Question

special right triangles date______
find the missing side lengths. leave your answers as radicals in simplest form.
1)
a) ( a = 2 ), ( b = 1 )
b) ( a = \frac{2sqrt{3}}{3} ), ( b = 1 )
c) ( a = \frac{2sqrt{3}}{3} ), ( b = 2 )
d) ( a = 1 ), ( b = \frac{2sqrt{3}}{3} )
2)
a) ( x = 4sqrt{2} ), ( y = \frac{4sqrt{3}}{3} )
b) ( x = 4sqrt{2} ), ( y = 4 )
c) ( x = 8 ), ( y = 4 )
d) ( x = 4 ), ( y = 8 )
3)
a) ( u = 2 ), ( v = sqrt{6} )
b) ( u = \frac{2sqrt{3}}{3} ), ( v = sqrt{2} )
c) ( u = \frac{2sqrt{3}}{3} ), ( v = sqrt{6} )
d) ( u = 2 ), ( v = sqrt{2} )
4)
a) ( a = 2 ), ( b = 2 )
b) ( a = 1 ), ( b = 1 )
c) ( a = sqrt{2} ), ( b = sqrt{2} )
d) ( a = \frac{4sqrt{3}}{3} ), ( b = \frac{4sqrt{3}}{3} )
5)
a) ( a = 7sqrt{2} ), ( b = \frac{7sqrt{6}}{4} )
b) ( a = 7 ), ( b = \frac{7sqrt{6}}{4} )
c) ( a = 7sqrt{2} ), ( b = \frac{7sqrt{2}}{2} )
d) ( a = 7 ), ( b = \frac{7sqrt{2}}{2} )
6)
a) ( x = 12 ), ( y = 6sqrt{3} )
b) ( x = 12 ), ( y = 12 )
c) ( x = 6sqrt{3} ), ( y = 24 )
d) ( x = 24 ), ( y = 6sqrt{3} )

Explanation:

Step1: Analyze the first triangle (30 - 60 - 90 triangle)

In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\). Let the side opposite 30° be \(x\), opposite 60° be \(x\sqrt{3}\), and hypotenuse be \(2x\). Given the side opposite 60° is \(\frac{\sqrt{3}}{3}\). So, \(x\sqrt{3}=\frac{\sqrt{3}}{3}\), then \(x = \frac{1}{3}\). Hypotenuse \(a = 2x=\frac{2}{3}\times\sqrt{3}\), and the side opposite 30° \(b=\frac{1}{3}\times2=\frac{2}{3}\times\sqrt{3}\div\sqrt{3} = \frac{1}{3}\times2 = 1\) (after simplification).

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

In a 45 - 45 - 90 triangle, the sides are in the ratio \(1:1:\sqrt{2}\). Given one leg is 4. So the other leg \(y = 4\), and hypotenuse \(x=4\sqrt{2}\).

Step3: Analyze the third triangle (45 - 45 - 90 triangle)

In a 45 - 45 - 90 triangle, with leg \(\sqrt{2}\). Hypotenuse \(u=\sqrt{2}\times\sqrt{2}=2\), and the other leg \(v=\sqrt{2}\).

Step4: Analyze the fourth triangle (45 - 45 - 90 triangle)

In a 45 - 45 - 90 triangle, hypotenuse \(2\sqrt{2}\). Let the legs be \(a\) and \(b\). Using \(c = a\sqrt{2}\) (\(c\) is hypotenuse), \(a=b = 2\).

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

In a 45 - 45 - 90 triangle, leg \(\frac{7\sqrt{2}}{2}\). Hypotenuse \(a=\frac{7\sqrt{2}}{2}\times\sqrt{2}=7\), and the other leg \(b=\frac{7\sqrt{2}}{2}\).

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

In a 30 - 60 - 90 triangle, the side opposite 60° is 6. Let the side opposite 30° be \(x\), then \(x\sqrt{3}=6\), \(x = 2\sqrt{3}\) (not in options, re - check: if the side opposite 30° is 6, then hypotenuse \(y = 12\), and side opposite 60° \(x = 6\sqrt{3}\) (using ratio \(1:\sqrt{3}:2\)).

Answer:

1. B; 2) B; 3) D; 4) A; 5) D; 6) A