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Math Expert V
Joined: 02 Sep 2009
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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 21% (03:14) correct 79% (02:30) wrong based on 97 sessions

### HideShow timer Statistics What is b in the circle shown above?

(1) bc = 5a
(2) The length of PQ is $$\sqrt{84}$$

Attachment: image004 (1).jpg [ 4.71 KiB | Viewed 1141 times ]

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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ  [#permalink]

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Bunuel wrote: What is b in the circle shown above?

(1) bc = 5a
(2) The length of PQ is $$\sqrt{84}$$

Attachment:
image004 (1).jpg

Point 1: PQR is a right triangle with right angle at Q
Point 2: The right Triangle PWR has been divided into two other right triangles
Point 3: All three right triangles are similar triangles
Using Similar triangle property between two smaller right triangles we can deduce that
b/a = 5/c

Question: What is b in the circle shown above?

Statement 1: bc = 5a

We already have this information hence Option A, C and D are out
NOT SUFFICIENT

Statement 2: The length of PQ is $$\sqrt{84}$$
i.e. $$a = √84$$
We canfurther use similar triangle between biggest second big right triangle to get the value of b

SUFFICIENT

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Re: What is b in the circle shown above? (1) bc = 5a (2) The length of PQ  [#permalink]

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GMATinsight Can you please elaborate explanation of 2nd statement ?
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Re: What is b in the circle shown above? (1) bc = 5a (2) The length of PQ  [#permalink]

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vipulshahi wrote:
GMATinsight Can you please elaborate explanation of 2nd statement ?

Hi vipulshahi!

Sorry for this "intrusion", but I will explain each statements alone, in detail, in the post below.

Regards,
Fabio.
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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ  [#permalink]

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Bunuel wrote: What is b in the circle shown above?

(1) bc = 5a
(2) The length of PQ is $$\sqrt{84}$$ Regards,
Fabio.
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Re: What is b in the circle shown above? (1) bc = 5a (2) The length of PQ  [#permalink]

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Could you further clarify the explanation for statement two?
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Re: What is b in the circle shown above? (1) bc = 5a (2) The length of PQ  [#permalink]

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rZAhyYJtj wrote:
Could you further clarify the explanation for statement two?

Hi, rZAhyYJtj !

Thank you for your interest in my solution.

01. By the similarity of the triangles mentioned (first line of statement (2)), we get the ratios´s equality in which only "b" (our FOCUS) is present!
02. Cross-multiplying and making the change-of-variables (k= b squared) we get a second-degree equation in the auxiliary k variable.
03. The "c/a" formula (Viète) gives us the product of the roots of a second-degree equation (when it has roots (*)).
(*) Obs.: it´s an examiner´s burden to guarantee that at least one b (therefore k) is viable... hence delta is nonnegative for sure, no need to check!
04. From the fact that the product of the two roots k1 and k2 is negative, only one of them is viable (the positive one), hence we know b^2 is this k.
05. From the fact that b is the square root of k or the opposite of the square root of k, we must take the first option (b>0). Therefore b viable is unique.

I hope you enjoy the arguments. If you believe you have the profile for a very high-level quant preparation, do not hesitate to do our test-drive!

Regards,
Fabio.
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Our high-level "quant" preparation starts here: https://gmath.net Re: What is b in the circle shown above? (1) bc = 5a (2) The length of PQ   [#permalink] 23 Sep 2018, 04:24
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