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# A trapezoid is symmetrical about a vertical center line. If

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27 Apr 2012, 12:26
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A trapezoid is symmetrical about a vertical center line. If a circle is drawn such that it is tangent to exactly three sides of the trapezoid and is enclosed entirely within the trapezoid, what is the diameter of the circle

(1) The parallel sides of the trapezoid are 10 inches apart.
(2) Of the parallel sides of the trapezoid, the shorter side is 15 inches long.

Will appreciate an algebraic proof.
[Reveal] Spoiler: OA

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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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27 Apr 2012, 13:20
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AbhiJ wrote:
A trapezoid is symmetrical about a vertical center line. If a circle is drawn such that it is tangent to exactly three sides of the trapezoid and is enclosed entirely within the trapezoid, what is the diameter of the circle

(1) The parallel sides of the trapezoid are 10 inches apart.
(2) Of the parallel sides of the trapezoid, the shorter side is 15 inches long.

Will appreciate an algebraic proof.

Given; the circle is is entirely enclosed within the isosceles trapezoid, so that it is tangent to exactly three sides

Below is the only case possible to satisfy the following conditions:
1. Height is 10.
2. Shorter base is 15.
Attachment:

Trapezoid.png [ 5.66 KiB | Viewed 5479 times ]
So, diameter of the circle is 10.

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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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27 Apr 2012, 20:39
I still have doubt.
1. If we consider, the height to be diameter, then wouldnt just A suffice

Bunuel wrote:
AbhiJ wrote:
A trapezoid is symmetrical about a vertical center line. If a circle is drawn such that it is tangent to exactly three sides of the trapezoid and is enclosed entirely within the trapezoid, what is the diameter of the circle

(1) The parallel sides of the trapezoid are 10 inches apart.
(2) Of the parallel sides of the trapezoid, the shorter side is 15 inches long.

Will appreciate an algebraic proof.

Given; the circle is is entirely enclosed within the isosceles trapezoid, so that it is tangent to exactly three sides

Below is the only case possible to satisfy the following conditions:
1. Height is 10.
2. Shorter base is 15.
Attachment:
Trapezoid.png
So, diameter of the circle is 10.

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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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27 Apr 2012, 20:46
I too thought the same , then with 1 you can fix the distance between the parallel sides of the trapezoid , but will not be able to restrict the size of the trapezoid until unless there is some extra info to do that , which here is the shorter side in 2.

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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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29 Apr 2012, 02:53
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If we consider the case when the circle touches the two equal sides of isosceles trapezium and the shorter side, here I am not able to prove rigorously that the distance between the two parallel sides must be more than 10.
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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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29 Apr 2012, 05:49
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devpanda wrote:
I still have doubt.
1. If we consider, the height to be diameter, then wouldnt just A suffice

Bunuel wrote:
AbhiJ wrote:
A trapezoid is symmetrical about a vertical center line. If a circle is drawn such that it is tangent to exactly three sides of the trapezoid and is enclosed entirely within the trapezoid, what is the diameter of the circle

(1) The parallel sides of the trapezoid are 10 inches apart.
(2) Of the parallel sides of the trapezoid, the shorter side is 15 inches long.

Will appreciate an algebraic proof.

Given; the circle is is entirely enclosed within the isosceles trapezoid, so that it is tangent to exactly three sides

Below is the only case possible to satisfy the following conditions:
1. Height is 10.
2. Shorter base is 15.
Attachment:
The attachment Trapezoid.png is no longer available
So, diameter of the circle is 10.

Consider the diagram below for statement (1):
Attachment:

1.png [ 6.92 KiB | Viewed 5355 times ]
As you can see we have two circles with different radii.
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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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29 Apr 2012, 06:08
I agree with Bunuel..
Both are needed to understand which combination of sides will be touching the circle.
Is it 1 Non-Parallel side and 2 Parallel sides OR 2 Non-Parallel sides and 1 Parallel side.
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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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29 Apr 2012, 06:18
I have a doubt..
there is also this statement in the question that "the trapezoid is symmetrical about a vertical line". But the solution is arrived by assuming it symmetrical about a horizontal line.??

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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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29 Apr 2012, 06:21
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Prashant85 wrote:
I have a doubt..
there is also this statement in the question that "the trapezoid is symmetrical about a vertical line". But the solution is arrived by assuming it symmetrical about a horizontal line.??

Vertical line: |

The trapezoid is symmetrical about a vertical line (|) means that the trapezoid is isosceles.
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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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29 Apr 2012, 06:23
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Nope.. The solutions is arrived by considering the statement that "the trapezoid is symmetrical about a vertical line".
It cannot have non parallel sides on the y-aixs if it was assumed that the trapezoid is symmetrical about a horizontal line because the non parallel sides will intersect at some point..
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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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29 Apr 2012, 06:27
varunmaheshwari wrote:
Nope.. The solutions is arrived by considering the statement that "the trapezoid is symmetrical about a vertical line".
It cannot have non parallel sides on the y-aixs if it was assumed that the trapezoid is symmetrical about a horizontal line because the non parallel sides will intersect at some point..

What you mean "nope"?

Stem says "a trapezoid is symmetrical about a vertical center line" which means that that the trapezoid is isosceles. All diagrams show trapezoids which are symmetrical about a vertical center line.
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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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29 Apr 2012, 06:30
Bunuel,
The nope was for Prashant85 question that the solution is arrived by assuming it symmetrical about a horizontal line..
I think we both started answering at the same time and that's why I missed you answer while posting mine..
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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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29 Apr 2012, 06:34
varunmaheshwari wrote:
Bunuel,
The nope was for Prashant85 question that the solution is arrived by assuming it symmetrical about a horizontal line..
I think we both started answering at the same time and that's why I missed you answer while posting mine..

That's why I didn't understand your response. I guess all is clear then.
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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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29 Apr 2012, 08:08
Thanks Bunuel and Varun...
Its clear now.

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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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01 May 2012, 12:50
Bunuel wrote:
AbhiJ wrote:
A trapezoid is symmetrical about a vertical center line. If a circle is drawn such that it is tangent to exactly three sides of the trapezoid and is enclosed entirely within the trapezoid, what is the diameter of the circle

(1) The parallel sides of the trapezoid are 10 inches apart.
(2) Of the parallel sides of the trapezoid, the shorter side is 15 inches long.

Will appreciate an algebraic proof.

Given; the circle is is entirely enclosed within the isosceles trapezoid, so that it is tangent to exactly three sides

Below is the only case possible to satisfy the following conditions:
1. Height is 10.
2. Shorter base is 15.
Attachment:
Trapezoid.png
So, diameter of the circle is 10.

Hi Bunuel,
Why can't you have a circle touching the isosceles side, in which case the radius is 15. I did not understand your solution. could you please explain.

thank you

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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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02 May 2012, 03:23
Bunuel wrote:
AbhiJ wrote:
A trapezoid is symmetrical about a vertical center line. If a circle is drawn such that it is tangent to exactly three sides of the trapezoid and is enclosed entirely within the trapezoid, what is the diameter of the circle

(1) The parallel sides of the trapezoid are 10 inches apart.
(2) Of the parallel sides of the trapezoid, the shorter side is 15 inches long.

Will appreciate an algebraic proof.

Given; the circle is is entirely enclosed within the isosceles trapezoid, so that it is tangent to exactly three sides

Below is the only case possible to satisfy the following conditions:
1. Height is 10.
2. Shorter base is 15.
Attachment:
Trapezoid.png
So, diameter of the circle is 10.

Hi Bunuel,
Why can't you have a circle touching the isosceles side, in which case the radius is 15. I did not understand your solution. could you please explain.

thank you

This is my doubt as well, I can see how it is true from the diagram. However when I try to prove it mathematically , i fail.
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Re: A trapezoid is symmetrical about a vertical center line. If [#permalink]

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04 Jul 2012, 06:03
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Bunuel wrote:
AbhiJ wrote:
A trapezoid is symmetrical about a vertical center line. If a circle is drawn such that it is tangent to exactly three sides of the trapezoid and is enclosed entirely within the trapezoid, what is the diameter of the circle

(1) The parallel sides of the trapezoid are 10 inches apart.
(2) Of the parallel sides of the trapezoid, the shorter side is 15 inches long.

Will appreciate an algebraic proof.

Given; the circle is is entirely enclosed within the isosceles trapezoid, so that it is tangent to exactly three sides

Below is the only case possible to satisfy the following conditions:
1. Height is 10.
2. Shorter base is 15.
Attachment:
Trapezoid.png
So, diameter of the circle is 10.

Hi Bunuel,
Why can't you have a circle touching the isosceles side, in which case the radius is 15. I did not understand your solution. could you please explain.

thank you

If we take only option B, then the diameter is >15 and not 15, i.e. it can be any number, so we can't say B.
Now for C If we have circles touching the isosceles sides, then the diameter will be >15 and not 10, i.e. the circle will not be inside the Trapezoid. This implies that circle will touch the parallel sides and not the non-parallel sides of the Trapezoid.Thus the two statements together are sufficient to answer the question.
Hope I am clear.
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13 Mar 2017, 01:03
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There are four basic ways this picture could look, as there are four sides of the trapezoid
that could serve as the non-tangent side:

(1) INSUFFICIENT: If the circle is tangent to both of the parallel sides (Figure A or B), then the diameter must be 10. If the circle is tangent to only one of the parallel sides (Figure C or D), then the diameter is less than 10. Since we are left with multiple possibilities for the diameter of the circle, we do not have enough information to answer the original question.

(2) INSUFFICIENT: Just knowing the length of the shorter parallel side is not enough to determine which of the basic figures above describes the correct situation. If Figure A or B represents the correct situation, the diameter of the circle is clearly determined solely by the distance between the parallel sides; the diameter is independent of the length of the shorter parallel side, so knowing that it is 15 inches long is not useful. If Figure C or D describes the correct situation, then the diameter would depend on not only the 15-inch side but also the longer parallel side, which has an unknown length.

(1) AND (2) SUFFICIENT: If the distance between the parallel sides is 10, and the shorter parallel side has a length of 15, then Figure A or B represents the correct situation and the diameter of the circle must equal 10.

Alternatively, notice that Figure D could not represent our situation, since the circle would have to have a diameter larger than 15 inches in order to be tangent to the short parallel side and the nonparallel sides of the trapezoid. We know that the parallel sides of the trapezoid are only 10 inches
apart, so the circle would be too large to be drawn entirely within the trapezoid as required. Similar logic explains why Figure C is also impossible when we consider (1) and (2) together.

Attachments

Trapezoid.PNG [ 222.45 KiB | Viewed 680 times ]

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