Q. If a chord of a circle is 10 cm long and the radius is 6 cm, what is the distance from the center of the circle to the chord?
A.
4 cm
B.
3 cm
C.
5 cm
D.
2 cm
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Solution
Using the Pythagorean theorem, the distance from the center to the chord is √(r² - (c/2)²) = √(6² - 5²) = √(36 - 25) = √11 cm.
Correct Answer:
B
— 3 cm
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Q. If a circle has a radius of 4 cm, what is the area of the circle?
A.
8π cm²
B.
12π cm²
C.
16π cm²
D.
20π cm²
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Solution
The area A of a circle is given by A = πr². Therefore, A = π(4)² = 16π cm².
Correct Answer:
C
— 16π cm²
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Q. If a circle has a radius of 4 cm, what is the diameter of the circle?
A.
8 cm
B.
12 cm
C.
16 cm
D.
10 cm
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Solution
The diameter of a circle is twice the radius. For a radius of 4 cm, the diameter is 2 * 4 = 8 cm.
Correct Answer:
A
— 8 cm
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Q. If a circle has a radius of 7 cm, what is the length of the circumference?
A.
14π cm
B.
21π cm
C.
49 cm
D.
14 cm
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Solution
The circumference of a circle is given by the formula C = 2πr. Therefore, C = 2π(7) = 14π cm.
Correct Answer:
A
— 14π cm
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Q. If a tangent and a chord intersect at a point on the circle, what is the relationship between the angle formed and the angle subtended by the chord at the opposite arc?
A.
They are equal
B.
The tangent angle is double
C.
The chord angle is double
D.
They are supplementary
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Solution
The angle formed by the tangent and the chord is equal to the angle subtended by the chord at the opposite arc.
Correct Answer:
A
— They are equal
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Q. If a tangent and a radius meet at a point on the circle, what is the angle between them?
A.
90 degrees
B.
45 degrees
C.
180 degrees
D.
0 degrees
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Solution
The tangent to a circle at any point is perpendicular to the radius at that point, forming a 90-degree angle.
Correct Answer:
A
— 90 degrees
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Q. If the diameter of a circle is 10 cm, what is the circumference of the circle?
A.
31.4 cm
B.
20 cm
C.
15.7 cm
D.
25 cm
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Solution
The circumference of a circle is given by the formula C = πd. For d = 10 cm, C = π(10) ≈ 31.4 cm.
Correct Answer:
A
— 31.4 cm
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Q. If two chords AB and CD of a circle intersect at point E, which of the following is true?
A.
AE * EB = CE * ED
B.
AE + EB = CE + ED
C.
AE = CE
D.
EB = ED
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Solution
The theorem states that if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal, i.e., AE * EB = CE * ED.
Correct Answer:
A
— AE * EB = CE * ED
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Q. If two chords in a circle are equal in length, what can be said about their distances from the center of the circle?
A.
They are equal
B.
One is longer than the other
C.
They are perpendicular to each other
D.
They are at different angles
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Solution
If two chords in a circle are equal in length, they are equidistant from the center of the circle.
Correct Answer:
A
— They are equal
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Q. If two tangents are drawn from an external point to a circle, what can be said about the lengths of the tangents?
A.
They are equal
B.
They are unequal
C.
One is longer
D.
One is shorter
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Solution
The lengths of the tangents drawn from an external point to a circle are equal.
Correct Answer:
A
— They are equal
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Q. In a circle, if a radius is drawn to a point where a tangent touches the circle, what is the angle between the radius and the tangent?
A.
90 degrees
B.
45 degrees
C.
60 degrees
D.
180 degrees
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Solution
The radius drawn to the point of tangency is perpendicular to the tangent, forming a 90-degree angle.
Correct Answer:
A
— 90 degrees
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Q. In a circle, if a tangent is drawn from a point outside the circle, what is the relationship between the tangent and the radius at the point of contact?
A.
They are equal
B.
They are perpendicular
C.
They are parallel
D.
They form an acute angle
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Solution
A tangent drawn from a point outside the circle is perpendicular to the radius at the point of contact.
Correct Answer:
B
— They are perpendicular
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Q. In a circle, if an angle inscribed in the circle intercepts an arc of 60 degrees, what is the measure of the angle?
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
120 degrees
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Solution
The inscribed angle is half the measure of the intercepted arc: 60 degrees / 2 = 30 degrees.
Correct Answer:
A
— 30 degrees
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Q. In a circle, if the radius is 10 cm, what is the length of an arc that subtends a central angle of 60 degrees?
A.
10.47 cm
B.
6.28 cm
C.
17.45 cm
D.
5.24 cm
Show solution
Solution
The length of an arc is given by L = (θ/360) * 2πr. Here, L = (60/360) * 2π(10) = (1/6) * 20π ≈ 10.47 cm.
Correct Answer:
A
— 10.47 cm
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Q. In a circle, if two angles subtended by the same arc are equal, what can be concluded about those angles?
A.
They are complementary
B.
They are equal
C.
They are supplementary
D.
They are proportional
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Solution
Angles subtended by the same arc at the circumference of a circle are equal.
Correct Answer:
B
— They are equal
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Q. In a circle, if two chords AB and CD are equal in length, what can be said about their distances from the center?
A.
They are equal
B.
One is longer
C.
One is shorter
D.
Cannot be determined
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Solution
If two chords are equal in length, they are equidistant from the center of the circle.
Correct Answer:
A
— They are equal
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Q. Two circles are tangent to each other. If the radius of the first circle is 3 cm and the second is 5 cm, what is the distance between their centers?
A.
2 cm
B.
8 cm
C.
3 cm
D.
5 cm
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Solution
The distance between the centers of two tangent circles is the sum of their radii: 3 cm + 5 cm = 8 cm.
Correct Answer:
B
— 8 cm
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Q. What is the length of an arc of a circle with a radius of 5 cm that subtends an angle of 60 degrees at the center?
A.
5.24 cm
B.
3.14 cm
C.
5.00 cm
D.
10.47 cm
Show solution
Solution
The length of an arc is given by the formula L = (θ/360) * 2πr. For r = 5 cm and θ = 60°, L = (60/360) * 2π(5) = (1/6) * 10π ≈ 5.24 cm.
Correct Answer:
A
— 5.24 cm
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Q. What is the measure of the angle subtended by an arc at the center of a circle compared to the angle subtended at any point on the remaining part of the circle?
A.
Half the angle at the center
B.
Equal to the angle at the center
C.
Twice the angle at the center
D.
None of the above
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Solution
The angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the remaining part of the circle.
Correct Answer:
A
— Half the angle at the center
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Q. What is the measure of the central angle that subtends an arc of 120 degrees in a circle?
A.
60 degrees
B.
120 degrees
C.
180 degrees
D.
90 degrees
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Solution
The measure of the central angle is equal to the measure of the arc it subtends, which is 120 degrees.
Correct Answer:
B
— 120 degrees
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Q. What is the relationship between the angles subtended by the same arc at the center and at any point on the circumference?
A.
The angle at the center is double
B.
The angle at the center is half
C.
They are equal
D.
They are supplementary
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Solution
The angle subtended at the center is always double the angle subtended at any point on the circumference.
Correct Answer:
A
— The angle at the center is double
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Q. What is the relationship between the angles subtended by the same arc at the center and at the circumference?
A.
They are equal
B.
The angle at the center is twice the angle at the circumference
C.
The angle at the circumference is twice the angle at the center
D.
They are complementary
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Solution
The angle subtended at the center is always twice the angle subtended at the circumference by the same arc.
Correct Answer:
B
— The angle at the center is twice the angle at the circumference
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Q. What is the relationship between the lengths of two tangents drawn from an external point to a circle?
A.
They are equal
B.
One is longer than the other
C.
They are perpendicular to the radius
D.
They are parallel
Show solution
Solution
The lengths of two tangents drawn from an external point to a circle are equal.
Correct Answer:
A
— They are equal
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Q. What is the relationship between the radius and diameter of a circle?
A.
The radius is twice the diameter.
B.
The diameter is twice the radius.
C.
The radius and diameter are equal.
D.
The radius is half the diameter.
Show solution
Solution
The diameter of a circle is twice the length of the radius.
Correct Answer:
B
— The diameter is twice the radius.
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Q. What is the relationship between the radius and the tangent at the point of contact on a circle?
A.
They are equal
B.
They are perpendicular
C.
They are parallel
D.
They are collinear
Show solution
Solution
The radius drawn to the point of contact is always perpendicular to the tangent at that point.
Correct Answer:
B
— They are perpendicular
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