Intersecting Chords Theorem
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 Intersect Chords
Q1. What is a Chord?
Answer: In plane geometry, a chord is referred to as the straight line segment joining two points on a curve. The term is often used to represent a line segment whose endpoints lie on a circle or the circular arc. All angles carved inside a circle and subtended by the same chord are equal in measure. The infinite line extension of a chord is a secant. Usually, a chord is a line segment connecting two points on any curve, for example, an ellipse.
Q2. What are the Properties of Intersecting Chords?
Answer: Following are the main properties of circle intersect chords:
If two chords intersect each other inside a circle, the products of their segments will be equivalent.
Each chord is divided into two segments at the point where they intersect.
One chord is cut into two line segments, suppose P and Q. The other is cut into the segments R and S.
The chords intersecting inside a circle theorem states that P × Q is always equal to R × S regardless of where the chords are.
As long as they intersect inside the circle, or if two intersecting chords of a circle, the two products will always be the same.
Q3. Is Diameter a Chord?
Answer: Yes, the diameter is the longest possible chord, since it also connects two points lying on the circumference of a circle.
A chord that crosses across the center of the circle is called the diameter chord, or simply a diameter.
