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If points A and C both lie on the circle with center B and the measurement of angle ABC is not a multiple of 30, what is the ratio of the area of the circle centered at point B to the area of triangle ABC?
- a)None of these
- b)2π
- c)4π
- d)π(BC)2/.5(BC)(AB)
- e)2π(AB)2/(BC)2
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
If points A and C both lie on the circle with center B and the measure...
Begin by finding the area of the circle:
Areacircle = πr2
Areacircle = π(AB)2 = π(BC)2
In dealing with triangle ABC, BC = AB since both are radii. At this point, some students make a mistake and assume that AB is the height of the triangle and BC is the base of the triangle (or vice versa). However, we cannot assume that BC is the base and AB is the height since we have not yet shown that ABC is a right triangle. You could only make BC the base and AB the height if triangle ABC were a right triangle (in which case AB would be a perpendicular segment drawn from a vertex, A, to the side opposite that vertex, B).
By definition, the height of a triangle is the length of a segment drawn from a vertex perpendicular to the side opposite that vertex. A line that is perpendicular to the side opposite a vertex will, by definition, form a 90 degree angle. Consequently, for line AB to be the height of triangle ABC, angle ABC must be a right angle (i.e., 90 degrees).
Since the question states that "the measurement of angle ABC is not a multiple of 30," angle ABC cannot be 30, 60, 90, 120, etc. Consequently, angle ABC is not a right angle and line AB is not the height of triangle ABC.
Without the height, you cannot determine the area of the triangle. Without the area of the triangle, you do not have enough information to solve the problem. The correct answer is It Cannot Be Determined.
Note: If the question omitted the words "the measurement of angle ABC is not a multiple of 30" and instead said that the length of line AC is 21/2 times larger than the radius, you would be dealing with a 45-45-90 right triangle with sides r, r, and r*21/2. In this instance with a right triangle, the area of the triangle would be (1/2)bh = (1/2)(r)(r) = .5r2 and the ratio of the area of the circle centered at point B to the area of triangle ABC would be 2π.
Areacircle = πr2
Areacircle = π(AB)2 = π(BC)2
In dealing with triangle ABC, BC = AB since both are radii. At this point, some students make a mistake and assume that AB is the height of the triangle and BC is the base of the triangle (or vice versa). However, we cannot assume that BC is the base and AB is the height since we have not yet shown that ABC is a right triangle. You could only make BC the base and AB the height if triangle ABC were a right triangle (in which case AB would be a perpendicular segment drawn from a vertex, A, to the side opposite that vertex, B).
By definition, the height of a triangle is the length of a segment drawn from a vertex perpendicular to the side opposite that vertex. A line that is perpendicular to the side opposite a vertex will, by definition, form a 90 degree angle. Consequently, for line AB to be the height of triangle ABC, angle ABC must be a right angle (i.e., 90 degrees).
Since the question states that "the measurement of angle ABC is not a multiple of 30," angle ABC cannot be 30, 60, 90, 120, etc. Consequently, angle ABC is not a right angle and line AB is not the height of triangle ABC.
Without the height, you cannot determine the area of the triangle. Without the area of the triangle, you do not have enough information to solve the problem. The correct answer is It Cannot Be Determined.
Note: If the question omitted the words "the measurement of angle ABC is not a multiple of 30" and instead said that the length of line AC is 21/2 times larger than the radius, you would be dealing with a 45-45-90 right triangle with sides r, r, and r*21/2. In this instance with a right triangle, the area of the triangle would be (1/2)bh = (1/2)(r)(r) = .5r2 and the ratio of the area of the circle centered at point B to the area of triangle ABC would be 2π.
Most Upvoted Answer
If points A and C both lie on the circle with center B and the measure...
Begin by finding the area of the circle:
Areacircle = πr2
Areacircle = π(AB)2 = π(BC)2
In dealing with triangle ABC, BC = AB since both are radii. At this point, some students make a mistake and assume that AB is the height of the triangle and BC is the base of the triangle (or vice versa). However, we cannot assume that BC is the base and AB is the height since we have not yet shown that ABC is a right triangle. You could only make BC the base and AB the height if triangle ABC were a right triangle (in which case AB would be a perpendicular segment drawn from a vertex, A, to the side opposite that vertex, B).
By definition, the height of a triangle is the length of a segment drawn from a vertex perpendicular to the side opposite that vertex. A line that is perpendicular to the side opposite a vertex will, by definition, form a 90 degree angle. Consequently, for line AB to be the height of triangle ABC, angle ABC must be a right angle (i.e., 90 degrees).
Since the question states that "the measurement of angle ABC is not a multiple of 30," angle ABC cannot be 30, 60, 90, 120, etc. Consequently, angle ABC is not a right angle and line AB is not the height of triangle ABC.
Without the height, you cannot determine the area of the triangle. Without the area of the triangle, you do not have enough information to solve the problem. The correct answer is It Cannot Be Determined.
Note: If the question omitted the words "the measurement of angle ABC is not a multiple of 30" and instead said that the length of line AC is 21/2 times larger than the radius, you would be dealing with a 45-45-90 right triangle with sides r, r, and r*21/2. In this instance with a right triangle, the area of the triangle would be (1/2)bh = (1/2)(r)(r) = .5r2 and the ratio of the area of the circle centered at point B to the area of triangle ABC would be 2π.
Areacircle = πr2
Areacircle = π(AB)2 = π(BC)2
In dealing with triangle ABC, BC = AB since both are radii. At this point, some students make a mistake and assume that AB is the height of the triangle and BC is the base of the triangle (or vice versa). However, we cannot assume that BC is the base and AB is the height since we have not yet shown that ABC is a right triangle. You could only make BC the base and AB the height if triangle ABC were a right triangle (in which case AB would be a perpendicular segment drawn from a vertex, A, to the side opposite that vertex, B).
By definition, the height of a triangle is the length of a segment drawn from a vertex perpendicular to the side opposite that vertex. A line that is perpendicular to the side opposite a vertex will, by definition, form a 90 degree angle. Consequently, for line AB to be the height of triangle ABC, angle ABC must be a right angle (i.e., 90 degrees).
Since the question states that "the measurement of angle ABC is not a multiple of 30," angle ABC cannot be 30, 60, 90, 120, etc. Consequently, angle ABC is not a right angle and line AB is not the height of triangle ABC.
Without the height, you cannot determine the area of the triangle. Without the area of the triangle, you do not have enough information to solve the problem. The correct answer is It Cannot Be Determined.
Note: If the question omitted the words "the measurement of angle ABC is not a multiple of 30" and instead said that the length of line AC is 21/2 times larger than the radius, you would be dealing with a 45-45-90 right triangle with sides r, r, and r*21/2. In this instance with a right triangle, the area of the triangle would be (1/2)bh = (1/2)(r)(r) = .5r2 and the ratio of the area of the circle centered at point B to the area of triangle ABC would be 2π.
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If points A and C both lie on the circle with center B and the measurement of angle ABC is not a multiple of 30, what is the ratio of the area of the circle centered at point B to the area of triangle ABC?a)None of theseb)2πc)4πd)π(BC)2/.5(BC)(AB)e)2π(AB)2/(BC)2Correct answer is option 'B'. Can you explain this answer?
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