The length of a rectangle's diagonal is 2sqrt29, and the length of the longer side is 10. What is the area of the rectangle?

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The length of a rectangle's diagonal is  2sqrt29, and the length of the longer side is 10. What is the area of the rectangle?

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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. 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. The formula is: c^2 = a^2 + b^2 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 2sqrt29, and the length (a) is 10. (2sqrt29)^2 = 10^2 + b^2 Now, let's calculate the square of the diagonal. (2sqrt29)^2 = (2^2) * (sqrt29)^2 = 4 * 29Calculate Diagonal Square: Calculate the square of the diagonal: 4 * 29 = 116 So, we have: 116 = 10^2 + b^2 Now, calculate the square of the length (10^2): 10^2 = 100Calculate 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 = 4Calculate 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: A = length * width A = 10 * 4Calculate the area of the rectangle: A = 10 * 4 = 40 The area of the rectangle is 40 square units.

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?

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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?