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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? A) 116 B) 20sqrt29 C) 10sqrt29 D) 40

The length of a rectangle's diagonal is 229 2 \sqrt{29} , and the length of the longer side is 1010 . What is the area of the rectangle?\newlineA) 116116\newlineB) 2029 20 \sqrt{29} \newlineC) 1029 10 \sqrt{29} \newlineD) 4040

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Full solution

Q. The length of a rectangle's diagonal is 229 2 \sqrt{29} , and the length of the longer side is 1010 . What is the area of the rectangle?\newlineA) 116116\newlineB) 2029 20 \sqrt{29} \newlineC) 1029 10 \sqrt{29} \newlineD) 4040
  1. Identify Rectangle Dimensions: To find the area of the rectangle, we need to know both the length and the width. We already know the length is 1010. We can use the Pythagorean theorem to find the width, since the diagonal (dd), length (ll), and width (ww) of a rectangle form a right triangle: d2=l2+w2d^2 = l^2 + w^2.
  2. Calculate Diagonal Squared: We are given that the diagonal dd is 2292\sqrt{29}. Let's square this to find d2d^2: (229)2=4×29=116(2\sqrt{29})^2 = 4 \times 29 = 116.
  3. Calculate Length Squared: We know the length ll is 1010, so let's square this to find l2l^2: 102=10010^2 = 100.
  4. Substitute into Pythagorean Theorem: Now we can substitute d2d^2 and l2l^2 into the Pythagorean theorem to find w2w^2: 116=100+w2116 = 100 + w^2.
  5. Solve for Width: Subtract 100100 from both sides to solve for w2w^2: 116−100=w2116 - 100 = w^2, so w2=16w^2 = 16.
  6. Find Width: Take the square root of both sides to find ww: w2=16\sqrt{w^2} = \sqrt{16}, so w=4w = 4.
  7. Calculate Area: Now that we have both the length (1010) and the width (44), we can find the area of the rectangle by multiplying the length by the width: Area=length×width=10×4\text{Area} = \text{length} \times \text{width} = 10 \times 4.
  8. Final Area Calculation: Calculate the area: 10×4=4010 \times 4 = 40.

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