Question 1179473: For each of the following, tell how many non congruent triangles PQR fit the given description, and find the size of angle Q. Make a separate diagram for each case. (a) p = 3, q = 5, angle P = 27 degrees (b) p = 8, q = 5, angle P = 57 degrees (c) p = 7, q = 8, angle P = 70 degrees (d) p = 10, q = 20, angle P = 30 degrees.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! To determine the number of non-congruent triangles and find the size of angle Q in each case, I'll use the Law of Sines and the Sine Rule. I'll also consider the ambiguous case of the Law of Sines, which can sometimes lead to multiple solutions.
**Case (a): p = 3, q = 5, angle P = 27 degrees**
Using the Sine Rule:
```
sin(Q) / q = sin(P) / p
sin(Q) / 5 = sin(27°) / 3
sin(Q) ≈ 0.756
```
Now, there are two possible angles for Q between 0° and 180° that have a sine of approximately 0.756:
* Q1 ≈ 49.1°
* Q2 ≈ 180° - 49.1° ≈ 130.9°
Since Q2 + P = 130.9° + 27° = 157.9° < 180°, both angles are valid.
Therefore, there are **two non-congruent triangles** in this case.
**Case (b): p = 8, q = 5, angle P = 57 degrees**
Using the Sine Rule:
```
sin(Q) / q = sin(P) / p
sin(Q) / 5 = sin(57°) / 8
sin(Q) ≈ 0.525
```
There is only one angle for Q between 0° and 180° that has a sine of approximately 0.525:
* Q ≈ 31.7°
Therefore, there is **one non-congruent triangle** in this case.
**Case (c): p = 7, q = 8, angle P = 70 degrees**
Using the Sine Rule:
```
sin(Q) / q = sin(P) / p
sin(Q) / 8 = sin(70°) / 7
sin(Q) ≈ 1.08
```
Since the sine of an angle cannot be greater than 1, there is **no triangle** possible in this case.
**Case (d): p = 10, q = 20, angle P = 30 degrees**
Using the Sine Rule:
```
sin(Q) / q = sin(P) / p
sin(Q) / 20 = sin(30°) / 10
sin(Q) = 1
```
There is only one angle for Q between 0° and 180° that has a sine of 1:
* Q = 90°
Therefore, there is **one non-congruent triangle** in this case (a right-angled triangle).
**Summary**
| Case | Non-congruent triangles | Angle Q |
|---|---|---|
| (a) | 2 | 49.1° or 130.9° |
| (b) | 1 | 31.7° |
| (c) | 0 | N/A |
| (d) | 1 | 90° |
I hope this helps! Let me know if you have any other questions.
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