Construction of traingle when base, vertical angle and difference of base and one side is given

Question

When it is not possible to construct a triangle ABC, if BC = 6 cm, angle A = 50 degrees and difference in BC & AC is a. 2.5 cm b. 4 cm c. 5 cm d. 6.4 cm

P.S. can not use cosine rule, as the question is intended for school children...:(

Answer 1

In a triangle, length of any edge should be larger than the difference of other two edges. Otherwise, it becomes a line. So the answer is D, but I do not know why the angle is given. Only reason I can think of is that since $A>0$ we know that it is not a line but it was triangle to begin with.

Sorry, misread the question.

I think all of the options are possible. At least, I was able to construct all of them in GeoGebra. But that may be wrong due to roundoff errors.

More rigorous, but definitely not elementary school level, explanation. Draw the line $BC$. Draw also the line $AC$ and a ray $AD$ such that the angle between $AC$ and $AD$ is $50$ degrees and $AD$ is directed in the general direction of the point $B$. If $AD$ intersects $BC$ at $B$, you are done. Otherwise, the intersection point is either before or after $B$. In that case, by continuously changing the angle $C$, or rotating $AC$ around $C$, we can continuously move the intersection point.

Since this explanation uses the continuity of the operation, it may not be possible to articulate in a way a kid would understand. However, I think that the question is wrong.

I will use law of sines but without actually calculating sines. Since $\angle BAC = 50>30$, $\sin(\angle BAC)>\sin(30)$ so $6/\sin(\angle BAC)<12$. But, necessarily $|AC|/\angle ABC = 6/\sin(\angle BAC)$. This is all good as long as $AC$ is shorter than $BC$. However, take $|AC|=12.4$. Then the minimum value $|AC|/\angle ABC$ can obtain is $12.4$ which is greater than $12$. So it is a contradiction. Correct answer is D.

Answer 2

The circumradius has been fixed with the given data, being equal to $$r = \frac{3}{\sin 50} = 3.92$$

If you think about it, all the possible points A will make an arc of a circle, and the chord AC will have maximum value when it is the diameter of the circle, i.e

$$ l(AC) = 2r = 7.83$$

Hence for the first three options, there will be a possible value of AC which is in (0,7.83) and satisfies the absolute difference, all except the last option