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AB is the Diameter and point O is the center of the circle shown above
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05 Oct 2016, 06:05
05 Oct 2016, 06:05
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Question Stats:
51% (02:24) correct
48% (02:38) wrong based on 286 sessions
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AB is the Diameter and point O is the center of the circle shown above
Quantity A |
Quantity B |
Area of Triangle ABC |
Twice the Area of Triangle OBD |
- A : Quantity A is greater
- B : Quantity B is greater
- C : Both quantities are equal
- D : Answer Cannot be determined
Re: AB is the Diameter and point O is the center of the circle shown above
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05 Oct 2016, 12:20
05 Oct 2016, 12:20
6
Expert Reply
Hey,
Let radius of the triangle be r.
Now since AB is the diameter of the circle, triangle ABC is a right angle triangle. And from triangle properties we know value of angle \(x = 30\) degrees.
Thus triangle OBD is a equilateral triangle.
Now AC= \(2*r*cos(x)\)=\(2*r*cos(30)\)=\(2*r*\sqrt{3}*\frac{1}{2}\)=\(\sqrt{3}r\).
And BC= \(2*r*sin(x)\)=\(2*r*sine(30)\)=\(2*r*\frac{1}{2}\)=\(r\)
Area of traingle ABC= \(\frac{1}{2}\)*AC*BC=\(\frac{1}{2}\sqrt{3}r^2\).
Now Area of the equilateral triangle OBD= \(\frac{1}{4}\sqrt{3}r^2\).
Clearly 2* Area of OBD= Area of ABC.
Hence C is the correct option.
Let radius of the triangle be r.
Now since AB is the diameter of the circle, triangle ABC is a right angle triangle. And from triangle properties we know value of angle \(x = 30\) degrees.
Thus triangle OBD is a equilateral triangle.
Now AC= \(2*r*cos(x)\)=\(2*r*cos(30)\)=\(2*r*\sqrt{3}*\frac{1}{2}\)=\(\sqrt{3}r\).
And BC= \(2*r*sin(x)\)=\(2*r*sine(30)\)=\(2*r*\frac{1}{2}\)=\(r\)
Area of traingle ABC= \(\frac{1}{2}\)*AC*BC=\(\frac{1}{2}\sqrt{3}r^2\).
Now Area of the equilateral triangle OBD= \(\frac{1}{4}\sqrt{3}r^2\).
Clearly 2* Area of OBD= Area of ABC.
Hence C is the correct option.
General Discussion
Re: AB is the Diameter and point O is the center of the circle shown above
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09 Oct 2016, 01:20
09 Oct 2016, 01:20
2
I personally recommend to avoid using Sine,Cosine and other trigonometric function to solve geometry related problems as other easier ways seem more handy and quick.Here is my solution:
<BCA is an inscribe angle and AB is the diameter,so <BCA = 90 degree.So as ABC is a right triangle so we can use Pythagorean Theorem on this triangle.Since sum of the interior angles of any triangle is equal to 180 degrees therefore
m<BCA+x+2x = 180 => 90+3x = 180 => x = 30 => Triangle ABC is a 30-60-90 triangle and we may use properties of 30-60-90 triangle to solve this question.Sides of a 30-60-90 triangle are in the ratio 1:\(sqrt3\):2.Since hypotenuse of Triangle ABC is equal to Diameter AB = 2r of the circle;where r is radius of the circle.So the sides of Triangle ABc are r,r\sqrt{3} and 2r.
=> Quantity A : Area of Triangle ABC = (1/2)*(base)*(height) = (1/2)*(BC)*(AC) = = (1/2)*(r)*(r\(sqrt3\)) = \((sqrt3/2)*r^2\)
Since 2x = 2*30 = 60 so Triangle OBD is an equilateral triangle and formula for area of an equilateral triangle is \((sqrt3/4)*s^2\),where s is the side of the equilateral triangle.Side of equilateral Triangle OBD is equal to radius r of the circle
Quantity B : Twice the Area of Triangle ABC = \(2*(sqrt3/4)*r^2\) = \((sqrt3/2)*r^2\) which is equal to Quantity => Quantity A = Quantity B.Hence answer is C.
This question has been taken from tests of Greatest Prep.Here is the link to the website of Greatest Prep : http://www.greatestprep.com
<BCA is an inscribe angle and AB is the diameter,so <BCA = 90 degree.So as ABC is a right triangle so we can use Pythagorean Theorem on this triangle.Since sum of the interior angles of any triangle is equal to 180 degrees therefore
m<BCA+x+2x = 180 => 90+3x = 180 => x = 30 => Triangle ABC is a 30-60-90 triangle and we may use properties of 30-60-90 triangle to solve this question.Sides of a 30-60-90 triangle are in the ratio 1:\(sqrt3\):2.Since hypotenuse of Triangle ABC is equal to Diameter AB = 2r of the circle;where r is radius of the circle.So the sides of Triangle ABc are r,r\sqrt{3} and 2r.
=> Quantity A : Area of Triangle ABC = (1/2)*(base)*(height) = (1/2)*(BC)*(AC) = = (1/2)*(r)*(r\(sqrt3\)) = \((sqrt3/2)*r^2\)
Since 2x = 2*30 = 60 so Triangle OBD is an equilateral triangle and formula for area of an equilateral triangle is \((sqrt3/4)*s^2\),where s is the side of the equilateral triangle.Side of equilateral Triangle OBD is equal to radius r of the circle
Quantity B : Twice the Area of Triangle ABC = \(2*(sqrt3/4)*r^2\) = \((sqrt3/2)*r^2\) which is equal to Quantity => Quantity A = Quantity B.Hence answer is C.
This question has been taken from tests of Greatest Prep.Here is the link to the website of Greatest Prep : http://www.greatestprep.com
