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705-805 (Hard)|   Geometry|      
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Pritishd
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The shaded portion represents what percent of the area? 31 Aug 2020, 08:01
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

75% (hard)

Question Stats:

44% (01:28) correct 56% (01:30) wrong based on 119 sessions

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In the figure below, the shaded portion represents what percent of the area of the triangle ABC?

1. D, E, F are midpoints of the sides of triangle
2. Triangle ABC is a equilateral triangle
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A

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The shaded portion represents what percent of the area? 31 Aug 2020, 10:07
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How is the answer for this A?
Shouldn't the triangle be equilateral as well to establish a proportion. Otherwise everything remains relative and dependant on the length of each side
Bunuel

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yashikaaggarwal
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The shaded portion represents what percent of the area? 31 Aug 2020, 10:13
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adibapat its not necessary for the triangle to be of equal sides. When 3 mid points of any triangle are connected, it divide triangle into 4 equal part by congruency of triangle. so the shaded region is 1/2 area of the whole triangle. you can draw any triangle by measure to justify the theory- scalene, isosceles, or equilateral triangle.
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The shaded portion represents what percent of the area? 31 Aug 2020, 10:42
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Statement 1-
Area of DEBF = (EB) * FB* (sin B) = 1/2 *BC *1/2*BA* sin B = 1/2 * [ 1/2 *BC*BA* sin B] = 1/2 * area of ABC

Sufficient

Statement 2- We have no idea about \(\frac{EB}{BC}\) and \(\frac{FB}{BA}\)

Insufficient


Pritishd
In the figure below, the shaded portion represents what percent of the area of the triangle ABC?

1. D, E, F are midpoints of the sides of triangle
2. Triangle ABC is a equilateral triangle
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The shaded portion represents what percent of the area? 31 Aug 2020, 12:11
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adibapat
How is the answer for this A?
Shouldn't the triangle be equilateral as well to establish a proportion. Otherwise everything remains relative and dependant on the length of each side
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Statement 1:
Attachment:
triangle BEF.png
triangle BEF.png [ 59.2 KiB | Viewed 4593 times ]
In the figure above, E is the midpoint of BC and F is the midpoint of AB.

TRIANGLES ABC AND BEF:
Triangles ABC and BEF share angle EFB.
Shared angle EFB is formed by sides in the same ratio (BC:BE = 2:1, AB:BF = 2:1).
Rule:
Triangles with a shared angle (EBF) formed by corresponding sides in the same ratio (BC:BE = 2:1, AB:BF = 2:1) are SIMILAR.
Thus, triangle ABC is similar to triangle BEF.

RULE FOR TWO SIMILAR TRIANGLES:
If each side in the larger triangle is \(k\) times the corresponding side in the smaller triangle, then the area of the larger triangle is \(k^2\) times the area of the smaller triangle.
Here, since each side of ABC is 2 times the corresponding side in BEF, the area of ABC is 4 times the area of BEF.
Thus, \(BEF = \frac{1}{4}ABC\).

The same line of reasoning can be used to show that \(ADF = \frac{1}{4}ABC\) and that \(CDE = \frac{1}{4}ABC\), with the result that \(DEF = \frac{1}{4}ABC\).
Thus:
shaded region \(= DEF + BEF = \frac{1}{4}ABC + \frac{1}{4}ABC = \frac{1}{2}ABC = 50\)%
SUFFICIENT.
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The shaded portion represents what percent of the area? 19 Dec 2022, 01:05
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