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# If arc PQR above is a semicircle, what is the length of diameter PR ?

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If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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13 Dec 2012, 07:14
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If arc PQR above is a semicircle, what is the length of diameter PR ?

(1) a = 4
(2) b = 1

[Reveal] Spoiler:
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Semicircle.png [ 8.12 KiB | Viewed 24933 times ]
[Reveal] Spoiler: OA

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If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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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: http://gmatclub.com/forum/triangles-106177.html, http://gmatclub.com/forum/geometry-problem-106009.html, http://gmatclub.com/forum/mgmat-ds-help-94037.html, http://gmatclub.com/forum/help-108776.html)

(1) a = 4. Sufficient.

(2) b = 1. Sufficient.

[Reveal] Spoiler:
Attachment:

Semicircle2.PNG [ 4.95 KiB | Viewed 26117 times ]

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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18 Dec 2012, 04:58
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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
Attachments

Relations.docx [160.17 KiB]

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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22 Dec 2012, 06:04
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.
Attachments

geometry solution.png [ 13.23 KiB | Viewed 24032 times ]

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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29 Mar 2014, 03:55
Can anybody explain why the area of the triangle PQR : Which is a right angle triangle in the figure above at Q, is not 1/2*QR*PQ ? Assuming this triangle was drawn without the semi-circle and if I slightly redraw the triangle keeping the base at QR, and PQ becomes the height and PR is the hypotenuse, then isnt the area of the triangle 1/2*QR*PQ?

Why is it that in these type of triangles which are drawn in such manner, that the hypotenuse is the base, the height is drawn from one vertex to another and then area is calculated?

This question is just to douse this silly doubt lingering in my head.

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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29 Mar 2014, 03:58
sudeeptasahu29 wrote:
Can anybody explain why the area of the triangle PQR : Which is a right angle triangle in the figure above at Q, is not 1/2*QR*PQ ? Assuming this triangle was drawn without the semi-circle and if I slightly redraw the triangle keeping the base at QR, and PQ becomes the height and PR is the hypotenuse, then isnt the area of the triangle 1/2*QR*PQ?

Why is it that in these type of triangles which are drawn in such manner, that the hypotenuse is the base, the height is drawn from one vertex to another and then area is calculated?

This question is just to douse this silly doubt lingering in my head.

The area of triangle PQR IS 1/2*PQ*QR but it's ALSO 1/2*PR*QT.
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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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06 May 2014, 22:21
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Another alternate, simpler solution is shown in the video, at 2:00.

Last edited by Dienekes on 26 Oct 2015, 15:06, edited 1 time in total.

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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16 Jul 2014, 23:12
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Bunuel wrote:

If arc PQR above is a semicircle, what is the length of diameter PR?

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

(1) a = 4. Sufficient.

(2) b = 1. Sufficient.

1/2*QT*PR=1/2*QP*QR
=> PR = (QP*QR)/2

With this equation, how statement 1 or 2 is being judged. Will you please fill up the gap?
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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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17 Jul 2014, 08:46
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musunna wrote:
Bunuel wrote:

If arc PQR above is a semicircle, what is the length of diameter PR?

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

(1) a = 4. Sufficient.

(2) b = 1. Sufficient.

1/2*QT*PR=1/2*QP*QR
=> PR = (QP*QR)/2

With this equation, how statement 1 or 2 is being judged. Will you please fill up the gap?

It's better to use ratios for this question.

For (1) use the following ratio: PT/QT = QT/RT --> 4/2 = 2/RT --> RT = 1 --> PR = PT + RT = 4 + 1 = 5.

For (2) use the the same ratio: PT/QT = QT/RT --> PT/2 = 2/1--> PT = 4 --> PR = PT + RT = 4 + 1 = 5.

Hope it's clear.
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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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02 Jun 2015, 01:02
Bunuel wrote:
Attachment:
Semicircle2.PNG
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.

we do not need to know this property. just make an equation in which b is a unknown number . we can solve

we can not remember this property in the test room.

the og explanation of this problem is good enough.

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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10 May 2016, 06:04
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Do you seriously think that question will ask what is a+b = ?
So that , from stat (1) + (2)
you can say a + b = 5 ;
GMAT will never ask addition of two numbers , so cut out the option C
and try to get the soln from A and from B alone.

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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31 Jul 2016, 01:01
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Attachment:
Semicircle.png
If arc PQR above is a semicircle, what is the length of diameter PR ?

(1) a = 4
(2) b = 1

I did this in a different way and still got the same answer, So if you look at the property of a semicircle, then if the triangle is formed using the diameter of the semicircle the the angle subtended by the angle formed at the arc is 90

Now given that Pythagorus theorem for the bog triangle PQR, $$PQ^2=QR^2+PR^2$$; $$PR^2= (a+b)^2$$ Now let PQ be x and QR be y then $$x^2+y^2= (a+b)^2$$---eq1

Smaller triangles PQa $$x^2=a^2+4$$; QRb $$y^2= b^2+4$$--- eq2,3
now eq 2+3
$$x^2+y^2=a^2+b^2+8$$
equate the same with eq 1 and get rid of x and y

$$(a+b)^2=a^2+b^2+8$$
$$a^2+b^2+2ab=a^2+b^2+8$$

2ab=8
ab=4

Now the question is actually asking if ab=4 what is a+b

knowing either the value of a or b we will know the other variable because of the multiplication constraint and get a+b

hence D

Happy GMATing!

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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22 Oct 2016, 04:40
PQT and QRT (which are also similar to big triangle PQR) is wrong. It should be similar to triangle PRQ. Makes a difference.

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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08 Feb 2017, 00:12
Since angle PQR is inscribed in a semicircle,it is a right triangle.
PQR is divided into two right triangles by the vertical line from Q to side PR.
Let PQ=x and QR=y

The larger right triangle has hypotenuse $$x$$, so $$x^2 = 4 + a^2$$
The smaller right triangle has hypotenuse $$y$$ , so $$y^2 = 4 + b^2$$

From triangle PQR $$(a+b)^2 = x^2 + y^2$$, by substituting
$$(a+b)^2= 4+a^2 + 4+b^2$$
$$a^2 + 2ab + b^2 = 8 + a^2 +b^2$$
$$2ab=8 ; ab =4$$

1) if a =4 and ab=4; b must be 1 SUFFICIENT
2) if b =1 and ab=4; a must be 4 SUFFICIENT

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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05 Aug 2017, 00:27
(1) a = 4
$$=> PQ^2 = 2^2 + a^2 = 20$$
Applying Pythagorean:
$$QR^2 = 2^2 + b^2 = 4 + b^2$$
$$QR^2 = (a+b)^2 – PQ^2 = (4+b)^2 -20 = (16 + 8b + b^2) – 20 = b^2 + 8b -4$$
$$=> 4 + b^2 = b^2 + 8b – 4$$
=> b => (a+b)
Sufficient

(2) Applying the same method as (1) => Sufficient

=> D

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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15 Aug 2017, 07:00
Hi! How do we know if PQR is a right triangle?

Thanks!

Bunuel wrote:
Attachment:
Semicircle2.PNG
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.

(1) a = 4. Sufficient.

(2) b = 1. Sufficient.

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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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15 Aug 2017, 09:25
RodrigoMi wrote:
Hi! How do we know if PQR is a right triangle?

Thanks!

Bunuel wrote:
Attachment:
Semicircle2.PNG
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.

(1) a = 4. Sufficient.

(2) b = 1. Sufficient.

Please check the highlighted part in the solution.
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Re: If arc PQR above is a semicircle, what is the length of diameter PR ? [#permalink]

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15 Aug 2017, 10:05
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D clearly

Let the perpendicular to PR be M

using similar triangle Triangle PQM~ Triangle QRM
using AAA property and Also Triangle PQR is right angle at Q since angle contained by diameter to the circle is 90.

PQ/QR = PM/QM = QM/RM

statement 1 is sufficient to find RM.
statement 2 is sufficient to find PM.

therefore D
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Re: If arc PQR above is a semicircle, what is the length of diameter PR ?   [#permalink] 15 Aug 2017, 10:05
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