Question

Solve for x. Show your work.

Answer 1

Answer:
`5√(6)`

Explanation:

Step 1: Draw a line to create 2 triangles (line AD) . One 45-45-90 triangle and one 30-60-90 triangle.

Step 2: find the length of AD by creating a proportion. You are told that the hypotenuse of the 45-45-90 triangle is 15. You should know that in a 45-45-90 triangle, the hypotenuse is
`2√(x)` long if the sides are x long. So if you solve for AD, you should get
`15=y√(2) \\`, divide
`√(2)` on both sides and simplify to get
`(15√(2) )/(2)`.

Step 3: Solve for x by creating a proportion. AD also happens to be the longer line of the 30-60-90 triangle. Knowing that, you can solve for x, which is the hypotenuse. In a 30-60-90 triangle, the hypotenuse the twice the length of the shorter side (line CD) and the longer side (line AD) is
`√(3)` time longer than the shorter side. Knowing that you can set up this proportion:

`((15√(2) )/(2) )/(x) =(n√(3) )/(2n) \\ cross-multiply\\xn√(3) =(2*15*n√(2))/(2) \\` simply the 2 on the right side and divide
`n√(3)` on both sides.
`x=(15√(2) )/(√(3) ) \\` Multiply numerator and denominator by
`√(3)` to simplify even more.
`(15√(2) )/(√(3) ) *(√(3) )/(√(3)) =(15√(6) )/(3) =5√(6)`