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In the diagram below, PQ is a diameter of the circle having center at

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In the diagram below, PQ is a diameter of the circle having center at  [#permalink]

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New post 23 Jul 2017, 22:18
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In the diagram below, PQ is a diameter of the circle having center at O. What is the measure of ∠PTQ?
Image

(1) ∠ROS = 40◦.
(2) ∠RPO = 55◦.

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2017-07-24_1017.png [ 12.91 KiB | Viewed 2016 times ]

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In the diagram below, PQ is a diameter of the circle having center at  [#permalink]

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New post 24 Jul 2017, 00:14
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Bunuel wrote:
In the diagram below, PQ is a diameter of the circle having center at O. What is the measure of ∠PTQ?
(1) ∠ROS = 40◦.
(2) ∠RPO = 55◦.


Attachment:
2017-07-24_1017.png
2017-07-24_1017.png [ 13.43 KiB | Viewed 1727 times ]


We have \(\widehat{POS}=180^o -2\widehat{SPO}\)

Also \(\widehat{POS}=\widehat{POR}+\widehat{ROS}\)

We also have \(\widehat{POR} = 180^o -2\widehat{RPO}\)

Hence \(\widehat{POS}=\widehat{POR}+\widehat{ROS} \\
\implies 180^o -2\widehat{SPO} = 180^o -2\widehat{RPO} +\widehat{ROS} \\
\implies \widehat{ROS} = 2\widehat{RPO} - 2\widehat{SPO} \\
\implies \widehat{ROS} = 2\widehat{RPS}\)

(1) If \(\widehat{ROS} = 40^o \implies \widehat{RPS} = 20^o\)

Since \(\widehat{PST} = 90^o \implies \widehat{PTS} = 90^o - \widehat{RPS} = 70^o\). Sufficient.

(2) We can't know the value of \(\widehat{PQT}\) so we can't know the value of \(\widehat{PTS}\). Insufficient.

The answer is A.
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Re: In the diagram below, PQ is a diameter of the circle having center at  [#permalink]

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New post 25 Oct 2018, 08:54
Is there any shorter way to solve this?
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Re: In the diagram below, PQ is a diameter of the circle having center at  [#permalink]

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New post 25 Oct 2018, 10:19
1
Bunuel wrote:
In the diagram below, PQ is a diameter of the circle having center at O. What is the measure of ∠PTQ?
Image

(1) ∠ROS = 40◦.
(2) ∠RPO = 55◦.

Statement (1) can be solved exploring two facts only:

(i) Angle PSQ (hence angle PST) is (are) right (because PQ is a diameter)

(ii) Angle RPS is inscribed with the corresponding 40-degrees central angle ROS

Statement (2) is not sufficient, as proven with the geometric bifurcation.

Image


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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In the diagram below, PQ is a diameter of the circle having center at  [#permalink]

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New post 25 Oct 2018, 11:41
1
fskilnik wrote:
Bunuel wrote:
In the diagram below, PQ is a diameter of the circle having center at O. What is the measure of ∠PTQ?
Image

(1) ∠ROS = 40◦.
(2) ∠RPO = 55◦.

Statement (1) can be solved exploring two facts only:

(i) Angle PSQ (hence angle PST) is (are) right (because PQ is a diameter)

(ii) Angle RPS is inscribed with the corresponding 40-degrees central angle ROS

Statement (2) is not sufficient, as proven with the geometric bifurcation.

Image


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Excuse me, why 40 divided by 2?, could you explain it, thank you.
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In the diagram below, PQ is a diameter of the circle having center at  [#permalink]

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New post 25 Oct 2018, 12:00
jorgetomas9 wrote:
Excuse me, why 40 divided by 2?, could you explain it, thank you.

Hi, jorgetomas9 !

Thank you for your interest in my solution.

Every inscribed angle (in this case RPS) is half the value of the central angle (in this case ROS) corresponding to the same arc (in this case arc RS).

Regards,
Fabio.
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Course release PROMO : finish our test drive till 30/Nov with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

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Re: In the diagram below, PQ is a diameter of the circle having center at  [#permalink]

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New post 26 Oct 2018, 00:02
That bifurcation method will come handy in various DS problems of geometry.
Thanks fskilnik.
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Re: In the diagram below, PQ is a diameter of the circle having center at  [#permalink]

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New post 26 Oct 2018, 03:20
castiel wrote:
That bifurcation method will come handy in various DS problems of geometry.
Thanks fskilnik.

Hi, castiel.

Thank you for the nice compliment (and for the kudos)!

You are absolutely right. Bifurcations are extremely useful and, to be honest, they change intuition (I feel...) into certainty (I am sure...)!

This "shield" is also possible in algebraic and geometric-algebraic bifurcations, by the way. (You can find them in my previous posts... and in my course.)

Regards and success in your studies,
Fabio.
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Re: In the diagram below, PQ is a diameter of the circle having center at  [#permalink]

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New post 26 Oct 2018, 05:06
1
fskilnik wrote:

This "shield" is also possible in algebraic and geometric-algebraic bifurcations, by the way. (You can find them in my previous posts... and in my course.)



I'll check them as well.
Thanks again!! :)
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Re: In the diagram below, PQ is a diameter of the circle having center at &nbs [#permalink] 26 Oct 2018, 05:06
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In the diagram below, PQ is a diameter of the circle having center at

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