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# A quadrilateral is inscribed in a circle. If the

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Updated on: 02 Feb 2012, 21:44
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54% (01:43) correct 46% (01:55) wrong based on 195 sessions

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A quadrilateral is inscribed in a circle. If the circumference of the circle is 5(pi), what is the area of the quadrilateral?

(1) The length of one of the diagonals of the quadrilateral is 5.
(2) The length of two opposite sides are 3 and 4 respectively.

Originally posted by Smita04 on 02 Feb 2012, 06:30.
Last edited by Smita04 on 02 Feb 2012, 21:44, edited 1 time in total.
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02 Feb 2012, 09:00
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I think the answer should be C. This one is difficult to visualize without drawing. We know the diameter of the circle is 5 and quadrilateral is inscribed

1. Since diagonal is 5, that means the quadrilateral is divided into two triangles with diagonal as the diameter of the circle. Now since they are inscribed. that means the angle both of these form with the Circumference is a right angle. But we do not know the sizes of the sides. Insufficient

2. I can draw various quadrilaterals which are inscribed and have opposite sides as 3 and 4. All of these will have different areas so insufficient.

Now combined I know that one side of one right angled triangle is 3 and of the other right angled triangle is 4 so the remaining two sides will be in the same ratio as well from the pythagorean theorem. Now that we know all 4 sides and the heights (since diagonal is 5 and is the hypotenuse of the right-angled triangle for both triangles that form the quadrilateral. Hence answer should be C.

Whats the OA?
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27 May 2017, 19:46
would someone please elaborate on why (2) is NS? I'm having trouble visualizing it. I thought the answer was B b/c we know the radius = 2.5
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27 May 2017, 22:36
omerrauf wrote:
I think the answer should be C. This one is difficult to visualize without drawing. We know the diameter of the circle is 5 and quadrilateral is inscribed

1. Since diagonal is 5, that means the quadrilateral is divided into two triangles with diagonal as the diameter of the circle. Now since they are inscribed. that means the angle both of these form with the Circumference is a right angle. But we do not know the sizes of the sides. Insufficient

2. I can draw various quadrilaterals which are inscribed and have opposite sides as 3 and 4. All of these will have different areas so insufficient.

Now combined I know that one side of one right angled triangle is 3 and of the other right angled triangle is 4 so the remaining two sides will be in the same ratio as well from the pythagorean theorem. Now that we know all 4 sides and the heights (since diagonal is 5 and is the hypotenuse of the right-angled triangle for both triangles that form the quadrilateral. Hence answer should be C.

Whats the OA?

The question stem states that the circumference is 5 and not the diameter.
Does that have any impact on the reasoning?
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27 May 2017, 22:51
I believe it says the circumference is 5(pie), meaning 2(pie)r = 5(pie) --> radius = 2.5 / diameter = 5

nainy05 wrote:
omerrauf wrote:
I think the answer should be C. This one is difficult to visualize without drawing. We know the diameter of the circle is 5 and quadrilateral is inscribed

1. Since diagonal is 5, that means the quadrilateral is divided into two triangles with diagonal as the diameter of the circle. Now since they are inscribed. that means the angle both of these form with the Circumference is a right angle. But we do not know the sizes of the sides. Insufficient

2. I can draw various quadrilaterals which are inscribed and have opposite sides as 3 and 4. All of these will have different areas so insufficient.

Now combined I know that one side of one right angled triangle is 3 and of the other right angled triangle is 4 so the remaining two sides will be in the same ratio as well from the pythagorean theorem. Now that we know all 4 sides and the heights (since diagonal is 5 and is the hypotenuse of the right-angled triangle for both triangles that form the quadrilateral. Hence answer should be C.

Whats the OA?

The question stem states that the circumference is 5 and not the diameter.
Does that have any impact on the reasoning?
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27 May 2017, 23:01
Apologies for the overlook. Thanks for pointing out!
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13 Jun 2017, 03:20
Why can't B be the answer?

It is given that circumference is 5 pi... that means the diameter is 5.. thus from B we get 2 triangles with measures 3, 4 and 5... Thus area of quad = Area of both triangles = 2 (1/2*3*4) = 12..
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02 Jul 2017, 03:03
Why can't B be the answer?

It is given that circumference is 5 pi... that means the diameter is 5.. thus from B we get 2 triangles with measures 3, 4 and 5... Thus area of quad = Area of both triangles = 2 (1/2*3*4) = 12..

Nothing is mentioned about the other two sides. What is this is a trapezium?

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02 Jul 2017, 03:06
omerrauf wrote:
I think the answer should be C. This one is difficult to visualize without drawing. We know the diameter of the circle is 5 and quadrilateral is inscribed

1. Since diagonal is 5, that means the quadrilateral is divided into two triangles with diagonal as the diameter of the circle. Now since they are inscribed. that means the angle both of these form with the Circumference is a right angle. But we do not know the sizes of the sides. Insufficient

2. I can draw various quadrilaterals which are inscribed and have opposite sides as 3 and 4. All of these will have different areas so insufficient.

Now combined I know that one side of one right angled triangle is 3 and of the other right angled triangle is 4 so the remaining two sides will be in the same ratio as well from the pythagorean theorem. Now that we know all 4 sides and the heights (since diagonal is 5 and is the hypotenuse of the right-angled triangle for both triangles that form the quadrilateral. Hence answer should be C.

Whats the OA?

So, if it is a right triangle with hypotenuse 5 and the height (drawn from the right angle to hypotenuse) being 2.5 (=radius)
Area of triangle = (1/2) *base* height = (1/2) * 5*2.5

Area of quadilateral = 2*Area of triangle
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02 Jul 2017, 09:57
Smita04 wrote:
A quadrilateral is inscribed in a circle. If the circumference of the circle is 5(pi), what is the area of the quadrilateral?

(1) The length of one of the diagonals of the quadrilateral is 5.
(2) The length of two opposite sides are 3 and 4 respectively.

if diagonal = diameter, quadrilateral is a rectangle with hypotenuse=5 ==> 3-4-5 right triangle?? so sides = 3 &4 --> Area can be calculated?
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03 Jul 2017, 05:22
Smita04 wrote:
A quadrilateral is inscribed in a circle. If the circumference of the circle is 5(pi), what is the area of the quadrilateral?

(1) The length of one of the diagonals of the quadrilateral is 5.
(2) The length of two opposite sides are 3 and 4 respectively.

The correct answer is C here although initially I thought its B. Here's why C is correct -

WKT $$2\pi$$R = $$5\pi$$
Hence diameter = 5

Fact 1: One of the diagonals is 5 - Clearly insuff [we only know that one of the diagonal is diameter]
Fact 2: Length of two opp side is 3 and 4 - We cannot determine the area only with this info since we dont know what other two opp sides measure, accordingly the area will change. Insuff

Fact 1 + Fact 2 : WKT that the diagonal measures 5 [remember the property that the triangle formed with one of the sides as diameter is a right angled triangle]. So we will have two right angled triangle with 3,4,5 measurement. Hence suff

Hope this helps!
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03 Jul 2017, 08:38
This question seems incorrect...
Statement 2 reads : The length of two opposite sides are 3 and 4 respectively.
shouldn't this be changed to the length of 2 adjacent sides ??
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03 Jul 2017, 08:50
darn wrote:
This question seems incorrect...
Statement 2 reads : The length of two opposite sides are 3 and 4 respectively.
shouldn't this be changed to the length of 2 adjacent sides ??

No, the question is correct.
Fact 2 does mean two opp sides and not adjacent. Even it were adjacent sides the data is still insuff without fact 1, which states one of diagonals is 5 [this would be the diameter] and only when we know that the diagonal is diameter we can deduce that the triangle is right angled
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03 Jul 2017, 10:53
Sash143 I made the silly mistake of assuming that the quadrilateral formed was a rectangle. Which is why i asked the question, since opposite sides of a rectangle cannot be unequal.
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03 Jul 2017, 13:52
Smita04 wrote:
A quadrilateral is inscribed in a circle. If the circumference of the circle is 5(pi), what is the area of the quadrilateral?

(1) The length of one of the diagonals of the quadrilateral is 5.
(2) The length of two opposite sides are 3 and 4 respectively.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

A quadrilateral is inscribed in a circle. If the circumference of the circle is $$5\pi$$, what is the area of the quadrilateral?

(1) The length of one of the diagonals of the quadrilateral is $$5$$.
(2) The length of two opposite sides are $$3$$ and $$4$$ respectively.

In order to identify a quadrilateral on a specific circle, we need the lengths of 3 sides. It means we have 3 variables and 0 equation. However the first condition has 1 equation and the second condition has 2 equations. Thus when we consider both conditions together, they are sufficient.

Normally for cases where we need 2 more equations, such as original conditions with 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer hence using 1) and 2) together. (It saves us time). Obviously there may be cases where the answer is A, B, D or E.

For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80 % chance that E is the answer, while C has 15% chance and A, B or D has 5% chance. Since E is most likely to be the answer using 1) and 2) together according to DS definition. Obviously there may be cases where the answer is A, B, C or D.
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Re: A quadrilateral is inscribed in a circle. If the   [#permalink] 03 Jul 2017, 13:52
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