Question

special right triangles
find the missing side lengths. leave your answers as radicals in simplest form.
1)
2)
3)
4)
5)
6)
7)
8)

Explanation:

Step 1: Identify the known side

The given leg length is \(b = 2\sqrt{2}\). In a 45 - 45-90 triangle, the two legs are equal, so \(a=b\).
\(a = 2\sqrt{2}\)
## Step 2: Find the hypotenuse
The formula for the hypotenuse \(c\) of a 45 - 45-90 triangle is \(c=\sqrt{2}a\). Substituting \(a = 2\sqrt{2}\), we get \(c=\sqrt{2}\times2\sqrt{2}=4\).
- # Answer:
\(a = 2\sqrt{2}\), \(b = 2\sqrt{2}\)
- **Problem 2**:
- # Explanation:
## Step 1: Identify the hypotenuse
The hypotenuse \(y = 4\). In a 45 - 45-90 triangle, if the length of a leg is \(x\), then \(y=\sqrt{2}x\).
Solving for \(x\), we have \(x=\frac{y}{\sqrt{2}}\).
## Step 2: Rationalize the denominator
\(x=\frac{4}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\). Since the two legs are equal in a 45 - 45-90 triangle, \(x = 2\sqrt{2}\), and the other leg has the same length.
- # Answer:
\(x = 2\sqrt{2}\), \(y = 4\)
- **Problem 3**:
- # Explanation:
## Step 1: Identify the known leg
The given leg length is \(x=\frac{3\sqrt{2}}{2}\). In a 45 - 45-90 triangle, the two legs are equal, so the other leg \(y=\frac{3\sqrt{2}}{2}\).
## Step 2: Find the hypotenuse
The hypotenuse \(z=\sqrt{2}x\). Substituting \(x=\frac{3\sqrt{2}}{2}\), we get \(z=\sqrt{2}\times\frac{3\sqrt{2}}{2}=3\).
- # Answer:
\(x=\frac{3\sqrt{2}}{2}\), \(y=\frac{3\sqrt{2}}{2}\)
- **Problem 4**:
- # Explanation:
## Step 1: Identify the known leg
The given leg length is \(x = 3\sqrt{2}\). In a 45 - 45-90 triangle, the two legs are equal, so \(y = 3\sqrt{2}\).
## Step 2: Find the hypotenuse
The hypotenuse \(c=\sqrt{2}x\). Substituting \(x = 3\sqrt{2}\), we get \(c=\sqrt{2}\times3\sqrt{2}=6\).
- # Answer:
\(x = 3\sqrt{2}\), \(y = 3\sqrt{2}\)
- **Problem 5**:
- # Explanation:
## Step 1: Identify the hypotenuse
The hypotenuse \(y = 6\). In a 45 - 45-90 triangle, if the length of a leg is \(x\), then \(y=\sqrt{2}x\). Solving for \(x\), we have \(x=\frac{y}{\sqrt{2}}\).
## Step 2: Rationalize the denominator
\(x=\frac{6}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{6\sqrt{2}}{2}=3\sqrt{2}\). Since the two legs are equal in a 45 - 45-90 triangle, \(x = 3\sqrt{2}\), and the other leg has the same length.
- # Answer:
\(x = 3\sqrt{2}\), \(y = 6\)
- **Problem 6**:
- # Explanation:
## Step 1: Identify the hypotenuse
The hypotenuse \(y = 2\sqrt{6}\). In a 45 - 45-90 triangle, if the length of a leg is \(x\), then \(y=\sqrt{2}x\). Solving for \(x\), we have \(x=\frac{y}{\sqrt{2}}\).
## Step 2: Rationalize the denominator
\(x=\frac{2\sqrt{6}}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{2\sqrt{12}}{2}=\frac{2\times2\sqrt{3}}{2}=2\sqrt{3}\). Since the two legs are equal in a 45 - 45-90 triangle, \(x = 2\sqrt{3}\), \(y = 2\sqrt{6}\).
- # Answer:
\(x = 2\sqrt{3}\), \(y = 2\sqrt{6}\)
2. **For a 30 - 60-90 triangle**:
- The ratio of the side lengths is \(a:b:c=1:\sqrt{3}:2\), where the side opposite the 30 - degree angle has length \(a\), the side opposite the 60 - degree angle has length \(\sqrt{3}a\), and the hypotenuse has length \(2a\).
- **Problem 7**:
- # Explanation:
## Step 1: Identify the hypotenuse
The hypotenuse \(y = 16\). In a 30 - 60-90 triangle, if the side opposite the 30 - degree angle is \(x\), then \(y = 2x\). Solving for \(x\), we get \(x=\frac{y}{2}=8\).
## Step 2: Find the other leg
The side opposite the 60 - degree angle \(z=\sqrt{3}x\). Substituting \(x = 8\), we get \(z = 8\sqrt{3}\).
- # Answer:
\(x = 8\), \(y = 16\)
- **Problem 8**:
- # Explanation:
## Step 1: Identify the side opposite the 30 - degree angle
The side opposite the 30 - degree angle \(u = 2\). In a 30 - 60-90 triangle, the hypotenuse \(v = 2u\), so \(v = 4\), and the side opposite the 60 - degree angle \(w=\sqrt{3}u=2\sqrt{3}\).
- # Answer:
\(u = 2\), \(v = 4\)

Answer:

1. For a 45 - 45-90 triangle:

• The ratio of the side lengths is \(a:b:c = 1:1:\sqrt{2}\), where the legs (the sides opposite the 45 - degree angles) have the same length, and the hypotenuse \(c\) is \(\sqrt{2}\) times the length of a leg.
• Problem 1:
• # Explanation:

Step 1: Identify the known side

The given leg length is \(b = 2\sqrt{2}\). In a 45 - 45-90 triangle, the two legs are equal, so \(a=b\).
\(a = 2\sqrt{2}\)

Step 2: Find the hypotenuse

The formula for the hypotenuse \(c\) of a 45 - 45-90 triangle is \(c=\sqrt{2}a\). Substituting \(a = 2\sqrt{2}\), we get \(c=\sqrt{2}\times2\sqrt{2}=4\).

• # Answer:

\(a = 2\sqrt{2}\), \(b = 2\sqrt{2}\)

• Problem 2:
• # Explanation:

Step 1: Identify the hypotenuse

The hypotenuse \(y = 4\). In a 45 - 45-90 triangle, if the length of a leg is \(x\), then \(y=\sqrt{2}x\).
Solving for \(x\), we have \(x=\frac{y}{\sqrt{2}}\).

Step 2: Rationalize the denominator

\(x=\frac{4}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{4\sqrt{2}}{2}=2\sqrt{2}\). Since the two legs are equal in a 45 - 45-90 triangle, \(x = 2\sqrt{2}\), and the other leg has the same length.

• # Answer:

\(x = 2\sqrt{2}\), \(y = 4\)

• Problem 3:
• # Explanation:

Step 1: Identify the known leg

The given leg length is \(x=\frac{3\sqrt{2}}{2}\). In a 45 - 45-90 triangle, the two legs are equal, so the other leg \(y=\frac{3\sqrt{2}}{2}\).

Step 2: Find the hypotenuse

The hypotenuse \(z=\sqrt{2}x\). Substituting \(x=\frac{3\sqrt{2}}{2}\), we get \(z=\sqrt{2}\times\frac{3\sqrt{2}}{2}=3\).

• # Answer:

\(x=\frac{3\sqrt{2}}{2}\), \(y=\frac{3\sqrt{2}}{2}\)

• Problem 4:
• # Explanation:

Step 1: Identify the known leg

The given leg length is \(x = 3\sqrt{2}\). In a 45 - 45-90 triangle, the two legs are equal, so \(y = 3\sqrt{2}\).

Step 2: Find the hypotenuse

The hypotenuse \(c=\sqrt{2}x\). Substituting \(x = 3\sqrt{2}\), we get \(c=\sqrt{2}\times3\sqrt{2}=6\).

• # Answer:

\(x = 3\sqrt{2}\), \(y = 3\sqrt{2}\)

• Problem 5:
• # Explanation:

Step 1: Identify the hypotenuse

The hypotenuse \(y = 6\). In a 45 - 45-90 triangle, if the length of a leg is \(x\), then \(y=\sqrt{2}x\). Solving for \(x\), we have \(x=\frac{y}{\sqrt{2}}\).

Step 2: Rationalize the denominator

\(x=\frac{6}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{6\sqrt{2}}{2}=3\sqrt{2}\). Since the two legs are equal in a 45 - 45-90 triangle, \(x = 3\sqrt{2}\), and the other leg has the same length.

• # Answer:

\(x = 3\sqrt{2}\), \(y = 6\)

• Problem 6:
• # Explanation:

Step 1: Identify the hypotenuse

The hypotenuse \(y = 2\sqrt{6}\). In a 45 - 45-90 triangle, if the length of a leg is \(x\), then \(y=\sqrt{2}x\). Solving for \(x\), we have \(x=\frac{y}{\sqrt{2}}\).

Step 2: Rationalize the denominator

\(x=\frac{2\sqrt{6}}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{2\sqrt{12}}{2}=\frac{2\times2\sqrt{3}}{2}=2\sqrt{3}\). Since the two legs are equal in a 45 - 45-90 triangle, \(x = 2\sqrt{3}\), \(y = 2\sqrt{6}\).

• # Answer:

\(x = 2\sqrt{3}\), \(y = 2\sqrt{6}\)

1. For a 30 - 60-90 triangle:

• The ratio of the side lengths is \(a:b:c=1:\sqrt{3}:2\), where the side opposite the 30 - degree angle has length \(a\), the side opposite the 60 - degree angle has length \(\sqrt{3}a\), and the hypotenuse has length \(2a\).
• Problem 7:
• # Explanation:

Step 1: Identify the hypotenuse

The hypotenuse \(y = 16\). In a 30 - 60-90 triangle, if the side opposite the 30 - degree angle is \(x\), then \(y = 2x\). Solving for \(x\), we get \(x=\frac{y}{2}=8\).

Step 2: Find the other leg

The side opposite the 60 - degree angle \(z=\sqrt{3}x\). Substituting \(x = 8\), we get \(z = 8\sqrt{3}\).

• # Answer:

\(x = 8\), \(y = 16\)

• Problem 8:
• # Explanation:

Step 1: Identify the side opposite the 30 - degree angle

The side opposite the 30 - degree angle \(u = 2\). In a 30 - 60-90 triangle, the hypotenuse \(v = 2u\), so \(v = 4\), and the side opposite the 60 - degree angle \(w=\sqrt{3}u=2\sqrt{3}\).

• # Answer:

\(u = 2\), \(v = 4\)