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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.
(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.
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.
The correct answer is C.
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.
The correct answer is C.
Attachments
Trapezoid.PNG [ 222.45 KiB | Viewed 8219 times ]
General Discussion
27 Apr 2012, 13:20
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AbhiJ
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 13277 times ]
Answer: C.
