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505-555 (Easy)|   Geometry|      
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In the figure above, the radius of circle with center O is 1 06 Jun 2016, 08:40
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Always remember: A circle inscribed in a semicircle with always be a right angle triangle.
In this case,
∠ABC = 90

Hence in the triangle ABC, we have base = BC = 1
Hypotenuse = CA = 2
Height = AB^2 = 2^2 + 1^2
Hence AB = √3

Therefore, area = 1/2*√3*1 = √3/2

Correct option: B
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In the figure above, the radius of circle with center O is 1 11 Feb 2017, 01:54
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How do you know that BC is the base and AC is the height? Is it because on right angle triangle the 90 degree angle is never the base or the height? Bunuel
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In the figure above, the radius of circle with center O is 1 11 Feb 2017, 03:03
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How do you know that BC is the base and AC is the height? Is it because on right angle triangle the 90 degree angle is never the base or the height? Bunuel
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The altitude (height) of a triangle is the perpendicular from the base to the opposite vertex (the base may need to be extended). Since there are three possible bases, there are also three possible altitudes. In case of a right triangle both legs are altitudes as well as perpendicular from right angle to hypotenuse.
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In the figure above, the radius of circle with center O is 1 26 May 2017, 09:11
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In the figure above, the radius of circle with center O is 1 and BC = 1. What is the area of triangular region ABC ?

Hi guys,
Just one doubt, how can we say that triangle ABC is a right angled triangle given the radius is one & other side BC is one.
I mean to say that is there any property for circle that if radius is equal to any one side of the triangle, then the inscribed triangle so formed will be right-angled one?

We know that AC is a diameter. There is a property of a right triangle inscribed in circle:

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 side, then that triangle is a right triangle and diameter is hypotenuse.
Attachment:
Math_Tri_inscribed.png

Hope it helps.
Show more

Usually it's always a GMAT rule that you should just assume ANYTHING...

The question only states that "the radius of the circle with center O is 1 and BC = 1".

How do I know that AC is the diameter? Yes, I see it on the graph but the graphs aren't accurate, are they? A could also be placed 1mm before the circle's line. It's not stated anywhere that the triangle is inscribed, is it?

Thanks for explaining!
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In the figure above, the radius of circle with center O is 1 26 May 2017, 09:31
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In the figure above, the radius of circle with center O is 1 and BC = 1. What is the area of triangular region ABC ?

Hi guys,
Just one doubt, how can we say that triangle ABC is a right angled triangle given the radius is one & other side BC is one.
I mean to say that is there any property for circle that if radius is equal to any one side of the triangle, then the inscribed triangle so formed will be right-angled one?

We know that AC is a diameter. There is a property of a right triangle inscribed in circle:

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 side, then that triangle is a right triangle and diameter is hypotenuse.
Attachment:
Math_Tri_inscribed.png

Hope it helps.

Usually it's always a GMAT rule that you should just assume ANYTHING...

The question only states that "the radius of the circle with center O is 1 and BC = 1".

How do I know that AC is the diameter? Yes, I see it on the graph but the graphs aren't accurate, are they? A could also be placed 1mm before the circle's line. It's not stated anywhere that the triangle is inscribed, is it?

Thanks for explaining!
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First of all, if it were the way you are saying, how would you solve the question? It's a PS question, so one of the answers should be correct. How you get it?

Next, GMAT will not use such tricks. Check what is OG saying about the figures.

OFFICIAL GUIDE:

Problem Solving
Figures: All figures accompanying problem solving questions are intended to provide information useful in solving the problems. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Data Sufficiency:
Figures:
• Figures conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2).
• Lines shown as straight are straight, and lines that appear jagged are also straight.
• The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero.
• All figures lie in a plane unless otherwise indicated.
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In the figure above, the radius of circle with center O is 1 19 Jul 2018, 19:00
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Hi all,

Why can't AC be the base, and a vertical line drawn down from B, the height? I keep getting answer A, Sqrt(2)/(2).

The vertical line from B forms a smaller 45-45-90 triangle to the right, with BC as the hypotenuse, and the height of the entire triangle becomes 1/sqrt(2). Since the base is then 2 times radius, or 2, the Area = b*h/2, so the answer is A. Where am I going wrong?
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In the figure above, the radius of circle with center O is 1 19 Jul 2018, 21:03
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Hi all,

Why can't AC be the base, and a vertical line drawn down from B, the height? I keep getting answer A, Sqrt(2)/(2).

The vertical line from B forms a smaller 45-45-90 triangle to the right, with BC as the hypotenuse, and the height of the entire triangle becomes 1/sqrt(2). Since the base is then 2 times radius, or 2, the Area = b*h/2, so the answer is A. Where am I going wrong?
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A triangle has three bases, so yes you can consider AC to be the base. The problem is that the perpendicular from B to AC does not create 45-45-90 triangle. Check the image below:



Show Spoiler
Attachment:
triangle1.png
triangle1.png [ 8.55 KiB | Viewed 6530 times ]
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In the figure above, the radius of circle with center O is 1 24 Nov 2018, 17:06
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A video explanation can be found here:
https://www.youtube.com/watch?v=QjP0s-vOXLY

Three of the many equations and facts you'll need to memorize for the GMAT are

- Area of a triangle = 1/2(base)(height)

- In a right triangle with angles 30-60-90 degrees, if the leg opposite the 30 degree angle has length a, then the hypotenuse has a length of 2a, and the other leg has a length of a√3 (see video below)

https://www.youtube.com/watch?v=QjP0s-vOXLY

- If a triangle inscribed in a circle has the diameter of the circle as one of its sides, that side will always be the hypotenuse, and the two legs that touch the circumference of the crcle will always interesect at 90 degrees (also best explained in video below)

Here we have hypotenuse = 2, therefore base = 1 and height = √3, therefore area = 1/2(1)(√3) = √3/2

Answer B.
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In the figure above, the radius of circle with center O is 1 10 Aug 2020, 04:37
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greatchap

In the figure above, the radius of the circle with center O is 1 and BC = 1. What is the area of triangular region ABC ?

(A) √2/2

(B) √3/2

(C) 1

(D) √2

(E) √3

Show Spoiler
Attachment:
tmp.JPG
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Solution:

Since triangle ABC’s longest side, AC, is also the diameter of the circle, triangle ABC is a right triangle with right angle at B. Now, If we draw radius BO, then we have triangle BOC and it’s an equilateral triangle since BO = CO = BC = 1. Since triangle BOC is equilateral, angle C is 60 degrees. Therefore, triangle ABC is a 30-60-90 right triangle with the shortest side BC = 1. Thus, AB = 1 x √3 = √3 and the area of triangle ABC is ½ x 1 x √3 = √3/2.

Alternate Solution:

Triangle ABC is a right triangle with hypotenuse equal to the diameter of the circle, which is 2. One leg (BC) of the triangle has length 1. Thus, the length of AB is:

(AB)^2 + 1^2 = 2^2

(AB) = √3

The area of right triangle ABC is:

A = ½ (√3) x 1 = √3/2

Answer: B
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In the figure above, the radius of circle with center O is 1 04 Jun 2023, 16:02
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Can we assume the hypotenuse of the triangle runs through the center of the circle if it is not stated?
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In the figure above, the radius of circle with center O is 1 05 Jun 2023, 00:04
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Can we assume the hypotenuse of the triangle runs through the center of the circle if it is not stated?
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AC is a diameter since it passes through the center of the circle. If a diameter forms one side of an inscribed triangle, then that triangle is necessarily a right triangle. The reverse is also true: in a right triangle inscribed in a circle, the hypotenuse will always coincide with the diameter of the circle.
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In the figure above, the radius of circle with center O is 1 06 Oct 2025, 12:33
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