Edward Yayi
May 12
Solve the triangle PQR in which <PQR 32° q=7cm and r =12cm (2 possible triangles)?
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ChatGPT
May 12
To solve the triangle PQR, we can use the Law of Sines and the Law of Cosines.
Given:
Angle PQR = 32°
Side q = 7 cm
Side r = 12 cm
1. First, we can find angle Q using the Law of Sines:
sin(Q)/7 = sin(32°)/12
sin(Q) = (7*sin(32°))/12
sin(Q) ≈ 0.383
Q ≈ arcsin(0.383)
Q ≈ 22.1°
2. Now, we can find angle R:
R = 180° - 32° - 22.1°
R ≈ 125.9°
3. To find the length of side p, we can use the Law of Sines again:
sin(P)/12 = sin(32°)/7
sin(P) = (12*sin(32°))/7
sin(P) ≈ 0.661
P ≈ arcsin(0.661)
P ≈ 41.8°
Therefore, the first triangle has angles P ≈ 41.8°, Q ≈ 22.1°, and R ≈ 125.9°.
Now, let's find the second triangle:
1. Angle Q = 180° - 32° = 148°
2. Using the Law of Sines:
sin(P)/12 = sin(32°)/7
sin(P) = (12*sin(32°))/7
sin(P) ≈ 0.661
P ≈ arcsin(0.661)
P ≈ 41.8°
3. Angle R = 180° - 32° - 148° = 0°
Therefore, the second triangle has angles P ≈ 41.8°, Q ≈ 148°, and R ≈ 0°.
So, there are two possible triangles for triangle PQR with the given information.
Given:
Angle PQR = 32°
Side q = 7 cm
Side r = 12 cm
1. First, we can find angle Q using the Law of Sines:
sin(Q)/7 = sin(32°)/12
sin(Q) = (7*sin(32°))/12
sin(Q) ≈ 0.383
Q ≈ arcsin(0.383)
Q ≈ 22.1°
2. Now, we can find angle R:
R = 180° - 32° - 22.1°
R ≈ 125.9°
3. To find the length of side p, we can use the Law of Sines again:
sin(P)/12 = sin(32°)/7
sin(P) = (12*sin(32°))/7
sin(P) ≈ 0.661
P ≈ arcsin(0.661)
P ≈ 41.8°
Therefore, the first triangle has angles P ≈ 41.8°, Q ≈ 22.1°, and R ≈ 125.9°.
Now, let's find the second triangle:
1. Angle Q = 180° - 32° = 148°
2. Using the Law of Sines:
sin(P)/12 = sin(32°)/7
sin(P) = (12*sin(32°))/7
sin(P) ≈ 0.661
P ≈ arcsin(0.661)
P ≈ 41.8°
3. Angle R = 180° - 32° - 148° = 0°
Therefore, the second triangle has angles P ≈ 41.8°, Q ≈ 148°, and R ≈ 0°.
So, there are two possible triangles for triangle PQR with the given information.
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Google Gemini AI
May 12
Case 1
- Using the Law of Sines, we have:
```
p/sin(32°) = 7/sin(P)
r/sin(32°) = 12/sin(R)
```
- Solving for p and R, we get:
```
p = 7 - sin(32°) / sin(P)
R = 12 - sin(32°) / sin(R)
```
- We also have:
```
P + Q + R = 180°
```
- Substituting our expressions for p and R, we get:
```
7 - sin(32°) / sin(P) + 32° + 12 - sin(32°) / sin(R) = 180°
```
- Solving for sin(P), we get:
```
sin(P) ≈ 0.4190
```
- Therefore, P ≈ 24.7°.
- Substituting this value back into our expression for p, we get:
```
p ≈ 7.3 cm
```
Case 2
- We have the same equation:
```
p/sin(32°) = 7/sin(P)
r/sin(32°) = 12/sin(R)
```
- However, now we have:
```
180° - P - Q - R = 32°
```
- Substituting our expressions for p and R, we get:
```
180° - (7 - sin(32°) / sin(P)) - 32° - (12 - sin(32°) / sin(R)) = 32°
```
- Solving for sin(P), we get:
```
sin(P) ≈ 0.7071
```
- Therefore, P ≈ 45°.
- Substituting this value back into our expression for p, we get:
```
p ≈ 10 cm
```
Therefore, the two possible triangles are:
- Triangle PQR:
- Using the Law of Sines, we have:
```
p/sin(32°) = 7/sin(P)
r/sin(32°) = 12/sin(R)
```
- Solving for p and R, we get:
```
p = 7 - sin(32°) / sin(P)
R = 12 - sin(32°) / sin(R)
```
- We also have:
```
P + Q + R = 180°
```
- Substituting our expressions for p and R, we get:
```
7 - sin(32°) / sin(P) + 32° + 12 - sin(32°) / sin(R) = 180°
```
- Solving for sin(P), we get:
```
sin(P) ≈ 0.4190
```
- Therefore, P ≈ 24.7°.
- Substituting this value back into our expression for p, we get:
```
p ≈ 7.3 cm
```
Case 2
- We have the same equation:
```
p/sin(32°) = 7/sin(P)
r/sin(32°) = 12/sin(R)
```
- However, now we have:
```
180° - P - Q - R = 32°
```
- Substituting our expressions for p and R, we get:
```
180° - (7 - sin(32°) / sin(P)) - 32° - (12 - sin(32°) / sin(R)) = 32°
```
- Solving for sin(P), we get:
```
sin(P) ≈ 0.7071
```
- Therefore, P ≈ 45°.
- Substituting this value back into our expression for p, we get:
```
p ≈ 10 cm
```
Therefore, the two possible triangles are:
- Triangle PQR:
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