Q. A tangent to a circle is drawn from a point outside the circle. If the distance from the point to the center of the circle is 10 cm and the radius of the circle is 6 cm, what is the length of the tangent?
A.
8 cm
B.
10 cm
C.
12 cm
D.
14 cm
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Solution
Using the Pythagorean theorem, the length of the tangent is √(10^2 - 6^2) = √(100 - 36) = √64 = 8 cm.
Correct Answer:
A
— 8 cm
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Q. If a tangent from point P touches the circle at point T, and the radius OT is 6 cm, what is the length of PT if PT is the tangent?
A.
6 cm
B.
12 cm
C.
8 cm
D.
10 cm
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Solution
The length of the tangent PT from point P to the point of tangency T is equal to the radius OT, which is 6 cm.
Correct Answer:
A
— 6 cm
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Q. If the angle subtended by an arc at the center of a circle is 60 degrees, what is the angle subtended at any point on the circumference?
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
120 degrees
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Solution
The angle subtended at the circumference is half of that at the center. Therefore, it is 60 degrees / 2 = 30 degrees.
Correct Answer:
A
— 30 degrees
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Q. If the diameter of a circle is 14 cm, what is the area of the circle?
A.
154 cm²
B.
196 cm²
C.
100 cm²
D.
120 cm²
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Solution
The radius r = diameter/2 = 14/2 = 7 cm. The area A = πr² = π(7)² = 49π ≈ 154 cm².
Correct Answer:
B
— 196 cm²
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Q. If the radius of a circle is halved, how does the area change?
A.
It remains the same
B.
It doubles
C.
It is halved
D.
It is quartered
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Solution
If the radius is halved, the area becomes π(r/2)² = π(r²/4) = (1/4) * πr², which means the area is quartered.
Correct Answer:
D
— It is quartered
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Q. If two chords AB and CD of a circle intersect at point E, and AE = 3 cm, EB = 5 cm, what is the length of CE if ED = 4 cm?
A.
6 cm
B.
8 cm
C.
12 cm
D.
10 cm
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Solution
According to the intersecting chords theorem, AE * EB = CE * ED. Thus, 3 * 5 = CE * 4, which gives CE = 15/4 = 3.75 cm.
Correct Answer:
B
— 8 cm
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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?
A.
They are equal
B.
They are unequal
C.
One is longer
D.
One is shorter
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Solution
In a circle, equal chords are equidistant from the center.
Correct Answer:
A
— They are equal
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Q. If two circles intersect at points A and B, and the line segment AB is the common chord, what can be said about the angles subtended by AB at the centers of the circles?
A.
They are equal
B.
They are supplementary
C.
They are complementary
D.
They are unequal
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Solution
The angles subtended by the common chord AB at the centers of the circles are equal.
Correct Answer:
A
— They are equal
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Q. If two tangents are drawn from a point outside a circle, what is the relationship between the lengths of the tangents?
A.
They are equal
B.
One is longer
C.
One is shorter
D.
They are unrelated
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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 the radius is 10 cm, what is the length of an arc that subtends an angle of 60 degrees at the center?
A.
10.47 cm
B.
6.28 cm
C.
17.45 cm
D.
5.24 cm
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Solution
The length of an arc is given by the formula L = (θ/360) * 2πr. Here, L = (60/360) * 2 * π * 10 = (1/6) * 20π = 10.47 cm.
Correct Answer:
A
— 10.47 cm
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Q. In a circle, if the radius is 5 cm, what is the length of the diameter?
A.
5 cm
B.
10 cm
C.
15 cm
D.
20 cm
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Solution
The diameter of a circle is twice the radius. Therefore, diameter = 2 * radius = 2 * 5 cm = 10 cm.
Correct Answer:
B
— 10 cm
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Q. In a circle, if the radius is halved, how does the area change?
A.
It remains the same
B.
It doubles
C.
It is halved
D.
It is quartered
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Solution
Area = π * r². If radius is halved, area becomes π * (r/2)² = π * (r²/4) = (1/4) * original area.
Correct Answer:
D
— It is quartered
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Q. In a right triangle inscribed in a circle, what is the relationship between the hypotenuse and the diameter of the circle?
A.
They are equal
B.
The hypotenuse is longer
C.
The hypotenuse is shorter
D.
They are unrelated
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Solution
In a right triangle inscribed in a circle, the hypotenuse is equal to the diameter of the circle.
Correct Answer:
A
— They are equal
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Q. In a right triangle, if one leg is 6 cm and the hypotenuse is 10 cm, what is the length of the other leg?
A.
8 cm
B.
7 cm
C.
5 cm
D.
4 cm
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Solution
Using the Pythagorean theorem, a² + b² = c², we have 6² + b² = 10². Thus, 36 + b² = 100, leading to b² = 64, so b = 8 cm.
Correct Answer:
A
— 8 cm
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Q. Two triangles are similar if their corresponding angles are equal. If triangle DEF is similar to triangle XYZ, and angle D = 50 degrees, what is the measure of angle X?
A.
50 degrees
B.
60 degrees
C.
70 degrees
D.
80 degrees
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Solution
Since the triangles are similar, corresponding angles are equal. Therefore, angle X = angle D = 50 degrees.
Correct Answer:
A
— 50 degrees
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Q. What is the circumference of a circle with a diameter of 12 cm? (Use π = 3.14)
A.
37.68 cm
B.
24 cm
C.
12 cm
D.
18.84 cm
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Solution
Circumference = π * diameter = 3.14 * 12 cm = 37.68 cm.
Correct Answer:
A
— 37.68 cm
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Q. What is the length of the radius of a circle if the area is 36π cm²?
A.
6 cm
B.
12 cm
C.
18 cm
D.
9 cm
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Solution
Using the area formula A = πr², we have 36π = πr², which simplifies to r² = 36, giving r = 6 cm.
Correct Answer:
A
— 6 cm
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Q. What is the measure of an inscribed angle that intercepts an arc of 80 degrees?
A.
40 degrees
B.
80 degrees
C.
160 degrees
D.
20 degrees
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Solution
The measure of an inscribed angle is half the measure of the intercepted arc. Therefore, it is 80 degrees / 2 = 40 degrees.
Correct Answer:
A
— 40 degrees
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Q. What is the measure of the central angle that subtends an arc of length 5 cm in a circle of radius 10 cm?
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
45 degrees
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Solution
The formula for the arc length is L = rθ, where θ is in radians. Thus, θ = L/r = 5/10 = 0.5 radians. Converting to degrees, θ = 0.5 * (180/π) ≈ 28.65 degrees, which is closest to 30 degrees.
Correct Answer:
B
— 60 degrees
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Q. What is the relationship between the angles subtended by an arc at the center and at any point on the remaining part of the circle?
A.
The angle at the center is half
B.
The angle at the center is double
C.
They are equal
D.
The angle at the center is zero
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Solution
The angle subtended by an arc at the center is always double the angle subtended at any point on the remaining part of the circle.
Correct Answer:
B
— The angle at the center is double
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