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If arc PQR above is a semicircle, what is the length of [#permalink]
 13 Dec 2012, 07:14
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60% (01:40) correct
39% (01:10) wrong based on 11 sessions
Attachment: 
Semicircle.png [ 8.12 KiB | Viewed 1260 times ]
If arc PQR above is a semicircle, what is the length of diameter PR ? (1) a = 4 (2) b = 1
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Re: If arc PQR above is a semicircle, what is the length of [#permalink]
13 Dec 2012, 07:16
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If arc PQR above is a semicircle, what is the length of diameter PR? You should know the following properties to solve this question: • A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle.So, as given that PR is a diameter then angle PQR is a right angle. • Perpendicular to the hypotenuse will always divide the triangle into two triangles with the same properties as the original triangle. Thus, the perpendicular QT divides right triangle PQR into two similar triangles PQT and QRT (which are also similar to big triangle PQR). Now, in these three triangles the ratio of the corresponding sides will be equal (corresponding sides are the sides opposite the same angles). For example: QR/PR=QT/PQ=TR/QR. This property (sometimes along with Pythagoras) will give us the following: if we know ANY 2 values from PR, PQ, QR, PT, QT, TR then we'll be able to find other 4. We are given that QT=2 thus to find PR we need to know the length of any other line segment. Also in such kind of triangles might be useful to equate the areas to find the length of some line segment, for example area of PQR=1/2*QT*PR=1/2*QP*QR (for more check: triangles-106177.html, geometry-problem-106009.html, mgmat-ds-help-94037.html, help-108776.html) (1) a = 4. Sufficient. (2) b = 1. Sufficient. Answer: D.
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Re: If arc PQR above is a semicircle, what is the length of [#permalink]
18 Dec 2012, 04:58
Walkabout wrote: Attachment: The attachment Semicircle.png is no longer available If arc PQR above is a semicircle, what is the length of diameter PR ? (1) a = 4 (2) b= 1 Better you learn some relations which hold true for such triangle. One which has 2 right angles as here in attachment. As per that 4 = a*b a+b=? Now. A) a = 4 so b =1 B) b=1 so a = 4 OA D
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Re: If arc PQR above is a semicircle, what is the length of [#permalink]
22 Dec 2012, 06:04
Walkabout wrote: Attachment: The attachment Semicircle2.png is no longer available If arc PQR above is a semicircle, what is the length of diameter PR ? (1) a = 4 (2) b= 1 Another approach that could be implemented in thsi question is: Since there is a perpendicular drawn to the hypotenuese, therefore the two triangles that are formed must be similar to each other and to the larger one.So if one side of a triangle reduces by a certain ratio, the other side must also reduce. In the diagram attached, if one considers any of the statement then he will be able to find out the other side. Consider statement 1) a=4 Look into the diagram. In the middle triangle, "a" or PI=4. We are given with the fact that IQ=2. Now in the smallest triangle, the corresponding side of PI=IQ. IQ=2. Therefore the factor with which PI has reduced is 2. Therefore other side must also reduce by the same factor. Hence IR=1. Sufficient Statement 2) b=1. "b" is the corresponding side of IQ. So IQ , in the middle traingle, has reduced by a factor of 2. In the smallest triangle IQ=2. Therefore PI must be 4. Sufficient.
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Re: If arc PQR above is a semicircle, what is the length of [#permalink]
22 Dec 2012, 06:05
Bunnel can you explain the below part little elaboarately For example: QR/PR=QT/PQ=TR/QR. This property (sometimes along with Pythagoras) will give us the following: if we know ANY 2 values from PR, PQ, QR, PT, QT, TR then we'll be able to find other 4. We are given that QT=2 thus to find PR we need to know the length of any other line segment. I really dont understand the concept
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Re: If arc PQR above is a semicircle, what is the length of [#permalink]
14 Jan 2013, 11:38
Marcab wrote: Walkabout wrote: Attachment: Semicircle2.png If arc PQR above is a semicircle, what is the length of diameter PR ? (1) a = 4 (2) b= 1 Another approach that could be implemented in thsi question is: Since there is a perpendicular drawn to the hypotenuese, therefore the two triangles that are formed must be similar to each other and to the larger one.So if one side of a triangle reduces by a certain ratio, the other side must also reduce. In the diagram attached, if one considers any of the statement then he will be able to find out the other side. Consider statement 1) a=4 Look into the diagram. In the middle triangle, "a" or PI=4. We are given with the fact that IQ=2. Now in the smallest triangle, the corresponding side of PI=IQ. IQ=2. Therefore the factor with which PI has reduced is 2. Therefore other side must also reduce by the same factor. Hence IR=1. Sufficient Statement 2) b=1. "b" is the corresponding side of IQ. So IQ , in the middle traingle, has reduced by a factor of 2. In the smallest triangle IQ=2. Therefore PI must be 4. Sufficient. Little complex for me... don u think bunuel's method is easier?
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Re: If arc PQR above is a semicircle, what is the length of
[#permalink]
14 Jan 2013, 11:38
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