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605-655 Level|   Geometry|      
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KarishmaB
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Points A, B, and C lie on a circle of radius 1. What is the area of tr 01 Apr 2015, 05:04
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buddyisraelgmat
VeritasPrepKarishma
qlx
if we draw a right triangle in a circle , does the triangle have to have Hypotenuse as the Diameter of the Circle?


From this question it does seem so.

Can we not have something as shown in the figure attached.

Attachment:
Triangle 2.jpg

Think about it logically - in your figure, say the hypotenuse is AC. Now arc AC subtends an inscribed angle of 90 degrees. So the central angle it subtends must be 180 degrees (since it is twice the inscribed angle). Angle of 180 degrees at the center means it is a straight angle and AC is the diameter. So no matter how you draw the figure. If you have made a right triangle in a circle, its hypotenuse will be the diameter.



Hi Karishma

1 doubt - maybe conceptual understanding

In Stmnt 1

I understand that AB is the Diameter and Angle C formed is Right angle.
Now Area of Triangle ABC = 1/2 base * height
If I consider base as AB (which is 2) and height as CO (O is centre) which makes CO = Radius = 1
This is sufficient. Isnt it?

Pls clarify . Thanks
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You can consider AB as base but is it necessary that CO will be the height? The height has to be perpendicular to the base. Draw a right triangle ABC in a circle. You will see that CO needn't be perpendicular to the diameter/hypotenuse. Hence we cannot say that it will be the height.
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Points A, B, and C lie on a circle of radius 1. What is the area of tr 22 Nov 2018, 07:50
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Bunuel
qlx
if we draw a right triangle in a circle , does the triangle have to have Hypotenuse as the Diameter of the Circle?


From this question it does seem so.

Can we not have something as shown in the figure attached.

Attachment:
Triangle 2.jpg

Yes, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle (the reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle)

Points A, B, and C lie on a circle of radius 1. What is the area of triangle ABC?

(1) \(AB^2 = BC^2 + AC^2\) --> triangle ABC is a right triangle with AB as hypotenuse --> \(area=\frac{BC*AC}{2}\). Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle (the reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle). So, hypotenuse AB=diameter=2*radius=2, but just knowing the length of the hypotenuse is not enough to calculate the legs of a right triangle thus we can not get the area. Not sufficient.

(2) \(\angle CAB\) equals 30 degrees. Clearly insufficient.

(1)+(2) From (1) ABC is a right triangle and from (2) \(\angle CAB=30\) --> we have 30°-60°-90° right triangle and as AB=hypotenuse=2 then the legs equal to 1 and \(\sqrt{3}\) --> \(area=\frac{BC*AC}{2}=\frac{\sqrt{3}}{2}\). Sufficient.

Answer: C.
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Hi Bunuel,

If ABC is right angle triangle inside a circle ,than AC must be the radius of that circle ,BAC has to be 30 degree ,could you please show me other cases than this?
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Points A, B, and C lie on a circle of radius 1. What is the area of tr 22 Nov 2018, 11:09
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Bunuel
qlx
if we draw a right triangle in a circle , does the triangle have to have Hypotenuse as the Diameter of the Circle?


From this question it does seem so.

Can we not have something as shown in the figure attached.

Attachment:
Triangle 2.jpg

Yes, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle (the reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle)

Points A, B, and C lie on a circle of radius 1. What is the area of triangle ABC?

(1) \(AB^2 = BC^2 + AC^2\) --> triangle ABC is a right triangle with AB as hypotenuse --> \(area=\frac{BC*AC}{2}\). Now, a right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle (the reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle). So, hypotenuse AB=diameter=2*radius=2, but just knowing the length of the hypotenuse is not enough to calculate the legs of a right triangle thus we can not get the area. Not sufficient.

(2) \(\angle CAB\) equals 30 degrees. Clearly insufficient.

(1)+(2) From (1) ABC is a right triangle and from (2) \(\angle CAB=30\) --> we have 30°-60°-90° right triangle and as AB=hypotenuse=2 then the legs equal to 1 and \(\sqrt{3}\) --> \(area=\frac{BC*AC}{2}=\frac{\sqrt{3}}{2}\). Sufficient.

Answer: C.




Hi Bunuel,

If ABC is right angle triangle inside a circle ,than AC must be the radius of that circle ,BAC has to be 30 degree ,could you please show me other cases than this?
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Hello

If a triangle ABC is a right angled triangle inside a circle, why is it necessary for AB to be the radius? In fact, the hypotenuse of the triangle is the diameter, as Bunuel has already explained in his post, which is quoted here.

Also its not necessary that angle BAC will be 30 degrees.

If you take any right angle triangle (or any triangle for that matter), it can always be inscribed in a circle. All triangles can be inscribed in a circle. So a right angle triangle with angles as 45-45-90 can be inscribed in a circle, or that with angles 20-70-90 can also be inscribed in a circle.
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Points A, B, and C lie on a circle of radius 1. What is the area of tr 22 Oct 2022, 05:19
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