Problem number 7. Two equilateral triangles are put together with their opposite sides parallel to form a hexagon. We know the length of four sides of this hexagon as shown in the diagram. What is the perimeter of the hexagon? So here we can see that we have four sides labeled 6, 15, 11, and 12 and we need to find the total length of the hexagon that is shaded in blue. Okay, so now since we have two equilateral triangles that intersect each other, we have tons of parallel lines and that creates more equilateral triangles. In fact, all of the tiny triangles are equilateral triangles because it's one equilateral triangle cutting off another equilateral triangle with parallel lines. So RQ is parallel to this side right here. So this side is parallel to QR and that means that this right here is also an equilateral triangle. That means that we have a lot more flexibility in terms of figuring out the other sides. We know that AR, for example, is equal to 6 and that allows us to figure out all the other sides. Since the two large triangles are equilateral and the opposite sides of the hexagons are parallel, each of the six small triangles are also equilateral. So we know that AB is equal to 6, plus 15, plus 11, which means it's equal to 32. Now since ABC is an equilateral triangle, BC must also be of length 32. So QP by elimination is equal to 32 minus 11 minus 12 equals 9, which is the length of CP. And QP is the same as CP. Now AC is also equal to 32. So RQ is equal to 32 minus 6 minus 9, which equals 17. And so now the perimeter of the hexagon is simply the sum of the sides that we just found, plus the sides we were already given. So 6 plus 15 plus 11 plus 12 is what we were given, plus 9 plus 17 are the sides we just figured out. And that gives us our answer of choice D.

hexagon perimeter

equilateral triangles

side lengths

geometry problem

perimeter calculation