Question
1. triangle:
45 degree
8√2
x
2. triangle:
30 degree
6
x
3. triangle:
45 degree
7√2
x
4. triangle:
60 degree
16
x
a. 7
b. 12
c. 16
d. 8√3
e. 8
45 degree
8√2
x
2. triangle:
30 degree
6
x
3. triangle:
45 degree
7√2
x
4. triangle:
60 degree
16
x
a. 7
b. 12
c. 16
d. 8√3
e. 8
Answers
GPT-4o mini
To find the lengths of the sides represented by \( x \) in these triangles, we can apply trigonometric principles and properties of special triangles.
### 1. Triangle with 45 degrees and side length \( 8\sqrt{2} \):
In a 45-45-90 triangle, the sides are in the ratio \( 1:1:\sqrt{2} \). If the hypotenuse is \( 8\sqrt{2} \), the legs can be calculated as follows:
\[
\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{8\sqrt{2}}{\sqrt{2}} = 8
\]
So, \( x = 8 \).
### 2. Triangle with 30 degrees and side length 6:
In a 30-60-90 triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \). If the shorter leg (opposite the 30-degree angle) is 6, then:
- The longer leg \( x \) (opposite the 60-degree angle) is given by:
\[
x = 6\sqrt{3}
\]
This value does not appear in the provided options.
### 3. Triangle with 45 degrees and side length \( 7\sqrt{2} \):
Using the same reasoning as the first triangle:
\[
\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{7\sqrt{2}}{\sqrt{2}} = 7
\]
So, \( x = 7 \).
### 4. Triangle with 60 degrees and side length 16:
Again, applying the ratios for a 30-60-90 triangle, if the side opposite the 60-degree angle is 16, the shorter leg (opposite the 30-degree angle) is:
\[
\text{Short leg} = \frac{16}{\sqrt{3}} \text{(which isn't needed as we seek the side opposite the 60-degree angle)}
\]
However, since 16 is already the length of the side opposite to the 60-degree angle, we confirm:
So in this case, \( x = 16 \).
### Summary of Values of \( x \):
1. \( x = 8 \)
2. \( x = 6\sqrt{3} \) (not in the options)
3. \( x = 7 \)
4. \( x = 16 \)
From the options provided:
- **a. 7** (Confirmed from triangle 3)
- **b. 12** (not applicable)
- **c. 16** (Confirmed from triangle 4)
- **d. 8√3** (not applicable)
- **e. 8** (Confirmed from triangle 1)
Thus, the values of \( x \) from the multiple-choice options are **7 (a), 8 (e), and 16 (c)**.
### 1. Triangle with 45 degrees and side length \( 8\sqrt{2} \):
In a 45-45-90 triangle, the sides are in the ratio \( 1:1:\sqrt{2} \). If the hypotenuse is \( 8\sqrt{2} \), the legs can be calculated as follows:
\[
\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{8\sqrt{2}}{\sqrt{2}} = 8
\]
So, \( x = 8 \).
### 2. Triangle with 30 degrees and side length 6:
In a 30-60-90 triangle, the sides are in the ratio \( 1:\sqrt{3}:2 \). If the shorter leg (opposite the 30-degree angle) is 6, then:
- The longer leg \( x \) (opposite the 60-degree angle) is given by:
\[
x = 6\sqrt{3}
\]
This value does not appear in the provided options.
### 3. Triangle with 45 degrees and side length \( 7\sqrt{2} \):
Using the same reasoning as the first triangle:
\[
\text{Leg} = \frac{\text{Hypotenuse}}{\sqrt{2}} = \frac{7\sqrt{2}}{\sqrt{2}} = 7
\]
So, \( x = 7 \).
### 4. Triangle with 60 degrees and side length 16:
Again, applying the ratios for a 30-60-90 triangle, if the side opposite the 60-degree angle is 16, the shorter leg (opposite the 30-degree angle) is:
\[
\text{Short leg} = \frac{16}{\sqrt{3}} \text{(which isn't needed as we seek the side opposite the 60-degree angle)}
\]
However, since 16 is already the length of the side opposite to the 60-degree angle, we confirm:
So in this case, \( x = 16 \).
### Summary of Values of \( x \):
1. \( x = 8 \)
2. \( x = 6\sqrt{3} \) (not in the options)
3. \( x = 7 \)
4. \( x = 16 \)
From the options provided:
- **a. 7** (Confirmed from triangle 3)
- **b. 12** (not applicable)
- **c. 16** (Confirmed from triangle 4)
- **d. 8√3** (not applicable)
- **e. 8** (Confirmed from triangle 1)
Thus, the values of \( x \) from the multiple-choice options are **7 (a), 8 (e), and 16 (c)**.