Question 1171512: 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! Let's analyze each case using the Law of Sines and the properties of triangles.
**Law of Sines:** p/sin(P) = q/sin(Q) = r/sin(R)
**(a) p = 3, q = 5, angle P = 27 degrees**
1. **Find sin(Q):**
* 3/sin(27°) = 5/sin(Q)
* sin(Q) = (5 * sin(27°)) / 3
* sin(Q) ≈ (5 * 0.4540) / 3 ≈ 0.7567
* Q ≈ arcsin(0.7567) ≈ 49.12 degrees
2. **Check for another possible Q:**
* Since sin(Q) = sin(180° - Q), there's another possible angle:
* Q' = 180° - 49.12° ≈ 130.88 degrees
3. **Check if both Q and Q' are valid:**
* For Q = 49.12°, P + Q = 27° + 49.12° = 76.12° < 180° (valid)
* For Q' = 130.88°, P + Q' = 27° + 130.88° = 157.88° < 180° (valid)
4. **Number of Triangles:**
* There are **two** non-congruent triangles.
5. **Size of Angle Q:**
* Q ≈ 49.12 degrees
* Q' ≈ 130.88 degrees
**(b) p = 8, q = 5, angle P = 57 degrees**
1. **Find sin(Q):**
* 8/sin(57°) = 5/sin(Q)
* sin(Q) = (5 * sin(57°)) / 8
* sin(Q) ≈ (5 * 0.8387) / 8 ≈ 0.5242
* Q ≈ arcsin(0.5242) ≈ 31.61 degrees
2. **Check for another possible Q:**
* Q' = 180° - 31.61° ≈ 148.39 degrees
3. **Check if both Q and Q' are valid:**
* For Q = 31.61°, P + Q = 57° + 31.61° = 88.61° < 180° (valid)
* For Q' = 148.39°, P + Q' = 57° + 148.39° = 205.39° > 180° (invalid)
4. **Number of Triangles:**
* There is **one** non-congruent triangle.
5. **Size of Angle Q:**
* Q ≈ 31.61 degrees
**(c) p = 7, q = 8, angle P = 70 degrees**
1. **Find sin(Q):**
* 7/sin(70°) = 8/sin(Q)
* sin(Q) = (8 * sin(70°)) / 7
* sin(Q) ≈ (8 * 0.9397) / 7 ≈ 1.0739
2. **Check for sin(Q) range:**
* Since sin(Q) > 1, there is no angle Q that satisfies this condition.
3. **Number of Triangles:**
* There are **zero** non-congruent triangles.
**(d) p = 10, q = 20, angle P = 30 degrees**
1. **Find sin(Q):**
* 10/sin(30°) = 20/sin(Q)
* sin(Q) = (20 * sin(30°)) / 10
* sin(Q) = (20 * 0.5) / 10 = 1
2. **Find angle Q:**
* Q = arcsin(1) = 90 degrees
3. **Check for another possible Q:**
* Q' = 180° - 90° = 90 degrees (same as Q)
4. **Number of Triangles:**
* There is **one** non-congruent triangle.
5. **Size of Angle Q:**
* Q = 90 degrees
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