Prove that if chords of congruent circles subtend equal angles at their centres, then chords are equal.
Given:
Chords of congruent circles subtend equal angles at their centres.
To do:
We have to prove that the chords are equal.
Solution:
Let $c_{1}$ and $C_{2}$ be two congruent circles, $AB$ and $PQ$ are their chords respectively.
Let us join $OA$ and $OB$ in circle $C_{1}$.
Similarly, in cirlcle $C_{2}$, join $MP$ and $MQ$.
In $\vartriangle OAB$ and $\vartriangle MPQ$.
$OA=MP$ [$\because$ Radius of the congruent circles are same]
$OB=MQ$ [$\because$ Radius of congruent circles are same]
$\angle AOB=\angle PMQ$ [It is given chords of congruent circles subtend equal angles at their centres]
$\Rightarrow \vartriangle OAB\cong \vartriangle MPQ$ [SAS rule of congruency]
$\therefore AB=PQ$ [By CPCT rule]
Hence, it has been proved that the chords are equal.
Related Articles
- Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
- Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.
- If two chords of a circle are equally inclined to the diameter through their point of intersection, prove the chords are equal.
- If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.
- Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.
- If the areas of two similar triangles are equal, prove that they are congruent.
- If two equal chords of a circle intersect within the circle, prove that the segments of one chord are equal to corresponding segments of the other chord.
- All the Congruent angles are equal in ______________
- If two circles intersect at two points, prove that their centres lie on the perpendicular bisector of the common chord.
- Prove that, in a parallelogram1)opposite sides are equal 2) opposite angles are equal 3) Each diagonal will divide the parallelogram into two congruent triangles
- Let the vertex of an angle \( \mathrm{ABC} \) be located outside a circle and let the sides of the angle intersect equal chords \( \mathrm{AD} \) and \( \mathrm{CE} \) with the circle. Prove that \( \angle \mathrm{ABC} \) is equal to half the difference of the angles subtended by the chords \( A C \) and \( D E \) at the centre.
- If from any point on the common chord of two intersecting circles, tangents be drawn to the circles, prove that they are equal.
- Show that if the diagonals of a quadrilateral are equal and bisect each other at right angles, then it is a square.
Kickstart Your Career
Get certified by completing the course
Get Started