# Given an angle and opposite side in a triangle, find constraints on the side adjacent to the angle

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.

• i get the maximum $b = 19/\sin 21^\circ = 53.016$ not the $56.79$ you got. – abel Dec 11 '14 at 4:26
• hmm, how did you get the numbers 19, and sin(21)? – oscilatorium Dec 11 '14 at 6:11

Your solution is basically correct. I think it'd be easier to draw a circle with center in the vertex $C$ and fixed radius = 38 and look at its intersections with the ray $AB$ while the vertex $C$ is moving from the vertex $A$ into infinity.
Also please think about boundary cases - for example, the triangle with maximal $b$ will be unique again.