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Triangle ABC is inscribed in a circle, such that AC is a diameter of

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Triangle ABC is inscribed in a circle, such that AC is a diameter of  [#permalink]

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New post 23 Nov 2019, 03:56
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Triangle ABC is inscribed in a circle, such that AC is a diameter of the circle and angle BAC is 45°. If the area of triangle ABC is 72 square units, how much larger is the area of the circle than the area of triangle ABC?

A. \(72\pi - 72\)
B. \(72\pi - 36\)
C. \(72\pi - 18\)
D. \(72\pi - 1\)
E. \(72\pi\)

How do I know that base = height in this triangle? Why can the diameter not be the base of the triangle?
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Re: Triangle ABC is inscribed in a circle, such that AC is a diameter of  [#permalink]

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New post 23 Nov 2019, 15:45
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Since the given triangle is a right angled triangle (Triangle inscribed in a circle such that one of its sides is the diameter of the circle is a right angled triangle), the sides would be in the ratio of 1:1:root2.

Given area of the triangle is 72.
I.e. 1/2*base*height = 72
Let base=height=x
Hence base=height=12.

And the length of side AC is 12 root 2.

Area of circle = pi r^2
. =Pi* (12 root 2)^2
. =72 pi

Area of circle - area of triangle
72 pi - 72

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Re: Triangle ABC is inscribed in a circle, such that AC is a diameter of  [#permalink]

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New post 24 Nov 2019, 00:09
Sahithi15 wrote:
Since the given triangle is a right angled triangle (Triangle inscribed in a circle such that one of its sides is the diameter of the circle is a right angled triangle), the sides would be in the ratio of 1:1:root2.

Given area of the triangle is 72.
I.e. 1/2*base*height = 72
Let base=height=x
Hence base=height=12.

And the length of side AC is 12 root 2.

Area of circle = pi r^2
. =Pi* (12 root 2)^2
. =72 pi

Area of circle - area of triangle
72 pi - 72

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Correct!
How do we know that base = height? Why is the hypothenuse for example not the base?
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Re: Triangle ABC is inscribed in a circle, such that AC is a diameter of  [#permalink]

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New post 24 Nov 2019, 14:34
Hi Luca,

Since the side opposite to 90 degrees is the hypotenuse of the triangle which in this case is the diameter of the circle, the other two legs are the base and height of the triangle.

Also, since the other two sides are at an angle of 45 degrees, it qualifies for an isosceles right angled triangle where the sides are in 1:1:root 2 ratio.

Since hypotenuse is the longer side it is root 2 and the rest are in 1:1 ratio.

Hope this helps!!

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Re: Triangle ABC is inscribed in a circle, such that AC is a diameter of  [#permalink]

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New post 25 Nov 2019, 22:36
Sahithi15 wrote:
Since the given triangle is a right angled triangle (Triangle inscribed in a circle such that one of its sides is the diameter of the circle is a right angled triangle), the sides would be in the ratio of 1:1:root2.

Given area of the triangle is 72.
I.e. 1/2*base*height = 72
Let base=height=x
Hence base=height=12.

And the length of side AC is 12 root 2.

Area of circle = pi r^2
. =Pi* (12 root 2)^2
. =72 pi

Area of circle - area of triangle
72 pi - 72

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But is AC the diameter ? Therefore, shouldn't the radius be 6root2 instead of 12root2?
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Re: Triangle ABC is inscribed in a circle, such that AC is a diameter of  [#permalink]

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New post 15 Dec 2019, 06:59
Given area of the triangle is 72.
I.e. 1/2*base*height = 72
Let base=height=x
Base * height = 144
Hence base=height=12.

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Re: Triangle ABC is inscribed in a circle, such that AC is a diameter of  [#permalink]

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New post 15 Dec 2019, 10:23
If you take the triangle which is 45 45 90. Bisecting the triangle and stacking it you get a square. I hope this makes sense. Area = 72 or sides 6root 2 since ac is the dia ac/2 is the radius
Pi r^2 = 72pi. Now take the difference 72(pi-1)

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Re: Triangle ABC is inscribed in a circle, such that AC is a diameter of   [#permalink] 15 Dec 2019, 10:23
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