The original question includes a diagram: A triangle with a side of length x - 1 opposite a 30 degree angle, and a side of length -x^2 + 6x - 5 adjacent to that angle.
A triangle has a side of length x − 1 opposite an angle of 30∘, and a side of length −x2 + 6x − 5 adjacent to that angle.
Find the complete set of values of x for which there are two non-congruent triangles with these side lengths and angle.
- A1 < x < 3
- B1 < x < 4
- C1 < x < 5
- D3 < x < 4
- E3 < x < 5
- F4 < x < 5
Show the answer and worked solution
answer · D
- A1 < x < 3
- B1 < x < 4
- C1 < x < 5
- D3 < x < 4
- E3 < x < 5
- F4 < x < 5
Write a = x − 1 for the side opposite the 30∘ angle and b = −x2 + 6x − 5 = (x−1)(5−x) for the adjacent side. Both must be positive, so 1 < x < 5. The ambiguous case gives two triangles exactly when bsin 30∘ < a < b. The left inequality is (x−1)(5−x)2 < x−1, which reduces to x > 3; the right is x − 1 < (x−1)(5−x), which reduces to x < 4. Together, 3 < x < 4.