Question
If α=30°
and a=9
, determine the measures for b
that will result in one nonright triangle, no triangle, and two triangles. Round b
values to the nearest hundredth as needed.
A. One nonright triangle exists when b≤
.
B. No triangles exist when b>
.
C. Two triangles exist when
<b<
.
and a=9
, determine the measures for b
that will result in one nonright triangle, no triangle, and two triangles. Round b
values to the nearest hundredth as needed.
A. One nonright triangle exists when b≤
.
B. No triangles exist when b>
.
C. Two triangles exist when
<b<
.
Answers
GPT-4o mini
To determine the measures for \( b \) that result in one nonright triangle, no triangle, and two triangles, we can use the Law of Sines.
Given:
- \( α = 30° \)
- \( a = 9 \)
### A. Determine b for one nonright triangle:
Using the Law of Sines:
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
Let’s denote \( B \) as the angle opposite side \( b \). From the Law of Sines, we have:
\[
\frac{9}{\sin 30°} = \frac{b}{\sin B}
\]
Since \( \sin 30° = \frac{1}{2} \):
\[
\frac{9}{\frac{1}{2}} = \frac{b}{\sin B}
\]
\[
18 = \frac{b}{\sin B}
\]
\[
b = 18 \sin B
\]
### 1. **One non-right triangle:**
For one non-right triangle, there must be a unique angle \( B \) such that \( 0 < B < 180° \) and the sine function is positive:
\[
b \leq 18 \quad (\text{when } B=90°)
\]
So,
\[\text{A. One nonright triangle exists when } b \leq 18. \]
---
### B. Determine b for no triangles:
For no triangle scenario, we use the fact that if side \( b \) is too short to form a triangle opposite \( A \), we check for that condition:
1. When \( b \) is less than the height from \( A \) to side \( a \):
- The height can be calculated using the formula:
\[
h = a \sin A = 9 \sin 30° = 9 \cdot \frac{1}{2} = 4.5
\]
Thus, for no triangle, we need \( b < 4.5 \).
2. However, if \( b \) goes beyond a certain value, it also results in no triangle. For that, we look at \( b \) where it is larger than the longest side, which in our case is \( a \) (since \( a = 9 \)):
- If \( b > 9 \): Triangle inequality fails
Putting this together, the critical no triangle case comes from:
\[\text{B. No triangles exist when } b > 9. \]
---
### C. Determine b for two triangles:
For two triangles, b must satisfy the conditions of the Ambiguous Case (SSA) in the Law of Sines. This occurs when \( b \) lies between the height of the triangle and the length of side \( a \):
1. From our previous calculation:
- \( b > 4.5 \) gives one possible triangle but is still ambiguous at certain values.
2. And it must also be less than the maximum boundary to create two triangles:
That is, \( 4.5 < b < 9 \).
Thus, rounding the values to the nearest hundredth:
\[\text{C. Two triangles exist when } 4.50 < b < 9.00. \]
### Summary of Responses:
- A: One nonright triangle exists when \( b \leq 18 \).
- B: No triangles exist when \( b > 9 \).
- C: Two triangles exist when \( 4.50 < b < 9.00 \).
Given:
- \( α = 30° \)
- \( a = 9 \)
### A. Determine b for one nonright triangle:
Using the Law of Sines:
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]
Let’s denote \( B \) as the angle opposite side \( b \). From the Law of Sines, we have:
\[
\frac{9}{\sin 30°} = \frac{b}{\sin B}
\]
Since \( \sin 30° = \frac{1}{2} \):
\[
\frac{9}{\frac{1}{2}} = \frac{b}{\sin B}
\]
\[
18 = \frac{b}{\sin B}
\]
\[
b = 18 \sin B
\]
### 1. **One non-right triangle:**
For one non-right triangle, there must be a unique angle \( B \) such that \( 0 < B < 180° \) and the sine function is positive:
\[
b \leq 18 \quad (\text{when } B=90°)
\]
So,
\[\text{A. One nonright triangle exists when } b \leq 18. \]
---
### B. Determine b for no triangles:
For no triangle scenario, we use the fact that if side \( b \) is too short to form a triangle opposite \( A \), we check for that condition:
1. When \( b \) is less than the height from \( A \) to side \( a \):
- The height can be calculated using the formula:
\[
h = a \sin A = 9 \sin 30° = 9 \cdot \frac{1}{2} = 4.5
\]
Thus, for no triangle, we need \( b < 4.5 \).
2. However, if \( b \) goes beyond a certain value, it also results in no triangle. For that, we look at \( b \) where it is larger than the longest side, which in our case is \( a \) (since \( a = 9 \)):
- If \( b > 9 \): Triangle inequality fails
Putting this together, the critical no triangle case comes from:
\[\text{B. No triangles exist when } b > 9. \]
---
### C. Determine b for two triangles:
For two triangles, b must satisfy the conditions of the Ambiguous Case (SSA) in the Law of Sines. This occurs when \( b \) lies between the height of the triangle and the length of side \( a \):
1. From our previous calculation:
- \( b > 4.5 \) gives one possible triangle but is still ambiguous at certain values.
2. And it must also be less than the maximum boundary to create two triangles:
That is, \( 4.5 < b < 9 \).
Thus, rounding the values to the nearest hundredth:
\[\text{C. Two triangles exist when } 4.50 < b < 9.00. \]
### Summary of Responses:
- A: One nonright triangle exists when \( b \leq 18 \).
- B: No triangles exist when \( b > 9 \).
- C: Two triangles exist when \( 4.50 < b < 9.00 \).