A line joining the midpoints of two chords pa | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $AB$ and $CD$ be two chords of a circle with centre $O$. Let $M$ and $N$ be the midpoints of chords $AB$ and $CD$ respectively.Given that the

Vedclass · Accepted Answer

The chords are parallel.

Vedclass · Answer

(A) Let $AB$ and $CD$ be two chords of a circle with centre $O$. Let $M$ and $N$ be the midpoints of chords $AB$ and $CD$ respectively.Given that the line $MN$ passes through the centre $O$.Since $M$ is the midpoint of $AB$,the line segment joining the centre to the midpoint of a chord is perpendicular to the chord. Therefore,$OM \perp AB$,which implies $\angle OMA = 90^{\circ}$.Similarly,since $N$ is the midpoint of $CD$,$ON \perp CD$,which implies $\angle ONC = 90^{\circ}$.Since the points $M, O, N$ are collinear (as the line $MN$ passes through the centre $O$),the line $MN$ acts as a transversal to the chords $AB$ and $CD$.The angles $\angle OMA$ and $\angle ONC$ are corresponding angles (or interior angles on the same side depending on the orientation). Specifically,$\angle OMA + \angle ONC = 90^{\circ} + 90^{\circ} = 180^{\circ}$ if they are interior angles on the same side.Since the sum of the interior angles on the same side of the transversal $MN$ is $180^{\circ}$,the lines $AB$ and $CD$ must be parallel.

$A$ line joining the midpoints of two chords passes through the centre of a circle. Prove that the chords are parallel.

Similar Questions

$O$ is the circumcentre of the triangle $ABC$ and $D$ is the mid-point of the base $BC$. Prove that $\angle BOD = \angle A$.

$ABCD$ is a cyclic quadrilateral such that $AB$ is a diameter of the circle circumscribing it and $\angle ADC = 140^{\circ}$,then $\angle BAC$ is equal to (in $^{\circ}$)

In a cyclic quadrilateral $PQRS$,$\angle QPR = 40^{\circ}$ and $\angle PSQ = 70^{\circ}$,then find $\angle PQR$. (in $^{\circ}$)

Write True or False and justify your answer in each of the following:
Two congruent circles with centres $O$ and $O^{\prime}$ intersect at two points $A$ and $B$. Then $\angle AOB = \angle AO^{\prime}B$.

In a circle with radius $13 \, cm$,find the length of a chord lying at a distance of $12 \, cm$ from the centre. (in $, cm$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module