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11 May 2021, 02:45
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Difficulty:
45%
(medium)
Question Stats:
62% (02:00) correct
38%
(02:23)
wrong
based on 34
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If P is the center of the circle shown above, and BAC=30º, and the area of triangle ABC is √3, what is the radius of the circle?
A. 1/2
B. 1/(√2)
C. 1
D. √2
E. 2
Attachment:
1.png [ 7.35 KiB | Viewed 1457 times ]
12 May 2021, 10:13
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The most important concept, perhaps, that one needs to know to start off solving this question is “The inscribed angle made by the diameter is a right angle”, which is also stated by many as “The angle in a semicircle is a right angle”.
Note that the inscribed angle is the angle made by an arc/chord at a point on the circumference on the circle.
This is where the data about P becomes so valuable. Knowing that P is the centre of the circle, it’s easy to infer that AC is the diameter. Therefore, angle B, which is the inscribed angle subtended by AC, is a right angle, which makes the triangle ABC is a right-angled triangle.
Additionally, we are told that angle BAC = 30 degrees. Therefore, triangle ABC is a 30-60-90 right-angled triangle, which is not surprising since this kind of right-angled triangle is quite frequent in GMAT Geometry questions.
In a 30-60-90 right-angled triangle, always hold on to a couple of elements – the 30-degree angle and the side opposite to it.
If the side opposite to the 30-degree angle is x units:
The side opposite to the 60-degree angle is x√3 units AND
The side opposite to the right angle i.e. you’re right, the hypotenuse is 2x units.
Let’s look at our right-angled triangle. Angle BAC = 30 degrees, angle ABC = 90 degrees, therefore angle ACB = 60 degrees.
The side opposite to angle BAC is BC and let this be x units. From our discussion, we can now deduce that AB = x√3 units and AC = 2x units. Therefore, the radius of the circle, PC = x units.
We have been told that the area of triangle ABC is √3 units. From our knowledge of calculating the area of a triangle,
√3 = ½ * AB * BC since AB is the base of triangle ABC and BC is the height. Substituting the values of AB and BC, we have,
√3 = ½ * x√3 * x. On simplifying, we get x = √2.
The correct answer is D.
Hope that helps!
Aravind B T
Note that the inscribed angle is the angle made by an arc/chord at a point on the circumference on the circle.
This is where the data about P becomes so valuable. Knowing that P is the centre of the circle, it’s easy to infer that AC is the diameter. Therefore, angle B, which is the inscribed angle subtended by AC, is a right angle, which makes the triangle ABC is a right-angled triangle.
Additionally, we are told that angle BAC = 30 degrees. Therefore, triangle ABC is a 30-60-90 right-angled triangle, which is not surprising since this kind of right-angled triangle is quite frequent in GMAT Geometry questions.
In a 30-60-90 right-angled triangle, always hold on to a couple of elements – the 30-degree angle and the side opposite to it.
If the side opposite to the 30-degree angle is x units:
The side opposite to the 60-degree angle is x√3 units AND
The side opposite to the right angle i.e. you’re right, the hypotenuse is 2x units.
Let’s look at our right-angled triangle. Angle BAC = 30 degrees, angle ABC = 90 degrees, therefore angle ACB = 60 degrees.
The side opposite to angle BAC is BC and let this be x units. From our discussion, we can now deduce that AB = x√3 units and AC = 2x units. Therefore, the radius of the circle, PC = x units.
We have been told that the area of triangle ABC is √3 units. From our knowledge of calculating the area of a triangle,
√3 = ½ * AB * BC since AB is the base of triangle ABC and BC is the height. Substituting the values of AB and BC, we have,
√3 = ½ * x√3 * x. On simplifying, we get x = √2.
The correct answer is D.
Hope that helps!
Aravind B T
