in this video we're going to focus on the 45-45-90 triangle now it's important that you understand how to use this triangle so that it can help you solve other triangles or other similar triangles this technique especially useful if you're studying for the sat or the ht exam so let me give you the 45 45 90 reference triangle the first thing you need to know is that it's a right triangle and two sides of the triangle are the same so let's say if one side is one the other is going to be one and the hypotenuse is going to be this number times the square root of two or simply square root two now let's say if you have the side across the 45 degree angle and you want to find the side across the 90 degree angle which is the hypotenuse all you need to do is multiply by the square root of 2. if you have the side that's across the 90 degree angle and you want to calculate the length of the side across the 45 degree angle you need to divide by the square root of two so i'm going to put this into practice and show you how to apply this information so consider this right triangle and you're given the length of one of the legs of the right triangle find the mist in two sides well we know this is a 45 45 90 degree triangle so these two sides have to be the same so if the bottom leg is eight the other one is eight now to find the hypotenuse which is always the longer side we have to multiply this side by the square root of two so this is going to be eight square root two so based on that example feel free to pause the video and try this example go ahead and find the missing sides so the other side is 12 units long and the hypotenuse is going to be this number times the square root of 2. so it's 12 roots two now go ahead and try these two examples so let's say that this side is five square root two and here's another one and let's say this side is 7 square root 2. so we know the other missing side has to be the same and that's not going to change now let's find the hypotenuse of the first triangle so we got to multiply by square root 2 if we want to find the length of the hypotenuse so the square root of 2 times the square root of 2 is the square root of 4 and the square root of 4 is 2 and 5 times 2 is 10. so the hypotenuse is 10 units long and it's very easy to find the missing side if you understand this particular special reference triangle so let's do the same thing for this one so if we multiply 7 square root 2 by the square root 2 this is going to be 7 times 2 which ends up being 14. so let me give you some other examples let's change it up a little so this time you're given the hypotenuse go ahead and find the value of the missing sides of these two triangles now this time we have this side across the 90 degree angle which is basically the hypotenuse and we want to find the side across the 45 degree angle so we're looking for the value of the shortest side so we need to divide by the square root of 2. so if we take the hypotenuse 13 square root 2 and divided by the square root 2. these two will cancel and we're just going to get 13. so the length of these two sides is 13. now if we do the same thing for the example on the right if we take 6 square root 2 divided by that well that's going to give us 6. and so that's how we could easily find the value of the missing side so consider these two examples so let's say this time the hypotenuse is 10 and for this one it's let's say 15. go ahead and calculate the value of the missing sides so we know if we're given the hypotenuse to find the legs of the triangle we got to divide by the square root of 2. so 10 divided by the square root of 2 is something that we need to rationalize so let's multiply the top and the bottom by the square root of 2. so on top we're going to have 10 square root 2 and on the bottom the square root of 2 times the square root of 2 is 2. and so we could divide 10 by 2 which will give us 5. so this is going to be 5 square root 2 and 5 square root 2. now let's do the same thing for the other triangle so first let's take the hypotenuse and divide it by the square root of 2 and then let's rationalize the denominator so in the numerator we're going to have 15 times the square root of 2 and the denominator we're just going to have 2. now we can't really reduce 15 over 2. i mean you could write it as 7.5 or you could just leave it as 15 square root 2 over 2. and so now you know how to solve a 45-45-90 degree triangle but now let's consider a practice problem that you might see on the sct exam let's see how we can put this information to good use segment ac is 16 units long b is the center of the circle what is the area of triangle abc and the figure shown below now feel free to take a few minutes pause the video and see if you can find the answer to this problem now what kind of triangle do we have is it a 30-60-90 triangle or is it a 45-45-90 triangle since this whole video is about 45-45-90 triangles we know that's going to be the case but how can we be sure that it is a 45-45-90 triangle now whenever you have a 45-45-90 triangle it's important to know that the sides across the 45 angles will be equal what we have is an isosceles right triangle and the reason why we know it's equal is because the triangle is on the circle b is the center of the circle the distance between the center of a circle and any point on a circle is known as the radius so a b is the radius of the circle and bc is the radius of the circle which means that those two sides must be equal to each other and so that's how we know we have a 45-45-90 degree triangle so once we know that we could find the missing sides of the right triangle now we know the hypotenuse segment ac is 16 units long now to find the other sides we're going from the hypotenuse to side across the 90 to the side across the 45 so we got to divide by the square root of 2. so 16 divided by the square root of 2 we need to rationalize the denominator so let's multiply the top and the bottom by the square root of 2 so it's 16 root 2 and the square root of 2 times the square root of 2 is 2 and then we need to divide 16 by 2 to get 8 square root 2. so segment bc and a b is equal to 8 square root 2. so now that we have the base and the height of the right triangle we can now calculate the area of that triangle so the area is one half base times height so this is the base of the triangle this is the height so it's going to be one half we have a base of eight square root two and a height of eight square root two so first let's multiply eight by eight eight times eight is 64. next let's multiply the square root of 2 times the square root of 2 which we know to be 2. now one half times 2 is 1 so we could cancel those two so the area of triangle abc is 64 square units in value so this is the answer to the problem so that's basically it for this video hopefully you find it to be useful now you may want to check out my next video on 30 60 90 special right triangles so i'm going to have some problems like this one which can help you for the sat and the ht exam at least the math sections for those exams so just keep this in mind whenever you're dealing with a 45 45 90 triangle if you have this side across the 45 and you want to find the side across the 90 or the hypotenuse multiply by the square root of 2 and if you have the hypotenuse and you want to find the length of the legs of the triangle or basically the side length across the 45 degree angle divide by the square root of two if you remember this you can mentally find the answer in your head it can save you a lot of time especially when you're taking the sat exam because you got to get the answers quickly you can't waste time so thanks again for watching this video you