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# Is the Triangle ABC right angled at C?

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Intern
Joined: 25 May 2015
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Is the Triangle ABC right angled at C?  [#permalink]

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23 Jun 2015, 03:08
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15% (low)

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70% (00:44) correct 30% (00:53) wrong based on 158 sessions

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Is the Triangle ABC right angled at C?

1) AB is not the longest chord that can be drawn in a circle.

2) The Length of AB is seven-fifth the radius of the circle.
Math Expert
Joined: 02 Sep 2009
Posts: 58434
Re: Is the Triangle ABC right angled at C?  [#permalink]

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23 Jun 2015, 03:51
monk16 wrote:
Is the Triangle ABC right angled at C?

1) AB is not the longest chord that can be drawn in a circle.

2) The Length of AB is seven-fifth the radius of the circle.

A right triangle's hypotenuse is a diameter of its circumcircle (circumscribed circle).
The reverse is also true: if one of the sides of an inscribed triangle is a diameter of the circle, then the triangle is a right angled
(right angel being the angle opposite the diameter/hypotenuse).

Is the Triangle ABC right angled at C?

According to the above if ABC is right angled at C, then AB (hypotenuse) must be the diameter of circumscribed circle.

(1) AB is not the longest chord that can be drawn in a circle. Diameter is the longest chord possible in a circle, thus AB is NOT a diameter, therefore ABC is not right angled at C. Sufficient.

(2) The Length of AB is seven-fifth the radius of the circle. If ABC were right angled at C, AB would be a diameter, thus equal to twice the radius of the circle. ABC is not right angled at C. Sufficient.

Attachment:

2015-06-23_1445.png [ 13.04 KiB | Viewed 2196 times ]

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Re: Is the Triangle ABC right angled at C?  [#permalink]

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23 Jun 2015, 03:59
1
Bunuel wrote:
monk16 wrote:
Is the Triangle ABC right angled at C?

1) AB is not the longest chord that can be drawn in a circle.

2) The Length of AB is seven-fifth the radius of the circle.

A right triangle's hypotenuse is a diameter of its circumcircle (circumscribed circle).
The reverse is also true: if one of the sides of an inscribed triangle is a diameter of the circle, then the triangle is a right angled
(right angel being the angle opposite the diameter/hypotenuse).

Is the Triangle ABC right angled at C?

According to the above if ABC is right angled at C, then AB (hypotenuse) must be the diameter of circumscribed circle.

(1) AB is not the longest chord that can be drawn in a circle. Diameter is the longest chord possible in a circle, thus AB is NOT a diameter, therefore ABC is not right angled at C. Sufficient.

(2) The Length of AB is seven-fifth the radius of the circle. If ABC were right angled at C, AB would be a diameter, thus equal to twice the radius of the circle. ABC is not right angled at C. Sufficient.

Attachment:
2015-06-23_1445.png

How do we know that C lies on circle and not in circle or out of circle?
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Joined: 02 Sep 2009
Posts: 58434
Re: Is the Triangle ABC right angled at C?  [#permalink]

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23 Jun 2015, 04:02
Harley1980 wrote:
Bunuel wrote:
monk16 wrote:
Is the Triangle ABC right angled at C?

1) AB is not the longest chord that can be drawn in a circle.

2) The Length of AB is seven-fifth the radius of the circle.

A right triangle's hypotenuse is a diameter of its circumcircle (circumscribed circle).
The reverse is also true: if one of the sides of an inscribed triangle is a diameter of the circle, then the triangle is a right angled
(right angel being the angle opposite the diameter/hypotenuse).

Is the Triangle ABC right angled at C?

According to the above if ABC is right angled at C, then AB (hypotenuse) must be the diameter of circumscribed circle.

(1) AB is not the longest chord that can be drawn in a circle. Diameter is the longest chord possible in a circle, thus AB is NOT a diameter, therefore ABC is not right angled at C. Sufficient.

(2) The Length of AB is seven-fifth the radius of the circle. If ABC were right angled at C, AB would be a diameter, thus equal to twice the radius of the circle. ABC is not right angled at C. Sufficient.

Attachment:
2015-06-23_1445.png

How do we know that C lies on circle and not in circle or out of circle?

Original question (I've seen it somewhere) has an image showing that ABC is inscribed in a circle.
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Math Expert
Joined: 02 Sep 2009
Posts: 58434
Re: Is the Triangle ABC right angled at C?  [#permalink]

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23 Jun 2015, 04:32
1
Harley1980 wrote:
Bunuel wrote:
monk16 wrote:
Is the Triangle ABC right angled at C?

1) AB is not the longest chord that can be drawn in a circle.

2) The Length of AB is seven-fifth the radius of the circle.

A right triangle's hypotenuse is a diameter of its circumcircle (circumscribed circle).
The reverse is also true: if one of the sides of an inscribed triangle is a diameter of the circle, then the triangle is a right angled
(right angel being the angle opposite the diameter/hypotenuse).

Is the Triangle ABC right angled at C?

According to the above if ABC is right angled at C, then AB (hypotenuse) must be the diameter of circumscribed circle.

(1) AB is not the longest chord that can be drawn in a circle. Diameter is the longest chord possible in a circle, thus AB is NOT a diameter, therefore ABC is not right angled at C. Sufficient.

(2) The Length of AB is seven-fifth the radius of the circle. If ABC were right angled at C, AB would be a diameter, thus equal to twice the radius of the circle. ABC is not right angled at C. Sufficient.

Attachment:
2015-06-23_1445.png

How do we know that C lies on circle and not in circle or out of circle?

Thank you for an idea of a question.
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Re: Is the Triangle ABC right angled at C?  [#permalink]

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23 Jun 2015, 06:22
Bunuel wrote:

Thank you for an idea of a question.

You are welcome.

This is really interesting topic. Hope I not steal your idea
vertices-a-and-b-of-triangle-abc-lie-on-circle-with-area-25pi-200369.html
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Re: Is the Triangle ABC right angled at C?  [#permalink]

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28 Feb 2018, 16:11
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Re: Is the Triangle ABC right angled at C?   [#permalink] 28 Feb 2018, 16:11
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