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Intersecting Chord Theorem

A chord is straight line drawn between 2 points on a circle or we can say it is a line segment whose endpoints lie on the circumference of the circle. Whereas, the 2 intersecting line segments inside the circle (each line segment starts and ends on the edge of the circle) are called as intersecting chords of a circle.

intersecting chords in a circle

In the above figure, we can see that two chords AD and BC intersects at point P. Hence these are called as Intersecting Chords

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Intersecting Chords in a Circle

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According to the theorem, if two chords intersects inside the circle, then the product of the length of the segments of one of the chord is equal to the product of the lengths of the segments of the second chord.

Intersecting Chord Theorem in a circle

That is A* B = C* D

Proof:

We can prove the theorem by using Similar Triangle Property.

Intersecting Chord Theorem

In the above figure, Triangle APB and Triangle CPD are similar triangle

∠ BAP = ∠ DCP. Since both the angles are suspended by same arc

∠ ABP= ∠ CDP. Since both the angles are suspended by same arc

∠ APB= ∠ CPD. Since both are vertical angles and vertical angles are equal

Hence, we can use the similar triangle property.

`(AP)/(PC)` = `(BP)/(PD)` = `(AB)/(CD)`

Using the first proportion,

`(AP)/(PC)` = `(BP)/(PD)`

On cross multiplying, we get,

AP* PD= PC* BP

Hence proved.

Angles Formed by Intersecting Chords

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When two chords intersect inside the circle, four angles are formed according to the theorem. The measure of the angle that is formed by the two chords that intersects inside the circle is equal to the half of the sum of the intercepted arcs

angles formed by intersecting chords in circle

In the above figure, we have 2 intercepted arcs as TE and GR. So, measurement of the angle x is given as

Measurement of ∠X = ½ ( sum of intercepted arcs)

Measurement of angle X = `(1)/(2)` (arc TE+ arc GR)

If the chords intersects outside the circle subtended by arcs x and y with x > y, then the angle formed by the intersection of chords is given as half of difference of the intercepted arcs

Angles formed by intersecting chords in a Circle

Angle formed by intersection of chords = `(1)/(2)` ( X- Y )

Example Problems on Intersecting Chords

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Below are some example problems on intersecting chords

Example 1:
In the given figure, Chord segments have lengths given as A= 8 C= 4 D= 6, Use the theorem to find the value of segment B.

Example proplems on Intersecting Chords in the Circle

Solution: According to the Intersecting Chord Theorem we have

If two chords intersects inside the circle then the product of length of segments of one chord is equal to the product of lengths of segments of other chord that is

A* B= C* D

since A= 8 C= 4 D= 6 subtituting the values

8* B= 4* 6

8B= 24

dividing both sides by 8

B= `(24)/(8)`

B= 3

Hence required length segment B=3

Example 2: Find the measure of angle x if the intercepted arcs are given as arc BD= 70° and arc CA= 170°

Example proplems on Intersecting Chords in the Circle

Solution:

We know from the theorem that the measure of angle X is given equal to half of sum of intercepted arcs so

arc BD= 70° and arc CA= 170°

substituting the values

Measure of angle x= `(1)/(2)` ( 70 + 170)

X= `(1)/(2)` ( 240)

X= 120°

Hence, Measure of angle X= 120 degree

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