[GMAT math practice question]
(Geometry) We have a square □ABCD, and △AEF is a triangle inscribed in □ABCD. What is the length of BE?
1) □ABCD is a square with AB = 10
2) △AEF is an equilateral triangle.
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Since we have a triangle and a square, we have 3 variables and 1 variable for the triangle and the square, respectively. Since we have 4 variables and 0 equations, E is most likely the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.
Conditions 1) & 2)
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Assume BE = x (0 < x < 10).
Then we have BE = DF since the triangle AEF is equilateral.
AE^2 = 100 + x^2 = EF^2 = (10 - x)^2 + (10 - x)^2.
Then we have
100 + x^2 = (10 - x)^2 + (10 - x)^2
100 + x^2 = 2(10 – x)^2 (adding like terms)
100 + x^2 = 2(10 – x)(10 – x)
100 + x^2 = 2(100 – 10x – 10x + x^2) (foiling out the brackets)
100 + x^2 = 2x^2 – 40x + 200 (multiplying 2 through the bracket and combining like terms)
x^2 – 40x + 100 = 0 (bringing all terms to one side)
Thus, we have x = 20 ± 10√3. (using the quadratic formula)
But we have x = 20 - 10√3 since 0 < x < 10.
Since both conditions together yield a unique solution, they are sufficient.
Therefore, C is the answer.
Answer: C
In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2) when taken together. Obviously, there may be occasions on which the answer is A, B, C, or D.
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