GRE Prep Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
AB is the Diameter and point O is the center of the circle shown above
[#permalink]
05 Oct 2016, 06:05
05 Oct 2016, 06:05
16
2
19
Bookmarks
00:00
Question Stats:
51% (02:23) correct
48% (02:38) wrong based on 288 sessions
Hide Show timer Statistics
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
[#permalink]
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
[#permalink]
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
