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The measure of one angle in an isosceles triangle is 120°. If the leng
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19 Mar 2021, 06:51
19 Mar 2021, 06:51
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The measure of one angle in an isosceles triangle is 120°. If the length of the side opposite this angle equals \(12 \sqrt{3}\) , then what is the perimeter of the triangle?
A. \(18 + 12 \sqrt{3}\)
B. \(21 + 12 \sqrt{3}\)
C. \(24 + 12 \sqrt{3}\)
D. \(30 + 12 \sqrt{3}\)
E. \(36 + 12 \sqrt{3}\)
Kudos for the right answer and explanation
Question part of the project GRE Quantitative Reasoning Daily Challenge - (2021) EDITION
GRE - Math Book
A. \(18 + 12 \sqrt{3}\)
B. \(21 + 12 \sqrt{3}\)
C. \(24 + 12 \sqrt{3}\)
D. \(30 + 12 \sqrt{3}\)
E. \(36 + 12 \sqrt{3}\)
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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The measure of one angle in an isosceles triangle is 120°. If the leng
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19 Mar 2021, 07:47
19 Mar 2021, 07:47
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Carcass wrote:
The measure of one angle in an isosceles triangle is 120°. If the length of the side opposite this angle equals \(12 \sqrt{3}\) , then what is the perimeter of the triangle?
A. \(18 + 12 \sqrt{3}\)
B. \(21 + 12 \sqrt{3}\)
C. \(24 + 12 \sqrt{3}\)
D. \(30 + 12 \sqrt{3}\)
E. \(36 + 12 \sqrt{3}\)
A. \(18 + 12 \sqrt{3}\)
B. \(21 + 12 \sqrt{3}\)
C. \(24 + 12 \sqrt{3}\)
D. \(30 + 12 \sqrt{3}\)
E. \(36 + 12 \sqrt{3}\)
Let ABC be that isosceles triangle where BC = \(12 \sqrt{3}\) and Angle BAC = 120°
Angle ABC + Angle BAC + Angle ACB = 180
2(Angle ABC) + 120° = 180°
Angle ABC = Angle ACB = 30°
Drop a Perpendicular from A to BC and name the point as D, such that BD = DC = \(6 \sqrt{3}\)
Angle ABC = 30°
Angle BAC = 60°
Angle BDA = 90°
In 30°-60°-90° triangle, the ratio of the sides is \(x : \sqrt{3}x : 2x \)
\(\sqrt{3}x = 6 \sqrt{3}\)
\(x = 6\)
Therefore,
BD = DC = 6
AB = AC = 12
Perimeter = \(12 + 12 + 12 \sqrt{3} = 24 + 12 \sqrt{3}\)
Hence, option C
General Discussion
Re: The measure of one angle in an isosceles triangle is 120°. If the leng
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24 Mar 2021, 03:36
24 Mar 2021, 03:36
An important property of an isoceles triangle is that the perpendicular drawn from the vertex between equal sides to the unequal side divides the angle at the vertex into two.
Let the isoceles triangle be ABC, with Angle A being 120 degrees and Angle B= Angle C=30 degrees
Now we draw a perpendicular from vertex A to side BC at point D, resulting in 2 right angle triangles ADB and ADC
Hence the Angle A will be split into two parts of 60 degrees each when we draw a perpendicular.
Applying some basic trigonometry in the resulting triangle ADB
cos30=\(\frac{\sqrt{3}}{2}\)=\(\frac{6\sqrt{3}}{AB}\)
Hence side AB=12
Similarly side AC=12
Perimeter= AB+BC+AC= \(24+12\sqrt{3}\)
Let the isoceles triangle be ABC, with Angle A being 120 degrees and Angle B= Angle C=30 degrees
Now we draw a perpendicular from vertex A to side BC at point D, resulting in 2 right angle triangles ADB and ADC
Hence the Angle A will be split into two parts of 60 degrees each when we draw a perpendicular.
Applying some basic trigonometry in the resulting triangle ADB
cos30=\(\frac{\sqrt{3}}{2}\)=\(\frac{6\sqrt{3}}{AB}\)
Hence side AB=12
Similarly side AC=12
Perimeter= AB+BC+AC= \(24+12\sqrt{3}\)
