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Triangle BDC is an equilateral triangle
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04 Jun 2021, 10:23
04 Jun 2021, 10:23
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Triangle BDC is an equilateral triangle and triangle ABC is a right triangle. If line segment DC is 6 units in length, what is the area of the shaded region?
A. 9
B. \(9\sqrt{3}\)
C. 18
D. \(18\sqrt{2}\)
E. \(18\sqrt{3}\)
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITION
GRE - Math Book
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Re: Triangle BDC is an equilateral triangle
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04 Jun 2021, 10:33
04 Jun 2021, 10:33
Carcass wrote:
Triangle BDC is an equilateral triangle and triangle ABC is a right triangle. If line segment DC is 6 units in length, what is the area of the shaded region?
A. 9
B. \(9\sqrt{3}\)
C. 18
D. \(18\sqrt{2}\)
E. \(18\sqrt{3}\)
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITION
GRE - Math Book
Since ∆BDC is an equilateral triangle, and since ∆ABC is a right triangle, we can add the following to our diagram.
Next, if we add the following perpendicular line . . .
. . . we can add more to our diagram
And we can add this.
At this point, we can simplify matters A LOT by recognizing that if we "cut" out the top triangle and rotate it . . .
And keep rotating . . .
. . . it fits nicely here.
At this point, we should see that the shaded area is actually a nifty EQUILATERAL triangle with sides of length 6
Area of an equilateral triangle = (√3)(side²/4)
So, the area of the shaded triangle = (√3)(6²/4)
= (√3)(36/4)
= 9√3
Answer: B
Cheers,
Brent
Triangle BDC is an equilateral triangle
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04 Jun 2021, 10:50
04 Jun 2021, 10:50
As BCD is an equilateral triangle, so BC=CD=DB=6, and all its angles = 60 degrees.
Its area= √3(side^2)/4 = √3*(6*6)/4 = 9√3
In triangle ADB, angle ADB = 180 - 60 = 120, so it is the largest angle of the triangle.
So the opposite side AB must be greater than side BD and therefore, BC.
Notice that triangle ABC is a 30-60-90 triangle, with BC=6 as the smallest side.
So side AB= √3* BC = 6√3
Area of triangle ABC = 0.5*6*6√3 = 18√3
Now, the area of shaded region= Triangle ABC - Triangle BCD = 18√3 - 9√3= 9√3
Hope this helps!
Its area= √3(side^2)/4 = √3*(6*6)/4 = 9√3
In triangle ADB, angle ADB = 180 - 60 = 120, so it is the largest angle of the triangle.
So the opposite side AB must be greater than side BD and therefore, BC.
Notice that triangle ABC is a 30-60-90 triangle, with BC=6 as the smallest side.
So side AB= √3* BC = 6√3
Area of triangle ABC = 0.5*6*6√3 = 18√3
Now, the area of shaded region= Triangle ABC - Triangle BCD = 18√3 - 9√3= 9√3
Hope this helps!
