Question

Analyze the diagram below and complete the instructions that follow. Find the unknown side lenght x. Write your answer in simplest radical form.

Answer 1

I'm going to work from left to right here, starting with the base of the largest triangle and ending with x. Pay close attention to the sides I am describing since there are no distinctive values for each vertex. The base of the largest triangle is also the hypotenuse of the one next to it. We will look at that missing side as the base of the large one to solve it. Since these are all right triangles, every single side/hypotenuse we solve for will use Pythagorean's theorem. The first missing side is found by $52^2-48^2=c^2$. Simplifying we get 2704 - 2304 = c^2. c^2 = 400 so c = 20. That first missing side is 20. Now we move to the next triangle in line. we have the hypotenuse of that triangle as 20 and the base as 12. The formula to find the missing side is $20^2-12^2=c^2$ and c^2=256. c = 16. 16 is the length of that whole side, but that whole side is actually the bases for those 2 small triangles. Before we can solve for x we have to find out what the base of the larger of the 2 triangles is and subtract that from 16 to find the base of the triangle with x in it. Got that? The hypotenuse is 13 and the height is 5, so the formula to find the base of that triangle is $13^2-5^2=c^2$ and $169-25=c^2$ so $144=c^2$ and c = 12. That means that the base of the triangle with x in it is 16 - 12 which is 4. Now we have the base as 4 of that triangle and the height as 5, so we solve for x: $4^2+5^2=x^2$ and $25+16=x^2$ so $x^2=41$ and finally we have that $x=\sqrt{41}$. That is the simplest radical form, so that's your answer!