Lesson: Circle Properties

Comment on Circle Properties

Sir the last question from bottom (Related sources: http://gre.myprepclub.com/forum/the-diameter-of-the-circle-is-3073.html)
There is a quadrilateral inscribed in a circle. The diameter is 10. I solved the question using the 2nd and 3rd property mentioned in the video.

2nd property : Inscribed angles holding chords are equal in length
3rd property : Inscribed angle holding a diameter is 90 degree or right angle triangle.

I solved the question by drawing an imaginary line or diameter from point C to A. In this way I got two equal right angle triangles i.e ABC and ADC.
Solving the area for one triangle and then multiplying it by 2, we will get the area for the whole quadrilateral.

Area of triangle ABC : 1/2(BasexHeight)
We know the length of line segment AC ( which is diameter) 10, but we don't know the length of AB and BC.
we can find it by using Pythagorian triple i.e
3-4-5 => 6-8-10
length of AB and BC is receptively 8 and 6.

Area of triangle ABC = 1/2(8*6) = 24
Area of Quadrilateral = 2x24 = 48.

QA>QB therfore A is answer, but the answer is D. I don't understand.
I looked through the explanation, the person kept changing the Quadrilateral shape and size and concluded NO relation can be established. How can he just change the shape and size of Quadrilateral like that? I don't understand the logic. Please help :)
greenlight-admin's picture

You're referring to this question: http://gre.myprepclub.com/forum/the-diameter-of-the-circle-is-3073.html

You made two BIG assumptions in your solution.

First, you assumed that AC is the diameter of the circle, but we can't make that assumption. Sure, it LOOKS like it might be the diameter, but looks can be deceiving :-)

Second, you took a triangle (which you assumed to be a right triangle), and given only one side length (hypotenuse = 10), you concluded that the other two sides must have lengths 6 and 8. This is another incorrect assumption. For example, the 3 sides could have lengths 5, 5√3 and 10 or the 3 sides could have lengths 5√2, 5√2 and 10, and so on.

I hope that helps.


can we generalize that, if 4 side polygon inscribed in a circle. Max area covered must be by a square.
greenlight-admin's picture

I didn't understand either 5th question in practice link in which quadrilateral is inscribed in a circle. Can we change quadrilateral's shape? How to solve it with only diameter 10 given. I don't see any other info
greenlight-admin's picture

Question link: http://gre.myprepclub.com/forum/the-diameter-of-the-circle-is-3073.html

The question doesn't tell us anything about the shape of the quadrilateral. All we can conclude is that ABCD is a quadrilateral that has all 4 points ON the circle - that's it.

Since ABCD is a quadrilateral, it COULD be a square (in which case the area of ABCD is MORE THAN 40).

Or it COULD be super thin rectangle, (in which case the area of ABCD is LESS THAN 40).

For more on the assumptions we can and can't make regarding Geometric diagrams, watch: https://www.greenlighttestprep.com/module/gre-geometry/video/863


Wouldn't it be more helpful if it was mentioned whether AB was a cord or not? It says that the figure is not drawn to scale, I was confused whether to assume AB as cord or diameter
greenlight-admin's picture

Question link: https://gre.myprepclub.com/forum/topic10093.html

AB is definitely not the diameter, but we have to perform several calculations, before we actually know this to be the case.

So, a large part of the question is determining whether AB is the diameter.

Does that help?


Why divide by 8pie-16 by 4
greenlight-admin's picture

Question link: https://gre.myprepclub.com/forum/in-the-figure-above-if-the-square-inscr...

From Sandy's calculations, we get:
Area of circle = 8π
Area of square = 16

So, the area of ALL FOUR partial circles = 8π - 16
However, the question asks for the area of ONE partial circle.
So, we must divide 8π - 16 by 4.


Kindly solve it Brent
greenlight-admin's picture

can the radius be smaller than the chord
greenlight-admin's picture

Yes, the radius can be shorter than a chord.
Here's an example: https://imgur.com/lqXPAop



This is one hell of a great exercise!
greenlight-admin's picture

Agreed! It certainly has the potential to eat up a lot of time on test day!

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