Question number 11. A, B, C, D is a square with side length equal to 1, while triangles B, C, F, and C, E, D are equilateral. What is the length of the segment F, E? Which is not drawn in here, so let's draw it in. In red, I'll have the segment connecting vertex F with vertex E, and let me make a copy of that triangle we just obtained. So we have roughly something like this, and the vertices are F over here, E, and C. And we're looking for the length of F, E, so let's call this X. Now, reading the question again, we see that A, B, C, D is a square, so all those segments are equal. So let's put dashes here to indicate that they're the same. We know that B, C, F is equilateral, so in fact all its edges are the same, and C, E, D is equilateral, so it turns out that all the segments here that are present in blue are of the same length, and that length is 1. That's the side length here of the square. So C, E is equal to 1, and F, C is also a segment of length 1. Now, we would like to know something about the angles here, so since the triangles are equilateral, the angles are all equal to 60 degrees. So here in black, these are 60 degree angles, and we can then compute the angle measure of this angle right here in blue. OK, so what we have is that that angle is the complement of the right angle at C, so that's 90 degrees minus 60 degrees, or 30 degrees. And so the angle C here in the triangle F, C, E measures exactly the complement, 30 degrees that we just computed, plus 60 degrees, the angle at vertex C of the triangle C, E, D. So this is a 90 degree or right angle, and we have a right triangle here. So now use the Pythagorean theorem to compute x. So x squared is 1 squared plus 1 squared, which is equal to 2, and so therefore x comes out to the square root of 2. And we have found our quantity that we're looking for. The length of segment FE is exactly equal to the square root of 2, and that gives us answer A to problem 11.

geometry

square

equilateral triangle

Pythagorean theorem

segment length