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

Question

https://gmatclub.com/forum/download/file.php?id=43605 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} image001.jpg

GMATinsight · Answer

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

soapbolt · Answer

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

Zoechrysostom · Answer

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 !!

TestPrepUnlimited · Answer

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

TestPrepUnlimited · Answer

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.

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

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

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