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805+ Level|   Geometry|      
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Bunuel
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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ 18 Sep 2018, 22:51
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95% (hard)

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23% (03:15) correct 77% (02:30) wrong based on 167 sessions

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What is b in the circle shown above?

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


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B
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Schools: IIM (A) ISB '24
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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ 20 Sep 2018, 05:02
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Bunuel

What is b in the circle shown above?

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


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Attachment:
image004 (1).jpg
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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

Answer: option B
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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ 21 Sep 2018, 11:13
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GMATinsight Can you please elaborate explanation of 2nd statement ?
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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ 21 Sep 2018, 12:36
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vipulshahi
GMATinsight Can you please elaborate explanation of 2nd statement ?
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Hi vipulshahi!

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

I hope you are able to clear your doubts. (Otherwise, I am sure GMATinsight will be most glad to help you!)

Regards,
Fabio.
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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ 21 Sep 2018, 12:40
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Bunuel

What is b in the circle shown above?

(1) bc = 5a
(2) The length of PQ is \(\sqrt{84}\)
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Please follow the image attached!



Regards,
Fabio.
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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ 22 Sep 2018, 20:48
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Could you further clarify the explanation for statement two?
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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ 23 Sep 2018, 04:24
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rZAhyYJtj
Could you further clarify the explanation for statement two?
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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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What is b in the circle shown above? (1) bc = 5a (2) The length of PQ 20 Oct 2022, 05:59
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GMATinsight
Bunuel

What is b in the circle shown above?

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


Show Spoiler
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

Answer: option B
Show more

Hi, Can you please explain how the 3 triangles are similar triangles ?
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