Full solution

Q. The length of a rectangle's diagonal is

2 \sqrt{29}

, and the length of the longer side is

10

. What is the area of the rectangle?

Find Rectangle Area: To find the area of the rectangle, we need to know both the length and the width. We already have the length (the longer side), which is $10$ . We can use the Pythagorean theorem to find the width, since we have the length of the diagonal. $\newline$ The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse ( $c$ ) is equal to the sum of the squares of the lengths of the other two sides ( $a$ and $b$ ). For a rectangle, the diagonal acts as the hypotenuse, and the sides of the rectangle are the other two sides of the triangle. $\newline$ The formula is: $c^2 = a^2 + b^2$ $\newline$ Here, $c$ is the diagonal, $a$ is the length, and $b$ is the width.
Use Pythagorean Theorem: First, let's plug in the values we know into the Pythagorean theorem. The diagonal $c$ is $2\sqrt{29}$ , and the length $a$ is $10$ . $(2\sqrt{29})^2 = 10^2 + b^2$ Now, let's calculate the square of the diagonal. $(2\sqrt{29})^2 = (2^2) \cdot (\sqrt{29})^2 = 4 \cdot 29$
Calculate Diagonal Square: Calculate the square of the diagonal: $\newline$ $4 \times 29 = 116$ $\newline$ So, we have: $\newline$ $116 = 10^2 + b^2$ $\newline$ Now, calculate the square of the length $10^2$ : $\newline$ $10^2 = 100$
Calculate Width Square: Subtract the square of the length from the square of the diagonal to find the square of the width $b^2$ : $116 - 100 = b^2$ $16 = b^2$ Now, find the width $b$ by taking the square root of both sides: $\sqrt{16} = b$ $b = 4$
Calculate Rectangle Area: Now that we have both the length and the width of the rectangle, we can calculate the area. The area $A$ of a rectangle is given by the formula: $\newline$ $A = \text{length} \times \text{width}$ $\newline$ $A = 10 \times 4$
Calculate Rectangle Area: Now that we have both the length and the width of the rectangle, we can calculate the area. The area $A$ of a rectangle is given by the formula: $\newline$ $A = \text{length} \times \text{width}$ $\newline$ $A = 10 \times 4$ Calculate the area of the rectangle: $\newline$ $A = 10 \times 4 = 40$ $\newline$ The area of the rectangle is $40$ square units.