SUMMARY: This chapter focuses on the properties and theorems related to circles, including angles, chords, and tangents. KEY TOPICS: circle definition, chord properties, arc and sector, angle subtended by a chord, cyclic quadrilaterals, tangent to a circle, theorems on tangents, perpendicular from the center to a chord, equal chords and their distances from the center.
DA straight line that extends infinitely in both directions.
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Correct answer: Option 1 — A set of points equidistant from a fixed point.
Q21 Mark
If two chords of a circle are equal in length, what can be said about their distances from the center of the circle?
AThey are equal.
BOne is greater than the other.
CThey are both zero.
DThey are unequal but positive.
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Correct answer: Option 1 — They are equal.
Q31 Mark
In a circle, if a tangent is drawn at a point on the circle, what is the angle between the tangent and the radius at that point?
A90 degrees
B180 degrees
C45 degrees
D0 degrees
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Correct answer: Option 1 — 90 degrees
Q41 Mark
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. This statement is known as:
AThe Angle Subtended by a Chord Theorem
BThe Tangent Theorem
CThe Cyclic Quadrilateral Theorem
DThe Perpendicular Chord Theorem
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Correct answer: Option 1 — The Angle Subtended by a Chord Theorem
Q51 Mark
In a cyclic quadrilateral, the sum of the opposite angles is:
A90 degrees
B180 degrees
C360 degrees
D0 degrees
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Correct answer: Option 2 — 180 degrees
Short Answer Questions5 questions
Q63 Marks
Define a circle and explain its basic components.
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A circle is a set of points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is called the radius, and a line segment connecting two points on the circle is called a chord.
Q73 Marks
What is the relationship between the angles subtended by a chord at the center and on the circumference of the circle?
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The angle subtended by a chord at the center of the circle is twice the angle subtended by the same chord on the circumference. This is known as the Angle at the Center Theorem.
Q83 Marks
State and prove the theorem related to the perpendicular from the center of a circle to a chord.
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The theorem states that the perpendicular from the center of a circle to a chord bisects the chord. To prove this, consider a circle with center O and chord AB. If OC is the perpendicular from O to AB, then triangles OAC and OBC are congruent by the Hypotenuse-Leg theorem, thus AC = BC, proving that OC bisects AB.
Q93 Marks
Explain the properties of equal chords in a circle.
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In a circle, equal chords are equidistant from the center. This means if two chords are equal in length, the perpendicular distances from the center of the circle to these chords will also be equal.
Q103 Marks
What is a tangent to a circle and how does it relate to the radius at the point of contact?
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A tangent to a circle is a straight line that touches the circle at exactly one point. At the point of contact, the tangent is perpendicular to the radius drawn to that point, illustrating the relationship between tangents and radii.
Long Answer Questions5 questions
Q116 Marks
Define a circle and explain the significance of the center and radius in its formation. How do these elements relate to the properties of chords within the circle?
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A circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. The distance from the center to any point on the circle is known as the radius. The properties of chords within the circle are closely related to the center and radius; for instance, the perpendicular from the center of the circle to a chord bisects the chord, and equal chords are equidistant from the center. This relationship helps in understanding the geometric structure of the circle and its internal segments.
Q126 Marks
Explain the concept of an arc and a sector in a circle. How do these concepts relate to the angle subtended by a chord at the center and at any point on the circle?
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An arc is a portion of the circumference of a circle, while a sector is the region enclosed by two radii and the arc between them. The angle subtended by a chord at the center of the circle is twice the angle subtended at any point on the remaining part of the circle. This relationship is crucial in circle geometry as it helps in solving problems related to angles and lengths of arcs and sectors, providing insights into the properties of circles.
Q136 Marks
Discuss the properties of cyclic quadrilaterals and how they are derived from the angles subtended by their opposite sides. Provide an example to illustrate your explanation.
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A cyclic quadrilateral is a quadrilateral whose vertices lie on the circumference of a circle. One of the key properties of cyclic quadrilaterals is that the sum of the opposite angles is supplementary, meaning they add up to 180 degrees. This property can be derived from the angles subtended by the sides of the quadrilateral at the circumference. For example, in a cyclic quadrilateral ABCD, if angle A and angle C are opposite angles, then angle A + angle C = 180 degrees. This property is useful in various geometric proofs and problem-solving.
Q146 Marks
What is the theorem related to the tangent to a circle? Explain how this theorem can be used to find the length of a tangent from a point outside the circle.
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The theorem related to the tangent to a circle states that a tangent at any point on a circle is perpendicular to the radius drawn to the point of tangency. This theorem can be used to find the length of a tangent from a point outside the circle by applying the Pythagorean theorem. If a point P is outside the circle and the radius to the point of tangency is R, and the distance from point P to the center of the circle is D, then the length of the tangent PT can be calculated using the formula PT = √(D² - R²). This is essential in solving problems involving tangents and circles.
Q156 Marks
Explain the significance of the distance of equal chords from the center of the circle. How can this property be used to prove that two chords are equal?
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The distance of equal chords from the center of the circle is significant because it establishes that if two chords are equal in length, they must be equidistant from the center of the circle. This property can be used to prove that two chords are equal by showing that their distances from the center are the same. For instance, if chord AB and chord CD are two chords in a circle, and the perpendicular distances from the center O to both chords are equal, then by the converse of the perpendicular bisector theorem, AB = CD. This property is fundamental in circle geometry and helps in various proofs and constructions.
Assertion–Reason Questions5 questions
Q161 Mark
Assertion (A): The angle subtended by a chord at the center of a circle is twice the angle subtended by the same chord at any point on the remaining part of the circle.
Reason (R): This property is derived from the Inscribed Angle Theorem.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q171 Mark
Assertion (A): If two chords of a circle are equal, then they are equidistant from the center of the circle.
Reason (R): This is a property of equal chords in a circle.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q181 Mark
Assertion (A): A tangent to a circle is perpendicular to the radius drawn to the point of contact.
Reason (R): This is a fundamental property of tangents to circles.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q191 Mark
Assertion (A): The sum of the opposite angles of a cyclic quadrilateral is equal to 180 degrees.
Reason (R): This property is true for all cyclic quadrilaterals.
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Correct answer: Option 1 —
Both A and R are true, and R is the correct explanation of A.
Q201 Mark
Assertion (A): If a line segment is drawn from the center of a circle to the midpoint of a chord, it bisects the chord.
Reason (R): This statement is false because the line segment may not necessarily be perpendicular to the chord.
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Correct answer: Option 3 —
A is true, but R is false.
Statement-Based Questions5 questions
Q211 Mark
Statement 1: A chord of a circle is a line segment whose endpoints lie on the circle.
Statement 2: The diameter of a circle is the longest chord of that circle.
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Correct answer: Option 1 —
Both statements are true.
Q221 Mark
Statement 1: The angle subtended by a chord at the center of the circle is always twice the angle subtended at any point on the remaining part of the circle.
Statement 2: In a cyclic quadrilateral, the sum of opposite angles is equal to 180 degrees.
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Correct answer: Option 1 —
Both statements are true.
Q231 Mark
Statement 1: If two chords are equal in length, then they are equidistant from the center of the circle.
Statement 2: A tangent to a circle is a line that intersects the circle at two points.
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Correct answer: Option 3 —
Only Statement 2 is true.
Q241 Mark
Statement 1: The perpendicular from the center of a circle to a chord bisects the chord.
Statement 2: All chords of a circle are equal in length if they are equidistant from the center.
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Correct answer: Option 1 —
Both statements are true.
Q251 Mark
Statement 1: A tangent to a circle is always perpendicular to the radius drawn to the point of tangency.
Statement 2: An arc is a part of the circumference of a circle.
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Correct answer: Option 1 —
Both statements are true.
