangle. An Asosceles triangle is a triangle with two sides that are congruent to each other. So for this diagram, let's say that if we have this triangle ABC, which is Asosceles, let's say that side length AC is congruent to side length BC. And let's say that both of these congruent sides are congruent or they're equal to 24 centimeters in length. Now with an Asosceles triangle, if we have two congruent sides, then it's also true that we'll have two congruent angles. That is the angles opposite of the two congruent sides and an Asosceles triangle will also be congruent to each other. And let's say that their shared angle measurement is 52 degrees. So angle A and the measure of angle B are both 52 degrees. Okay, so with that in mind, we want to now compute the length of the base and the altitude of this Asosceles triangle. So what do these words even mean for us for a moment? What does altitude for example mean? So an altitude with respect to a triangle is the unique line that connects a vertex to the opposite side of the triangle so that those two lines form a perpendicular angle of a right angle. That is, these two lines are perpendicular. So for example, the altitude of A would look something like this. This is the line that passes through the vertex A and is perpendicular to the opposite side. So we get something like a right angle right here. The altitude of angle B would look something similar. You get something like this. It's the line segment that connects vertex B to the opposite side of B and forms a right angle. That is, these two lines are perpendicular. Now when it comes to an Asosceles triangle, two of the angles are congruent to each other. So you have this third angle. The altitude with respect to that third angle is critical for an Asosceles triangle and you can see it illustrated here on the screen. The reason why this one's so interesting is for the following reasons. You'll notice that with this triangle segment AC is congruent to segment BC because it's an Asosceles triangle. Angle A is congruent to angle B because it's an Asosceles triangle. And likewise, we have that this line segment. Let's say that the point of intersection between the altitude of C with respect to the line segment AB. Let's call that point D. This is known as the foot of the perpendicular line. You'll notice that the line segment CD of course is congruent to itself. I don't know what it is, but it's congruent to itself. And so if you look at the two triangles formed by cutting the Asosceles triangles in half by this altitude, you see one triangle over here ADC. You also see another one over here. You have BCD like so. You see these two triangles. Now these triangles are going to be congruent to each other. You'll notice that angle A and angle B are congruent. Angle C, excuse me, side B and side A are congruent to each other. Of course, the segment CD is congruent to itself. These two angles right here are right angles so they're congruent to each other. And it turns out that this is the bisector of that angle. It cuts it into equal pieces. And the foot is also going to form the midpoint of the line segment AB. So these are congruent to each other as well. So we have these congruent triangles. An Asosceles triangle can be cut into two pieces and those two triangles themselves are right triangles. That's what I meant by Asosceles triangles are very much related to right triangles. Basically, an Asosceles triangle is just a double right triangle. If you take a right triangle like this one over here and you just reflect it over one of its sides, you just form an Asosceles triangle and you have that connection there. So what we need to do is we need to find the base and altitude of this triangle. Why is this significant? Why might we want to care about that? Well, recall that the area of a triangle is one half base times height, which the base is the length of that one of the sides of the triangle. And then the height is then the length of the associated altitude of that side. So if our base here is the line segment AB, then the height of the triangle is the length of its associated altitude. So if we know the base and the altitude, we can compute the area of the triangle. In general, this can be challenging, but finding the area of an right triangle is pretty easy. And then for an Asosceles triangle, well, since it's just two right triangles, we can find the area of an Asosceles triangle fairly easy. So how are we going to do this? So the altitude is the height of the triangle. The base of the triangle is going to be B here. What I'm going to do is introduce a new symbol. We're going to call it X. Little X is the distance between A and D, which A and D, this right here is going to be half of the base. So the base is going to be 2 times X, whatever X turns out to be. Now to find X, what we can do is we can do a right triangle relationship on the triangle ADC. You'll notice that X is the adjacent side with respect to angle A, and we know the hypotenuse 24. And so that tells us that if we take cosine of 24, excuse me, cosine of 52 degrees, this is equal to X over 24. Clearing the denominators, we get that X equals 24 times cosine of 52 degrees. Well, since the base is just 2 times the length of X, that means the base is going to equal 2 times 24, which is 48 times cosine of 52 degrees. Consulting our calculator, we can then compute cosine of 52 degrees in our calculator times that by 48. And we'll get that the base of the triangle is approximately 30 centimeters. You'll notice that there was no instructions on how to round or any significant digits or anything like that. So what I'm going to do is I'm going to match the original style. Since the measurements of the side length of the triangle are given to the nearest centimeter, I'm going to round my answers to the nearest centimeters well. So the base is 30 centimeters. What about the height? We're looking at this right triangle again with respect to angle A, the height is just the opposite side, and we still know the hypotenuse is 24. So if we use the sine ratio this time, sine of 52 degrees, this will equal H over 24. Clear the denominators, we end up with H is equal to 24 times sine of 52 degrees. This is the exact answer. If we use our calculator to approximate the sine of 52 degrees and then times that by 24, make sure you're in degree mode when you do this and round to the nearest centimeter, we'll end up with 19 centimeters as the length of the altitude, the so-called height of the triangle. And so if we have the height and we have the base, then we could find the area of the triangle. We would just take 30 times 19 and divide that by 2. And so we see that the area of the triangle is approximately one half base times height, like I said, is one half 30, which half of 30 is 15 times that by 19, you would end up with 285 square units of area.