2003 CEMC Pascal Problems/Problem 15

Problem

In the diagram, square $ABCD$ is made up of $36$ squares, each with side length $1$ . The area of the square $KLMN$ , in square units, is

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

$\text{ (A) }\ 12 \qquad\text{ (B) }\ 16 \qquad\text{ (C) }\ 18 \qquad\text{ (D) }\ 20 \qquad\text{ (E) }\ 25$

Solution 1

Since the side lengths of a square are all equal to each other, we only need to find one of the side lengths of square $KLMN$ .

The angles of squares are also all $90^{\circ}$ , so we know that triangle $KBL$ is a right triangle, which means that we can use the pythagorean theorem.

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

$KB = 4$ , since the side lengths of the tiny squares are all $1$ . Using this logic, we also get $BL = 2$ .

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

Using the pythagorean theorem, we have:

$KB^2 + BL^2 = KL^2$

$4^2 + 2^2 = KL^2$

$KL^2 = 16 + 4 = 20$

$KL$ is one of the side lengths of square $KLMN$ , and the area of a square is the square of its side length. We can see that $KL^2 = \boxed {\textbf {(D) } 20}$ .

~anabel.disher

Solution 2

To find the area of square $KLMN$ , we can subtract the area of the triangles from the total area.

Since the angles of a square are all right angles, we know that the triangles are all right triangles.

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

We can also see $AN = KB = MD = CL = 4$ and $AK = BL = CM = ND = 2$ using the same logic found in solution 1.

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

Because the legs of the triangles are equal to each other, the areas are the same, and we only need to find one of the areas. We can multiply this area by $4$ to get the total area of the triangles:

The area of a triangle is its height multiplied by one of its sides, which is the length of the legs multiplied together in a right triangle, divided by $2$ :