These are special triangles. The triangle with a hypotenuse of 11 is a 30-60-90 triangle, and the other triangle in the diagram is a 45-45-90 triangle. For such triangles, the following properties apply.
For 30-60-90 triangles:
If the short leg is x -
· the hypotenuse is 2x
· the long leg is x√(3)
For 45-45-90 triangles:
Their legs are congruent. If their legs are x -
· the hypotenuse is x√(2)
We can find x by determining the length of the legs of the 45-45-90 triangle and using the above property. Notice that one of the legs of the 45-45-90 triangle is also the long leg of the 30-60-90 triangle. By finding the length of the long leg of the 30-60-90 we can determine the length of the hypotenuse of the 45-45-90 triangle.
The hypotenuse measures 11. The long leg is √(3) times the length of the short leg. The short leg is half the hypotenuse, thus the short leg is 5.5. The long leg is 5.5√(3) or [tex] \frac{11\sqrt{3}}{2} [/tex]. Since this is the length of the legs of the 45-45-90 triangle, the hypotenuse (x) is [tex] \frac{11\sqrt{3}}{2} \cdot\sqrt{2} [/tex].
[tex] \frac{11\sqrt{3}}{2} \cdot\sqrt{2} [/tex]
Simplify.
[tex] \frac{11\sqrt{3}\sqrt{2}}{2} [/tex]
[tex] \frac{11\sqrt{6}}{2} [/tex]
Answer:
D. [tex] \frac{11\sqrt{6}}{2} [/tex]
