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Questions and Answers
What is the unique property of a circle in relation to the distance from the center to any point on the circle?
What is the unique property of a circle in relation to the distance from the center to any point on the circle?
What is the angle subtended by a chord at a point on the minor arc if the chord is twice the radius of the circle?
What is the angle subtended by a chord at a point on the minor arc if the chord is twice the radius of the circle?
Which of the following best describes the diameter of a circle?
Which of the following best describes the diameter of a circle?
If a chord of a circle is equal to half the radius of the circle, what is the central angle subtended by this chord?
If a chord of a circle is equal to half the radius of the circle, what is the central angle subtended by this chord?
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If a circle has a diameter of 12 units, what would be its radius?
If a circle has a diameter of 12 units, what would be its radius?
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In a circle of radius 6 units, what would be the length of a chord that subtends an angle of 120° on the minor arc?
In a circle of radius 6 units, what would be the length of a chord that subtends an angle of 120° on the minor arc?
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Which formula can be used to find the circumference of a circle?
Which formula can be used to find the circumference of a circle?
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If a chord in a circle is equal to its diameter, what would be the angle subtended by this chord on the major arc?
If a chord in a circle is equal to its diameter, what would be the angle subtended by this chord on the major arc?
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What property of a circle does the formula $\text{Area} = \pi r^2$ help in calculating?
What property of a circle does the formula $\text{Area} = \pi r^2$ help in calculating?
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For a circle with radius 8 units, if a chord has a length of 10 units, what is the central angle it subtends?
For a circle with radius 8 units, if a chord has a length of 10 units, what is the central angle it subtends?
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If a circle has an area of $16\pi$ square units, what is its radius?
If a circle has an area of $16\pi$ square units, what is its radius?
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Given a circle with radius 7 units and a chord that is 14 units long, what would be the angle subtended by this chord at a point on the minor arc?
Given a circle with radius 7 units and a chord that is 14 units long, what would be the angle subtended by this chord at a point on the minor arc?
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Study Notes
Circles
A circle is a two-dimensional shape where all points are equidistant from a central point, called the center. This central point is the center of the circle, and the distance from the center to any point on the circle's edge is called the radius. The diameter is the length of the line passing through the center and touching two opposite points on the circle's edge. The distance around a circle is called the circumference, and the area enclosed by a circle is called the area.
Radius and Diameter
A circle has a unique property where the distance from the center to any point on the circle is the same—the radius. The diameter is twice the radius.
Question: Which of the segments in the circle below is a radius?
Answer: The segment AB is a radius.
Question: Which of the segments in the circle below is a diameter?
Answer: The segment CD is a diameter.
Circumference
The circumference of a circle is the distance around the circle. It can be found using the formula:
Circumference = 2πr
where r is the radius of the circle.
Question: Find the circumference of the circle with a radius of 5 units.
Answer: The circumference of the circle is 10π units.
Area
The area of a circle is the region enclosed by the circle. It can be found using the formula:
Area = πr²
where r is the radius of the circle.
Question: Find the area of the circle with a radius of 3 units.
Answer: The area of the circle is 9π square units.
Chords and Arc Length
A chord is a line segment that passes through the center of a circle and intersects the circle at two points. The length of a chord is equal to the radius of the circle.
An angle subtended by a chord at a point on the circle's minor arc is half the central angle that the chord intersects. An angle subtended by a chord at a point on the major arc is equal to the central angle.
Question: A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc, and also at a point on the major arc.
Answer: At a point on the minor arc, the angle subtended by the chord is 90°. At a point on the major arc, the angle subtended by the chord is 180°.
Practice Questions
Question: Given a circle with radius 4 units, find the length of the chord that touches the circle.
Answer: The length of the chord that touches the circle is 4 units.
Question: Two circles intersect at two points A and B. A line PQ is drawn parallel to the line OO' through A (or B) intersecting the circles at P and Q. Prove that PQ = 2OO'.
Answer: Since PQ is parallel to OO', the distance between P and Q will be equal to the distance between O and O', which is 2OO'. Therefore, PQ = 2OO'.
Question: A point A is at a distance of 13 cm from the center O of a circle of radius 5 cm. AP and AQ are the tangents to the circle at P and Q. If a tangent BC is drawn at a point R lying on the minor arc PQ to intersect AP at B and AQ at C, find the perimeter of the triangle ABC.
Answer: Since the circle has radius 5 cm and AP and AQ are tangents, the lengths of AP and AQ will be equal to 5 cm. The triangle ABC will have sides 5 cm, 5 cm, and the circumference of the circle, which is 2π(5) = 10π cm. Therefore, the perimeter of the triangle ABC is 10π cm.
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Description
Test your knowledge on circle properties including radius, diameter, circumference, area, chords, arc length, and practice questions related to circles. Understand the concepts and formulas associated with circles.
