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In a parallelogram ABCD, if the height from A on CD is equal to 20, then what is the perimeter of the parallelogram?
(1) Area of the parallelogram = \(\frac{800}{\sqrt{3}}\)
(2) Largest angle in triangle ABC is equal to \(60^{\circ}\)
(1) Area of the parallelogram = \(\frac{800}{\sqrt{3}}\)
(2) Largest angle in triangle ABC is equal to \(60^{\circ}\)
20 Jun 2023, 12:36
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In parallelogram ABCD, height_A = 20.
We need to answer the question:
perimeter_ABCD = ?
AB + BC + CD + AD = ?
Statement One Alone:
=> Area of the parallelogram = 800/√3
Since the area of a parallelogram is base × height, we have:
AB × 20 = 800/√3
AB = 40/√3, and thus CD = 40/√3 (because in any parallelogram, opposite sides are always equal).
We cannot determine the values of BC and AD, but we can make the following inference:
BC = AD ≥ height_A
BC = AD ≥ 20
Clearly, we can’t determine the perimeter of the parallelogram. Statement one is not sufficient. Eliminate answer choices A and D.
Statement Two Alone:
=> Largest angle in triangle ABC is equal to 60 degrees.
The sum of the other two angles in triangle ABC must be equal to 180 – 60 = 120 degrees. In the only valid case, both of these other angles are equal to 60 degrees, since none of them can be greater than 60 degrees and their sum must be equal to 120 degrees.
Therefore, triangle ABC is an equilateral triangle, so AB = BC = AC. Since in any parallelogram, opposite sides are always equal, it must be true that AB = BC = CD =AD.
We see that triangle ACD is equilateral as well. Since height_A (20) of parallelogram ABCD is also the height_A (20) of equilateral triangle ACD, we could determine the length of side CD, which would help us to determine the perimeter in question:
perimeter_ABCD = 4 × CD
Statement two is sufficient.
Answer: B
Note: In any equilateral triangle, height = [(√3)/2] × base. In statement two, we know that in equilateral triangle ACD, height_ A = 20, so:
[(√3)/2] × CD = 20
CD = 40/√3, and thus perimeter_ABCD = 4 × 40/√3 = 160/√3
