Question

Two equal chords of a circle intersect within the circle Then the corresponding segments of the chords are

• A. not always equal.
• B. not equal
• C. not related anyway.
• D. equal

Answer 1

The correct option is D equal

Let $PQ$ and $RS$ be the two equal chords of a given circle and they are intersecting each other at point $T$. Draw perpendiculars $OV$ and $OU$ on these chords.

In $△OVT$ and $△OUT$,

$OV=OU$ (Equal chords of a circle are equidistant from the center)

$\angle OVT=\angle OUT$ (Each 90 degree)

$OT=OT$ (Common)

By SAS congruence, $△OVT\cong △OUT$

Therefore, $VT=UT$ (By CPCT).......... (1)

It is given that $PQ=RS$......... (2)

$\Rightarrow \frac{1}{2}PQ=\frac{1}{2}RS$

$\Rightarrow PV=RU$....... (3)

On adding equations 1 and 3, we get,

$PV+VT=RU+UT$

$\Rightarrow PT=RT$......... (4)

On subtracting equation 4 from 2 we get,

$PQ-PT=RS-RT$

$\Rightarrow QT=ST$........... (5)

Equations 4 and 5 indicate that the corresponding segments of chords $PQ$ and $RS$ are congruent to each other.

Adding equations 4 and 5, we get,

$PT+QT=RT+ST$

$\Rightarrow PQ=RS$

Hence, the corresponding segments of the chords are equal.