In triangle ABC, we are given an angle A = 42 degrees, and its opposite side length, a = 38. i) For what values of adjacent side b such that we have one unique triangle ? ii) For what values b do we have 2 different triangles?
I was drawing a bunch of triangles with different cases where side b varies, and came up with the following, but I'm not sure if this is correct and am wondering if someone more knowledgeable could verify, and/or correct my misunderstandings:
i) Here we can draw an altitude h from C meeting AB at point X, making two triangles, ACX and BCX. What I noticed is that if side a > side b, then we only get one unique triangle (since if we reflect BCX across h, it lies beyond point A). Hence the constraint should be b < 38.
ii) Again if we draw the same altitude h from C meeting AB at point X to make two triangles ACX and BCX, I notice that if a < b, then when reflecting BCX across h, BX will lie between AX, creating a second (smaller triangle) that fits the given criteria. (However, if a is too small compared to b, it will not make any triangle at all, i.e if a < h, then it won't reach AB). So a must be greater than h, but smaller than b, => h < a < b. So now I express h in terms of b using right triangle ACX: sin(42) = h/b, => h = bsin(42). Since h < a, substituting we get bsin(42) < 38, => b < 38/sin(42), giving b < 56.79 But since a < b, we have b > 38, so the constraint should be 38 < b < 56.79
Please let me know if any of this doesn't make sense, or I'm missing anything.