26. The large square in the diagram to the right is divided into four smaller squares. The circle touches the right-hand side of the square at its midpoint. What is the side length of the large square? Note that the diagram is not drawn to scale, so don't try to take out your ruler and try to actually measure the diagram. Instead, let's draw in the radius r shown in blue. And then we can see that we actually have a right triangle that's actually going to be really useful for us. If we label one half of the side of the big square as x, we can actually make some equations. We can create a right triangle with the legs x-6 and x-r and with hypotenuse equal to r. And once we have that, what we can do is the diameter of the circle plus the line segment of length 8 is the side length of the square. So 2r plus 8 is equal to 2x. So r is equal to x minus 4. And that means we have two equations, two variables, and that means that we can probably solve for each of them. Using the Pythagorean theorem on the right triangle gives us x minus r squared plus x minus 6 squared is equal to r squared. Solve to get x equals 9 and that gives us that 2x equals 18. Since we're looking for the side length of the large square, our answer is a.

large square

circle

right triangle

Pythagorean theorem

side length