The perimeter of a rectangle is 40 and the length of one of its diagonals is 16. Find the area of the rectangle.

Let's denote the length of the rectangle as "l" and the width as "w". We are given that the perimeter (total length of all sides) is 40 units and the diagonal is 16 units.

 

Perimeter: Perimeter of a rectangle is calculated by adding the lengths of all its sides. In this case, Perimeter = 2l + 2w = 40.

 

Diagonal: The diagonal of a rectangle divides it into two congruent right triangles. We can use the Pythagorean theorem to relate the lengths of the sides of the rectangle and the diagonal.

 

The Pythagorean theorem states: a^2 + b^2 = c^2, where a and b are the lengths of the shorter sides (length and width in this case), and c is the length of the hypotenuse (diagonal). In this case, we are given c (diagonal) as 16.

 

We can approach this problem in two ways:

 

Method 1: Solve for l and w directly

 

From the perimeter equation, we can rewrite it to isolate "l": l = (40 - 2w) / 2.

 

Substitute this expression for "l" in the Pythagorean theorem equation: [(40 - 2w) / 2]^2 + w^2 = 16^2.

 

Solve this equation for "w" (tedious but solvable). Once you find "w", you can plug it back into the equation for "l" to find the length.

 

Method 2: Exploit the relationship between l and w using the diagonal

 

We know the diagonal is 16, and a rectangle's diagonals cut each other in half, creating right triangles with legs equal to half the length and half the width.

 

Therefore, in each right triangle formed by the diagonal, one leg (half the length) is 8 (half of 16).

 

Since a rectangle has opposite sides equal, the other leg of the right triangle (half the width) must also be 8.

 

Now we know that half the width (w/2) is 8. Solve for the actual width (w) by multiplying by 2: w = 16.

 

Finding the Area

 

Once you have the width (w = 16 from Method 2), you can find the length (l) using the perimeter equation (l = (40 - 2 * 16) / 2 = 4).

 

Now that you know both the length (l = 4) and width (w = 16), the area of the rectangle can be calculated using the formula: Area = l * w = 4 * 16 = 64 square units.

 

Therefore, the area of the rectangle is 64 square units.