Question

special right triangles practice
find the missing values.
1.
2.
3.
4.
5.
6.

Explanation:

Step1: Analyze the first triangle (45 - 45 - 90)

In a 45 - 45 - 90 triangle, the legs are equal. So \(y = 10\) m. The hypotenuse \(x=\text{leg}\times\sqrt{2}\), so \(x = 10\sqrt{2}\) m.

Step2: Analyze the second triangle (45 - 45 - 90)

In a 45 - 45 - 90 triangle, if the hypotenuse \(h = 12\sqrt{2}\) cm, then the legs \(x=y=\frac{h}{\sqrt{2}}=\frac{12\sqrt{2}}{\sqrt{2}} = 12\) cm.

Step3: Analyze the third triangle (30 - 60 - 90)

In a 30 - 60 - 90 triangle, the side opposite 30° is the shorter leg. Here, the shorter leg (opposite 30°) is \(7\) ft. The longer leg \(x=\text{shorter leg}\times\sqrt{3}=7\sqrt{3}\) ft. The hypotenuse \(y = 2\times\text{shorter leg}=14\) ft.

Step4: Analyze the fourth triangle (30 - 60 - 90)

In a 30 - 60 - 90 triangle, if the hypotenuse \(h = 18\) in, then the shorter leg \(y=\frac{h}{2}=9\) in. The longer leg \(x=\frac{h\sqrt{3}}{2}=9\sqrt{3}\) in.

Step5: Analyze the fifth triangle (45 - 45 - 90)

In a 45 - 45 - 90 triangle, if the hypotenuse \(h = 12\sqrt{2}\) ft, then the legs \(x=y=\frac{h}{\sqrt{2}}=\frac{12\sqrt{2}}{\sqrt{2}} = 12\) ft.

Step6: Analyze the sixth triangle (30 - 60 - 90)

In a 30 - 60 - 90 triangle, if the longer leg is \(8\) m. The shorter leg \(y=\frac{8}{\sqrt{3}}=\frac{8\sqrt{3}}{3}\) m. The hypotenuse \(x=\frac{16\sqrt{3}}{3}\) m.

Answer:

1. \(x = 10\sqrt{2}\) m, \(y = 10\) m
2. \(x = 12\) cm, \(y = 12\) cm
3. \(x = 7\sqrt{3}\) ft, \(y = 14\) ft
4. \(x = 9\sqrt{3}\) in, \(y = 9\) in
5. \(x = 12\) ft, \(y = 12\) ft
6. \(x=\frac{16\sqrt{3}}{3}\) m, \(y=\frac{8\sqrt{3}}{3}\) m