Q. A circle is inscribed in a triangle. What is the relationship between the radius of the incircle and the area of the triangle?
A.
r = A/s
B.
r = s/A
C.
r = 2A/s
D.
r = s/2A
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Solution
The radius of the incircle (r) is given by the formula r = A/s, where A is the area of the triangle and s is the semi-perimeter.
Correct Answer:
A
— r = A/s
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Q. If a circle has a circumference of 31.4 cm, what is the radius?
A.
5 cm
B.
10 cm
C.
15 cm
D.
7.5 cm
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Solution
Circumference = 2πr, thus r = Circumference / (2π) = 31.4 / (2π) ≈ 5 cm.
Correct Answer:
B
— 10 cm
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Q. If a circle has a diameter of 14 cm, what is the length of the radius?
A.
7 cm
B.
14 cm
C.
28 cm
D.
3.5 cm
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Solution
The radius of a circle is half of the diameter. Therefore, the radius is 14 cm / 2 = 7 cm.
Correct Answer:
A
— 7 cm
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Q. If the center of a circle is at (2, 3) and the radius is 5, what is the equation of the circle?
A.
(x - 2)² + (y - 3)² = 25
B.
(x + 2)² + (y + 3)² = 25
C.
(x - 2)² + (y + 3)² = 5
D.
(x + 2)² + (y - 3)² = 25
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Solution
The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
Correct Answer:
A
— (x - 2)² + (y - 3)² = 25
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Q. If the radius of a circle is 7 cm, what is the area of the circle?
A.
154 cm²
B.
49 cm²
C.
28 cm²
D.
44 cm²
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Solution
The area of a circle is given by the formula A = πr². Thus, A = π(7)² = 49π ≈ 154 cm².
Correct Answer:
A
— 154 cm²
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Q. If two chords AB and CD intersect at point E inside a circle, what is the relationship between the segments AE, EB, CE, and ED?
A.
AE * EB = CE * ED
B.
AE + EB = CE + ED
C.
AE = CE
D.
EB = ED
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Solution
According to the intersecting chords theorem, AE * EB = CE * ED.
Correct Answer:
A
— AE * EB = CE * ED
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Q. If two chords AB and CD of a circle intersect at point E, what is the relationship between AE, EB, CE, and ED?
A.
AE * EB = CE * ED
B.
AE + EB = CE + ED
C.
AE - EB = CE - ED
D.
AE / EB = CE / ED
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Solution
According to the intersecting chords theorem, AE * EB = CE * ED.
Correct Answer:
A
— AE * EB = CE * ED
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Q. In a circle, if the angle at the center is 120 degrees, what is the angle at the circumference subtended by the same arc?
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
120 degrees
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Solution
The angle at the circumference is half the angle at the center, so it is 120/2 = 60 degrees.
Correct Answer:
A
— 30 degrees
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Q. In a circle, if the angle subtended by an arc at the center is 120 degrees, what is the angle subtended by the same arc at any point on the remaining part of the circle?
A.
60 degrees
B.
120 degrees
C.
30 degrees
D.
90 degrees
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Solution
The angle subtended at the circumference is half of the angle subtended at the center, so it is 120/2 = 60 degrees.
Correct Answer:
A
— 60 degrees
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Q. In a circle, if the radius is doubled, how does the circumference change?
A.
It doubles
B.
It triples
C.
It quadruples
D.
It remains the same
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Solution
The circumference of a circle is given by C = 2πr. If the radius is doubled, the new circumference is C = 2π(2r) = 4πr, which is double the original circumference.
Correct Answer:
A
— It doubles
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Q. In a circle, if two tangents are drawn from an external point to the circle, what can be said about the lengths of the tangents?
A.
They are equal
B.
They are unequal
C.
One is longer than the radius
D.
They are both zero
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Solution
The lengths of the tangents drawn from an external point to a circle are always equal.
Correct Answer:
A
— They are equal
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Q. Two chords AB and CD of a circle intersect at point E. If AE = 3 cm, EB = 5 cm, and CE = 4 cm, what is the length of ED?
A.
6 cm
B.
8 cm
C.
5 cm
D.
7 cm
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Solution
Using the intersecting chords theorem: AE * EB = CE * ED, we have 3 * 5 = 4 * ED, thus ED = 15/4 = 3.75 cm.
Correct Answer:
B
— 8 cm
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Q. Two circles intersect at points A and B. If the line segment AB is the common chord, what can be said about the perpendicular from the center of either circle to AB?
A.
It bisects AB
B.
It is equal to AB
C.
It is longer than AB
D.
It is shorter than AB
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Solution
The perpendicular from the center of a circle to a chord bisects the chord.
Correct Answer:
A
— It bisects AB
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Q. Two circles intersect at points A and B. What is the relationship between the angles ∠AOB and ∠APB, where O is the center of one circle and P is the center of the other?
A.
∠AOB = ∠APB
B.
∠AOB = 2∠APB
C.
∠AOB = ½∠APB
D.
∠AOB + ∠APB = 180 degrees
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Solution
The angle ∠AOB is twice the angle ∠APB because of the inscribed angle theorem, which states that the angle at the center is twice the angle at the circumference.
Correct Answer:
B
— ∠AOB = 2∠APB
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Q. What is the distance between the center of a circle and a point on its circumference?
A.
Diameter
B.
Radius
C.
Chord
D.
Arc
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Solution
The distance between the center of a circle and any point on its circumference is defined as the radius.
Correct Answer:
B
— Radius
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Q. What is the length of the arc of a circle with a radius of 10 cm that subtends an angle of 60 degrees at the center?
A.
10.47 cm
B.
6.28 cm
C.
17.45 cm
D.
10.00 cm
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Solution
The length of an arc is given by the formula 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. What is the measure of the angle subtended by a diameter at any point on the circle?
A.
90 degrees
B.
60 degrees
C.
45 degrees
D.
180 degrees
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Solution
According to the inscribed angle theorem, the angle subtended by a diameter at any point on the circle is always 90 degrees.
Correct Answer:
A
— 90 degrees
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Q. What is the radius of a circle if the circumference is 31.4 cm?
A.
5 cm
B.
10 cm
C.
15 cm
D.
20 cm
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Solution
The circumference C = 2πr. Thus, r = C/(2π) = 31.4/(2π) ≈ 5 cm.
Correct Answer:
B
— 10 cm
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Q. What is the relationship between the lengths of the tangents drawn from an external point to a circle?
A.
They are equal
B.
They are unequal
C.
They depend on the radius
D.
They are always zero
Show solution
Solution
The lengths of the tangents drawn from an external point to a circle are always equal.
Correct Answer:
A
— They are equal
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