Intersecting Chords Theorem formula proof and solved examples
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 Circle Intersecting Chords Theorem Explained with Proof
1. What is the intersecting chords theorem in a circle?
The intersecting chords theorem states that when two chords intersect inside a circle, the product of the segments of one chord equals the product of the segments of the other chord. If chord AB and chord CD intersect at point P inside the circle, then AP × PB = CP × PD. This theorem is also called the circle intersecting chords property and is widely used to find unknown segment lengths in geometry problems.
2. What is the formula for intersecting chords in a circle?
The formula for intersecting chords inside a circle is AP × PB = CP × PD. Here:
- AP and PB are the two segments of the first chord.
- CP and PD are the two segments of the second chord.
3. How do you solve problems using the intersecting chords theorem?
To solve problems using the intersecting chords theorem, set up the equation AP × PB = CP × PD and solve for the unknown value.
- Step 1: Identify the four chord segments.
- Step 2: Multiply the two segments of one chord.
- Step 3: Set it equal to the product of the other two segments.
- Step 4: Solve the equation.
3 × 4 = 2 × PD → 12 = 2PD → PD = 6.
4. Why does the intersecting chords theorem work?
The intersecting chords theorem works because of similar triangles formed by the intersecting chords inside the circle. When two chords intersect, they create two pairs of vertical angles and equal inscribed angles, forming similar triangles. Since corresponding sides of similar triangles are proportional, this leads to the relationship AP × PB = CP × PD.
5. Does the intersecting chords theorem apply outside the circle?
No, the formula AP × PB = CP × PD applies only when chords intersect inside the circle. If lines intersect outside the circle, the rule changes to the secant-secant power theorem, where the product of the whole secant and its external segment is equal for both secants.
6. Can you give an example of intersecting chords in a circle?
Yes, an example of intersecting chords is when two chords cross at a point inside the circle and satisfy AP × PB = CP × PD.
- Suppose AP = 5 and PB = 2.
- Suppose CP = 4 and PD = x.
5 × 2 = 4 × x → 10 = 4x → x = 2.5. This shows how to find unknown chord lengths using the circle intersecting chords formula.
7. What is the difference between intersecting chords and secants?
The main difference is that intersecting chords meet inside the circle, while secants extend outside the circle. For intersecting chords, the rule is AP × PB = CP × PD. For two secants intersecting outside, the rule is (external segment × whole secant) = (external segment × whole secant). The formulas are similar but apply in different geometric situations.
8. What happens if two chords intersect at the center of the circle?
If two chords intersect at the center, they are diameters and are equal in length, and the intersecting chords theorem still holds. Since the center divides each diameter into two equal radii, the products become equal automatically because r × r = r × r. This is a special case of the intersecting chords property.
9. What are common mistakes when using the intersecting chords theorem?
A common mistake is using the formula incorrectly or applying it outside the circle instead of inside.
- Mixing up segment pairs (wrong multiplication).
- Using the entire chord instead of individual segments.
- Applying AP × PB = CP × PD when lines intersect outside the circle.
10. How is the intersecting chords theorem related to the power of a point?
The intersecting chords theorem is a special case of the Power of a Point Theorem for a point inside a circle. The power of a point states that for any line through a point inside the circle, the product of the two segment lengths is constant. Therefore, AP × PB = CP × PD represents the equal power of the same interior point with respect to the circle.
