How To Apply the Intersecting Chords Theorem in Geometry Problems
In plane geometry, when two chords of a circle intersect inside the circle, then the measure of the angle thus formed will be one half of the sum of the measurement of the two intercepted arcs made by the angle and its vertical angle. This is to say, in a circle, the two chords ⌢AB and ⌢CD intersect inside the circle.
m∠1= ½ (m⌢AB + m⌢CD)
m∠2= ½ (m⌢BC + m⌢AD)
Because vertical angles are congruent, thus
m∠1 = m∠3
m∠2 = m∠4
This is the theorem on two intersecting chords.
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Tangent and Intersecting Chord Theorem Outside Circle
A tangent can be defined as a perpendicular line drawn from a radius of a circle thus, intersecting a circle in exactly one point. This intersecting point is known as the tangency point.
With respect to tangent and intersecting chord theorem, if a chord and a tangent intersect outside a circle, then the product of the lengths of the segment of the chord is equivalent to the square of the length of the tangent from the point of contact till the point of intersection. This is also in sync to the intersecting chord theorem outside the circle.
Unequal Chords in a Circle
Chords which are not at an equal distance from the center of a circle are called unequal chords.
Unequal Chords Theorem
For two unequal chords of a circle, the greater chord is supposed to be closer to the center than the smaller chord.
This must be clearly evident when you see the figure given below.
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As your chord goes closer to the center, it increases in length.
The figure shown depicts two chords AB and CD of a circle having center O, such that AB > CD.
OX and OY are perpendicular to the two chords from the center; this implies that they will intersect the two chords AB and CD respectively. Thus, we have the two intersecting chords i.e AB and CD.
Now, we need to prove that OX < OY.
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We will Join OA and OC.
Unequal Chords Theorem Proof
The proof involves a simple use of the Pythagoras theorem.
We note that in ΔOAX,
OX² + XA² = OA²
Likewise, in ∆OCY,
OY² + YC² = OC²
Since OA = OC (radii of the circle), we have
OX² + XA² = OY² + YC²
Both the sides above contain a sum of two terms.
If a term on one side is larger than the corresponding term on the other side, then the other term on the 1st side should be less than the corresponding term on the 2nd side.
That is, because XA > YC, this shall mean that
OX < OY
Thus, AB is closer to the center than CD.
Two Secants
A secant is a line of the segment which intersects a circle in exactly two points. When two secants (a tangent and a secant), or two tangents intersect outside a circle, then the measure of the angle formed is exactly one-half the positive difference of the measures of the intercepted arcs.
Secants Intersecting Inside a Circle
If two segments of secant are drawn to a circle from an outside point, then the product of the measures of one secant and its external secant will be equivalent to the product of the measures of the other secant and its external secant.
Solved Example
Example:
In the circle shown, if m ⌢AB = 92°m and m⌢CD = 110°, then find m∠3
Solution:
Substituting the values, we get
m∠3= ½ (m⌢AB + m⌢CD)
½ = (92° + 110°)
½ = (202°)
= 101°
Hence, m∠3 = 101°
Example:
In the circle shown below, O is the center having a radius of 5 cm.
Evaluate the length of chord AB if the length of the perpendicular drawn from the center of the circle measures 4 cm.
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Solution:
Given that OM is perpendicular to AB,
Thus, △AOM is a right-angled triangle.
In △AOM, (from the Pythagoras theorem)
OA² = OM² + AM²
After transposition of terms, we obtain
AM² = OA² − OM²
On substituting the values, we get
AM² = 5² − 4²
AM = √9
Thus, Chord ⌢AB = 2 × 3
= 6
Hence, Chord = 6cm
Fun Facts
A line is drawn from the center of a circle to bisect a chord is perpendicular to the chord.
The perpendicular from the center to the chord intersects the chord.
While establishing a comparison of the length of two arcs, the length of the chord belonging to smaller arc length is smaller and the greater arc length is greater.
The diameter is the longest possible chord in any circle.
FAQs on Intersecting Chords in a Circle: Theorem, Proof, and Examples
1. What is the fundamental definition of a chord in a circle?
A chord is a straight line segment whose two endpoints both lie on the circumference of a circle. Unlike a secant, which is an infinite line that cuts through the circle, a chord is contained entirely within the circle. The longest possible chord in any circle is its diameter, as it passes directly through the center.
2. What is the Intersecting Chords Theorem and what is its formula?
The Intersecting Chords Theorem describes a powerful relationship when two chords cross each other inside a circle. It states that if two chords, say AB and CD, intersect at a point P, then the product of the segments of one chord is equal to the product of the segments of the other chord. The formula is expressed as: AP × PB = CP × PD. This allows you to find an unknown segment length if the other three are known.
3. How are the angles formed by two intersecting chords inside a circle calculated?
The measure of the angle formed by two intersecting chords is half the sum of the measures of the arcs intercepted by the angle and its opposite (vertical) angle. For instance, if chords AB and CD intersect at point P, the measure of angle APC is calculated as: ∠APC = ½ × (measure of arc AC + measure of arc BD). This provides a direct way to relate the angles inside the circle to the arcs on its circumference.
4. What is the key difference between a chord, a diameter, and a secant?
While related, these three terms describe different lines associated with a circle. Understanding their differences is crucial for geometry problems:
Chord: A line segment with both endpoints on the circle's circumference.
Diameter: A specific type of chord that is the longest possible because it passes through the center of the circle.
Secant: An infinite line that passes through a circle, intersecting it at two distinct points. A chord is essentially the segment of a secant that lies inside the circle.
5. How does the Intersecting Chords Theorem apply if the intersecting chords are equal in length?
This is a special case with a unique property. If two equal chords, AB and CD, intersect at point P, the theorem proves that their corresponding segments are also equal. This means the larger segment of chord AB will be equal to the larger segment of chord CD, and the smaller segment of AB will be equal to the smaller segment of CD. This property is often used in proofs to establish congruence between parts of a circle.
6. How does the intersection rule change if the lines intersect outside the circle?
When the intersection point is outside the circle (i.e., when two secants or a tangent and a secant intersect), the rule changes. This is described by the Secant-Secant Theorem or Tangent-Secant Theorem. For two secants drawn from an external point P that cut the circle at A, B, C, and D, the formula becomes: PA × PB = PC × PD. Here, you multiply the external part of the secant by its entire length, which is a different calculation from the internal intersection theorem (part × part).
7. Why is the Intersecting Chords Theorem so important in solving geometry problems related to circles?
The theorem's importance lies in its ability to establish a relationship between segment lengths without needing information about the circle's radius or angles. It simplifies complex problems by allowing us to set up a direct algebraic equation (AP × PB = CP × PD) to find unknown lengths. This makes it a fundamental tool for solving a wide range of problems and proofs involving the internal geometry of a circle, as per the CBSE 2025-26 syllabus guidelines.
