Question
Find the value of x in the triangle below.
sides are 45 and the two angles are 35 ll2 we are look for the second longest side
(5 points)
Responses
31.9
31.9
30.4
30.4
29.0
29.0
27.8
sides are 45 and the two angles are 35 ll2 we are look for the second longest side
(5 points)
Responses
31.9
31.9
30.4
30.4
29.0
29.0
27.8
Answers
GPT-4o mini
To find the value of \( x \) in this triangle, we can use the Law of Sines. The Law of Sines states that the ratio of a side of a triangle to the sine of its opposite angle is constant.
Given:
- One side \( a = 45 \)
- One angle \( A = 35^\circ \)
- Another angle \( B = 112^\circ \)
First, we can find the third angle \( C \) using the triangle sum property, which states that the sum of the angles in a triangle is \( 180^\circ \):
\[
C = 180^\circ - A - B = 180^\circ - 35^\circ - 112^\circ = 33^\circ
\]
Now we have:
- \( A = 35^\circ \)
- \( B = 112^\circ \)
- \( C = 33^\circ \)
- \( a = 45 \)
Next, we apply the Law of Sines to find the side opposite angle \( B \) (which is the longest side since angle \( B \) is the largest angle). We will denote this side as \( b \):
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
Substituting the known values:
\[
\frac{45}{\sin(35^\circ)} = \frac{b}{\sin(112^\circ)}
\]
Now, solving for \( b \):
\[
b = 45 \cdot \frac{\sin(112^\circ)}{\sin(35^\circ)}
\]
Calculating \( \sin(35^\circ) \) and \( \sin(112^\circ) \):
Using a calculator:
- \( \sin(35^\circ) \approx 0.5736 \)
- \( \sin(112^\circ) \approx 0.9272 \)
Now plug in the values:
\[
b = 45 \cdot \frac{0.9272}{0.5736} \approx 45 \cdot 1.6207 \approx 72.9
\]
Now we need to find the side \( c \) opposite angle \( C \):
Using the Law of Sines again:
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Solving for \( c \):
\[
c = 45 \cdot \frac{\sin(33^\circ)}{\sin(35^\circ)}
\]
Calculating \( \sin(33^\circ) \) using a calculator:
- \( \sin(33^\circ) \approx 0.5446 \)
Substituting values:
\[
c = 45 \cdot \frac{0.5446}{0.5736} \approx 45 \cdot 0.9503 \approx 42.7
\]
Thus, we have:
- Longest side \( b \approx 72.9 \)
- Second longest side \( a = 45 \)
- Shortest side \( c \approx 42.7 \)
The problem asks for the second longest side, which is:
\[
x \approx 45
\]
However, none of the provided options match this calculation. It might be worth double-checking the angles or input numbers if needed. For the choices given:
The closest option for the second longest side could be \( 31.9, 30.4, 29.0, \) or \( 27.8 \). Without further information or context about what specifically these numbers relate to, I can't conclusively select one. However, if just looking for side \( c \), the options appear too far off. Please clarify if there are adjustments or additional constraints.
Given:
- One side \( a = 45 \)
- One angle \( A = 35^\circ \)
- Another angle \( B = 112^\circ \)
First, we can find the third angle \( C \) using the triangle sum property, which states that the sum of the angles in a triangle is \( 180^\circ \):
\[
C = 180^\circ - A - B = 180^\circ - 35^\circ - 112^\circ = 33^\circ
\]
Now we have:
- \( A = 35^\circ \)
- \( B = 112^\circ \)
- \( C = 33^\circ \)
- \( a = 45 \)
Next, we apply the Law of Sines to find the side opposite angle \( B \) (which is the longest side since angle \( B \) is the largest angle). We will denote this side as \( b \):
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
Substituting the known values:
\[
\frac{45}{\sin(35^\circ)} = \frac{b}{\sin(112^\circ)}
\]
Now, solving for \( b \):
\[
b = 45 \cdot \frac{\sin(112^\circ)}{\sin(35^\circ)}
\]
Calculating \( \sin(35^\circ) \) and \( \sin(112^\circ) \):
Using a calculator:
- \( \sin(35^\circ) \approx 0.5736 \)
- \( \sin(112^\circ) \approx 0.9272 \)
Now plug in the values:
\[
b = 45 \cdot \frac{0.9272}{0.5736} \approx 45 \cdot 1.6207 \approx 72.9
\]
Now we need to find the side \( c \) opposite angle \( C \):
Using the Law of Sines again:
\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]
Solving for \( c \):
\[
c = 45 \cdot \frac{\sin(33^\circ)}{\sin(35^\circ)}
\]
Calculating \( \sin(33^\circ) \) using a calculator:
- \( \sin(33^\circ) \approx 0.5446 \)
Substituting values:
\[
c = 45 \cdot \frac{0.5446}{0.5736} \approx 45 \cdot 0.9503 \approx 42.7
\]
Thus, we have:
- Longest side \( b \approx 72.9 \)
- Second longest side \( a = 45 \)
- Shortest side \( c \approx 42.7 \)
The problem asks for the second longest side, which is:
\[
x \approx 45
\]
However, none of the provided options match this calculation. It might be worth double-checking the angles or input numbers if needed. For the choices given:
The closest option for the second longest side could be \( 31.9, 30.4, 29.0, \) or \( 27.8 \). Without further information or context about what specifically these numbers relate to, I can't conclusively select one. However, if just looking for side \( c \), the options appear too far off. Please clarify if there are adjustments or additional constraints.