The question is as follows:
We know that in order to create 1, 2, 3 and 4 congruent unit equilateral triangles on a flat plane, we need 3, 5, 7 and 9 matchsticks respectively. What is the minimum number of matchsticks on a flat plane needed to create 7 congruent unit equilateral triangles? 13, 14, 15, or 16?
I am thinking that the answer is 14 because I have drawn out 12 matchsticks that would create six equilateral triangles, and by an addition of two more matchsticks, I was able to get seven equilateral triangles. My diagram may be wrong, therefore, I am asking about what the correct answer could be.
We need 3 matchsticks to make a triangle, but a matchstick in the interior of our figure can contribute to two triangles.
If we want 7 triangles, those have 21 sides; to do this with 13 matchsticks, we need 8 of the matchsticks to be inside the figure (contributing to 16 sides), and 5 on the outside (contributing to 5 more sides, 21 in total).
But a shape with perimeter 5 has area at most $\frac{25}{4\pi} \approx 1.99$: the area of a circle with circumference 5. This is not enough to fit 7 equilateral triangles, which have total area $\frac{7\sqrt3}{4} \approx 3.03$.
So there cannot be a 13-matchstick construction, and at least 14 matchsticks are required. As you've noticed, there is a 14-matchstick construction, so that is the best value possible.