Finding a number that, when added to 13600, forms a perfect square does not seem to be an easy task. Let's solve this geometrically. Consider a rectangle of area 13600 with one side equal to 136. Divide it into a square \( A \) and a rectangle \( B \). Cut the rectangle \( B \) into two equal rectangles, both denoted by \( C \). Position the rectangles \( C \) on two consecutive sides of the square \( A \).
![](https://cdn.mathpix.com/cropped/2024_05_01_20684006194a352111ccg-21.jpg?height=520&width=1290&top_left_y=682&top_left_x=478)

(a) What is the area of the rectangle \( C \) ?

(b) Notice that if we add the square \( X \), we complete a larger square. What must be the side length of the square \( X \) ?

(c) After answering the two previous items, determine a number that, when added to 13600, results in a perfect square and determine the square root of that perfect square.
Solution

(a) Since one of the sides is 136 and the area is 13600, the other side is 100. Furthermore, since $A$ is a square, it has a side length of 100. Consequently, the shorter side of rectangle $B$ is $136-100=36$ and its longer side is 100, as indicated in the following figure:

![](https://cdn.mathpix.com/cropped/2024_05_01_20684006194a352111ccg-21.jpg?height=371&width=906&top_left_y=2182&top_left_x=675)

One of the sides of rectangle $C$ is 100 and the other is half the shorter side of rectangle $B$, which is $\frac{36}{2}=18$. We conclude that the area of rectangle $C$ is $18 \cdot 100=1800$.
(b) The side length of square $X$ is equal to the shorter side of rectangle $C$, which is 18.

(c) If we add $18^{2}=324$ to the number 13600, the result will be $13924=118^{2}$. Therefore, the sought root is 118.