If two equal chords of a circle intersect, prove that the parts of one chord are separately equal to the parts of the other chord.
Solution:
Consider two equal chords AB and CD of a circle
The chords AB and CD meet at a point E
We have to prove that the parts of one chord are separately equal to the parts of the other chord.
Draw OM perpendicular to AB and ON perpendicular to CD
Join OE where O is the centre of the circle.
Considering triangles OME and ONE,
We know that the equal chords are equidistant from the centre of a circle.
OM = ON
Common side = OE
∠OME = ∠ONE = 90°
RHS criterion states that if the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent.
By RHS criterion, the triangles OME and ONE are similar.
The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem states that when two triangles are similar, then their corresponding sides and angles are also congruent or equal in measurements.
By CPCTC,
EM = EN --------------------------- (1)
Now, AB = CD
Dividing both sides by 2,
AB/2 = CD/2
AM = CN ------------------------------- (2)
We know that the perpendicular drawn from the centre of the circle to a chord bisects the chord.
So, AM = MB
CN = DN
Adding (1) and (2),
EM + AM = EN + CN
From the figure,
EM + AM = AE
EN + CN = CE
So, AE = CE ---------------------------- (3)
Now, AB = CD
Subtracting AE from both sides,
AB - AE = CD - AE
From the figure,
AB - AE = BE
So, BE = CD - AE
From (3),
BE = CD - CE
Therefore, BE = DE
✦ Try This: Prove that, if two lines containing chords of a circle intersect each other outside the circle, then the measure of angle between them is half the difference in measures of the arcs intercepted by the angle.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 10
NCERT Exemplar Class 9 Maths Exercise 10.4 Problem 1
If two equal chords of a circle intersect, prove that the parts of one chord are separately equal to the parts of the other chord.
Summary:
If two equal chords of a circle intersect, it is proven that the parts of one chord are separately equal to the parts of the other chord
☛ Related Questions: