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What is the length of diameter PR in the circle shown above?

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Bunuel
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What is the length of diameter PR in the circle shown above? [#permalink] New post   13 Sep 2018, 02:33
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What is the length of diameter PR in the circle shown above?

(1) PQ has length of 6.
(2) QR has length of \(\sqrt{12}\)


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D
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Re: What is the length of diameter PR in the circle shown above? [#permalink] New post   13 Sep 2018, 03:10
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Bunuel wrote:

What is the length of diameter PR in the circle shown above?

(1) PQ has length of 6.
(2) QR has length of \(\sqrt{12}\)


Show Spoiler
Attachment:
The attachment image001.jpg is no longer available


Question: Diameter = a+b = ?

We must know that we see here three right angle triangles and all triangles are similar triangles (Proof attached)
and \(3/a = b/3\)

Statement 1: PQ has length of 6
i.e. \(a = √(6^2-3^2) = 3√3\)
And now \(b = √3\)
SUFFICIENT

Statement 2: QR has length of \(\sqrt{12}\)
Now b can be calculated using pythagorus theorem and then a can be calculated as a*b=9 hence
SUFFICIENT

Answer: option D
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Geometry 7 Similar Triangle Latest.jpg
Geometry 7 Similar Triangle Latest.jpg [ 195.88 KiB | Viewed 2601 times ]

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Re: What is the length of diameter PR in the circle shown above? [#permalink] New post   01 Dec 2018, 06:25
https://gmatclub.com/forum/if-arc-pqr-above-is-a-semicircle-what-is-the-length-of-diameter-pr-144057.html

Bunnel and other experts posted great detailed solution for similar problem in the above link.

Hope it helps :)
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Re: What is the length of diameter PR in the circle shown above? [#permalink] New post   22 Jun 2020, 06:04
Hi :)

Can anyone explain how you find that these triangles are similar? We only know that they each have a right angle but that's not enough?

Thanks !!
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Re: What is the length of diameter PR in the circle shown above? [#permalink] New post   22 Jun 2020, 10:56
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Bunuel wrote:

What is the length of diameter PR in the circle shown above?

(1) PQ has length of 6.
(2) QR has length of \(\sqrt{12}\)


Show Spoiler
Attachment:
image001.jpg


Note from the graph we have 3 similar triangles. Thus we can take the proportions b/3 = 3/a or ab = 9. We are trying to find \(a + b\) but finding either \(a\) or \(b\) will give the other. So we actually only need \(a\) or \(b\).

Statement 1:
Can solve for \(a\) using the Pythagorean theorem. Then using ab = 9 we can solve for b, sufficient.

Statement 2:
Can solve for \(b\) using the Pythagorean theorem. Sufficient.

Ans: D
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Re: What is the length of diameter PR in the circle shown above? [#permalink] New post   22 Jun 2020, 11:09
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Zoechrysostom wrote:
Hi :)

Can anyone explain how you find that these triangles are similar? We only know that they each have a right angle but that's not enough?

Thanks !!


Hi Zoechrysostom I'm happy to answer your question and any following ones!

First let's call the height QM. Angle PQR is a right angle because it is inscribed in a circle while PR is the diameter. Triangle RQP and triangle QMP are similar because they share angle P and a right angle, hence all of their angles are the same and they have the same shape. With the same logic, triangle QMR and triangle PQR are similar. Finally we can transfer the similarity and get the two smaller triangles are similar.
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PQR.png
PQR.png [ 21.72 KiB | Viewed 1684 times ]

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Re: What is the length of diameter PR in the circle shown above? [#permalink]
22 Jun 2020, 11:09
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