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Chord of Circle: Theorems, Properties, Definitions, Videos

Chord of Circle is a line segment that joins any two points of the circle. The endpoints of this line segments lie on the circumference of the circle. Secant is the extension of the Chord.

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What is a Chord?

A Chord is a line segment that joins any two points of the circle. The endpoints of this line segments lie on the circumference of the circle. Diameter is the Chord that passes through the center of the circle. It is the longest chord possible in a circle. Chord is derived from a Latin word “Chorda” which means “Bowstring“. T A Segment of the circle is the region that lies between the Chord and either of Arcs. You already know about the concepts of arc and circumference. Let us now look at the theorems related to chords of a circle.

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Chord of a Circle Theorems

  1. Theorem 1: Equal chords of a circle subtend equal angles at the center.

chords

2. Theorem 2: This is the converse of the previous theorem. It implies that if two chords subtend equal angles at the center, they are equal.

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3. Theorem 3: A perpendicular dropped from the center of the circle to a chord bisects it. It means that both the halves of the chords are equal in length.

chords

4. Theorem 4: The line that is drawn through the center of the circle to the midpoint of the chords is perpendicular to it. In other words, any line from the center that bisects a chord is perpendicular to the chord.

5. Theorem 5: If there are three non-collinear points, then there is just one circle that can pass through them.

6. Theorem 6: Equal chords of a circle are equidistant from the center of a circle.

7. Theorem 7: This is the converse of the previous theorem. It states that chords equidistant from the center of a circle are equal in length.

chords

8. Theorem 8: The angle subtended by an arc at the center of a circle is double that of the angle that the arc subtends at any other given point on the circle.

9. Theorem 9: Angles formed in the same segment of a circle are always equal in measure.

10. Theorem 10: If the line segment joining any two points subtends equal angles at two other points that are on the same side, they are concyclic. This means that they all lie on the same circle.

These are some of the basic theorems on the chords and arcs of a circle. We will read about some other theorems in the next chapter. Let us look at some solved examples based on these theorems.

Solved Example For You

Question 1: In the following diagram, calculate the measure of ∠POQ when the value of ∠PRQ is given 620.

Chords

Answer : According to the theorem of chords of a circle, the angle subtended at the center of the circle by an arc is twice the angle subtended by it at any other point on the circumference. Hence, ∠POQ is equal to two times of ∠PRQ. Therefore, ∠POQ = 2 x ∠PRQ

⇒ ∠POQ  = 2 x 620 = 1240

Question 2: What is a Chord?

Answer: A Chord refers to a line segment that is joining any two points of the circle. The endpoints of these line segments lie on the circle’s circumference. Diameter refers to the chord that passes through the centre of the circle. In fact, it is also the longest chord possible in a circle. This term is taken from the Latin word “Chorda” that means “Bowstring“.

Question 3: How do you find the radius of a circle with a chord?

Answer: It is easy to find the radius of a circle if the length and height of a chord of that circle are known. We simply need to multiply the height of the chord times four. For example, if the height is three, we will multiply three times four to get twelve. Finally, just square the length of the chord.

Question 4: Can a diameter be a chord?

Answer: A chord of a circle refers to a straight line segment whose endpoints both lie on the circle. Similarly, a chord that is passing through a circle’s centre point will be the diameter of the circle. Every diameter is a chord, however not every chord can be a diameter.

Question 5: What is the arc of a circle?

Answer: The arc of a circle refers to a portion of the circumference of a circle. The formula for finding out the arc length in radians has r as the radius of the circle and θ as the measure of the central angle in radians.

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