Puzzles

An arbitrary point (p) lies inside a square. Lines are drawn from (p) to the midpoints of the four sides, dividing the square into four quadrilaterals. If the areas of three of the quadrilaterals are 18, 24, and 48 square units, determine the area of the fourth quadrilateral.

Since the only information given about the quadrilateral is the areas of three of the four regions and the fact that each region has two sides of length (s), the available data are insufficient for a direct solution. Therefore, we must adopt a different approach to determine the area of the fourth region.

Next, connect each vertex of the outer square to point P with dashed blue line segments. Each of the four quadrilaterals is thereby divided into two triangles, giving a total of eight triangles, each with a base of length s.

Finally, draw perpendiculars from point P to the sides of the square (shown in green). These perpendiculars represent the heights of the eight triangles.

Let us focus on the lower side of the square. The two triangles (PAB) and (PBC) have the same base length, (s), and the same height, (h₁). Therefore, they have equal areas. Let the area of each triangle be denoted by (a).

Similarly, we denote the areas of the corresponding pairs of triangles along the other three sides of the square by (b), (c), and (d).

Using these definitions, we obtain the following area equations:

(1) a + b = 18

(2) b + c = 24

(3) c + d = 48

(4) d + a = x

Notice that we have only three equations but four unknowns. Therefore, the areas of the individual triangles cannot be determined uniquely. Fortunately, this is not necessary, since our goal is only to find the value of (a + d).

Subtract equation (2) from equation (1) to eliminate (b):

a-c=18-24=-6

Finally, add equations (3) and (5) to eliminate c, thereby obtaining the area of the fourth quadrilateral, (a + d).

a+d=-6+48=42