Given the triangle below, determine the missing side(s) without using the Pythagorean theorem (make sure your answer is fully simplified).

x = 5 3 , y = 10 x=5\sqrt3,y=10 x = 5 3 <path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"> , y = 10

Given the triangle below, determine the missing side(s) without using the Pythagorean theorem (make sure your answer is fully simplified).

So let's try solving this problem. So in this problem, we are told, given the triangle below, determine the missing size without using the Pythagorean theorem, and make sure your answer is fully simplified. Now since we're specifically told not to use the Pythagorean theorem, what that means is we probably have a special case triangle of some kind. But let's see if we can observe and figure out what this triangle is. So I noticed that we have a 30 degree angle right here. And whenever you have a 30 degree angle in a right triangle, then the other angle has to be 60 degrees. And the reason why is because all the angles in any triangle have to add to 180, and 30 plus 60 plus 90 is 180. So this means we do have the special 30, 60, 90 triangle. Now recall for the 30, 60, 90 triangles, then the hypotenuse of the triangle is going to be equal to the short leg multiplied by 2. And then the long leg of the triangle is going to be equal to the short leg multiplied by the square root of 3. So using these relationships, we can find the 2 missing sides. Now I'll start with the hypotenuse right there. And I can see that we have the short leg of the triangle which is 5, and the hypotenuse is y. So we have that y is equal to the short leg, which in this case is 5 multiplied by 2, and 5 times 2 is 10. So that means this is the hypotenuse of our right triangle. Now lastly we need to find the long leg, And to find the long leg, we can use this relationship. Because the long leg, which we see here is x, is going to be equal to the short leg, which we figured out is 5 multiplied by the square root of 3. And 5 times the square root of 3 is as simplified as it will get, so we can just write this as 5 times the radical 3. So we have 5 square root 3 as our solution for x and the long leg of this triangle. So that is how you can find all the missing sides of this 30, 60, 90 triangle. Hope you found this video helpful. Thanks for watching.