Before I explain how to solve, let me give each side a name.
Hypotenuse (longest side) of big triangle = A
Longest leg of big triangle (5√6) = B
Shortest leg of big triangle = C
Hypotenuse of small triangle = C
Top leg of small triangle = D
lower leg of small triangle = E (x)
For this question we need to use the properties of special right triangles.
Quick review on what those are:
In a 30-60-90 triangle
- side opposite 30º is y units long
- side opposite 60º is y√3 units long
- side opposite 90º is 2y units long
In a 45-45-90 triangle
- sides opposite 45º are both y units long
- sides opposite 90º is y√2 units long
In the big triangle the side opposite 60º is 5√6 units long. We need to find out what the other lengths are. Remember, the side associated with 60 is y√3 so we can set up an equation to find y (which will help us find the other sides)
5√6 = y√3
5√6 / √3 = y
y = 5√6√3 / 3
y = 5√18 / 3
y = 5√2
Now we know y is 5√2 so we can find the side C of the big triangle (which is also the hypotenuse of the small triangle)
Side C is opposite 30º so it's equation is y.
We already know y = 5√2 so side C is 5√2.
Since side C on the big triangle is the hypotenuse of the small triangle we can say that the hypotenuse of the small triangle is 5√2.
Now we can use the properties of a 45-45-90 triangle to solve for x.
We already know what the hypotenuse is,(5√2), and the equation for the hypotenuse is y√2. This means the y in the small triangle is 5.
Remember the legs of a 45-45-90 triangle = y. The y of the small triangle is 5 so that side is just 5 units long. That side also happens to be x so
x = 5
