Question

find the value of the missing sides. leave in rationalized and simplified form.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)

Explanation:

Step1: Recall 45 - 45 - 90 triangle ratio

In a 45 - 45 - 90 triangle, the ratio of the sides is $a:a:a\sqrt{2}$, where $a$ is the length of each of the legs and $a\sqrt{2}$ is the length of the hypotenuse.

Step2: Solve for each triangle

For triangle 1:

The legs are 17. Using the hypotenuse formula $h = a\sqrt{2}$, with $a = 17$, the hypotenuse $x=17\sqrt{2}$.

For triangle 2:

The hypotenuse $h = 2\sqrt{2}$. Since $h=a\sqrt{2}$, then $a\sqrt{2}=2\sqrt{2}$, so $a = 2$ (each leg is 2).

For triangle 3:

The leg $a = 4\sqrt{2}$. The hypotenuse $h=a\sqrt{2}=4\sqrt{2}\times\sqrt{2}=4\times2 = 8$.

For triangle 4:

The leg $a = 20$. The hypotenuse $h=a\sqrt{2}=20\sqrt{2}$.

For triangle 5:

The leg $a = 10\sqrt{2}$. The hypotenuse $h=a\sqrt{2}=10\sqrt{2}\times\sqrt{2}=20$.

For triangle 6:

The hypotenuse $h = 3\sqrt{6}$. Since $h=a\sqrt{2}$, then $a=\frac{3\sqrt{6}}{\sqrt{2}}=\frac{3\sqrt{6}\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{3\sqrt{12}}{2}=\frac{3\times2\sqrt{3}}{2}=3\sqrt{3}$ (each leg is $3\sqrt{3}$).

For triangle 7:

The hypotenuse $h = 6$. Since $h=a\sqrt{2}$, then $a=\frac{6}{\sqrt{2}}=\frac{6\sqrt{2}}{2}=3\sqrt{2}$ (each leg is $3\sqrt{2}$).

For triangle 8:

The leg $a = 1$. The hypotenuse $h=a\sqrt{2}=\sqrt{2}$.

For triangle 9:

The leg $a = 12$. The hypotenuse $h=a\sqrt{2}=12\sqrt{2}$.

For triangle 10:

The leg $a = 5\sqrt{8}=5\times2\sqrt{2}=10\sqrt{2}$. The hypotenuse $h=a\sqrt{2}=10\sqrt{2}\times\sqrt{2}=20$.

For triangle 11:

The leg $a = 6\sqrt{8}=6\times2\sqrt{2}=12\sqrt{2}$. The hypotenuse $h=a\sqrt{2}=12\sqrt{2}\times\sqrt{2}=24$.

For triangle 12:

The leg $a = 9\sqrt{2}$. The hypotenuse $h=a\sqrt{2}=9\sqrt{2}\times\sqrt{2}=18$.

Answer:

1. $x = 17\sqrt{2}$
2. Legs: 2
3. Hypotenuse: 8
4. Hypotenuse: $20\sqrt{2}$
5. Hypotenuse: 20
6. Legs: $3\sqrt{3}$
7. Legs: $3\sqrt{2}$
8. Hypotenuse: $\sqrt{2}$
9. Hypotenuse: $12\sqrt{2}$
10. Hypotenuse: 20
11. Hypotenuse: 24
12. Hypotenuse: 18