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Geometry Difficulty 2.6 Junior Find the answer

If the length of a diagonal of a square is a+ba + b, then the area of the square is:
(A) (a+b)2\mathrm{(A) \ (a+b)^2 }(B) 12(a+b)2\mathrm{(B) \ \frac{1}{2}(a+b)^2 }(C) a2+b2\mathrm{(C) \ a^2+b^2 }(D) 12(a2+b2)\mathrm{(D) \ \frac {1}{2}(a^2+b^2) }(E) none of these\mathrm{(E) \ \text{none of these} }

Multiple choice: answer with the letter of the option you want.

Solution

Let a side be ss; then by the Pythagorean Theorem, s2+s2=2s2=(a+b)2s^2 + s^2 = 2s^2 = (a+b)^2. The area of a square is s2=(a+b)22⇒(B)s^2 = \frac{(a+b)^2}{2} \Rightarrow \mathrm{(B)}.
Alternatively, using the area formula for a kite, the area is 12d1d2=12(a+b)2\frac{1}{2}d_1d_2 = \frac{1}{2}(a+b)^2.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.