# Prove that if chords of congruent circles subtend equal angles at their centres, then the 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.

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Updated on: 10-Oct-2022

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The passage contains a geometric proof that if chords of congruent circles subtend equal angles at their centers, then the chords are equal. The proof is structured with a given statement, a to-do statement, and a solution that proves the chords are equal.
no_question

the text does not contain a mathematical question but rather a proof of a geometric statement. therefore, it does not fit into the categories that involve questions or answers to questions.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Prove that if chords of congruent circles subtend equal angles at their centres, then the 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.

Tutorialspoint

Simply Easy Learning

Updated on: 10-Oct-2022

27 Views
user
The passage contains a geometric proof that if chords of congruent circles subtend equal angles at their centers, then the chords are equal. The proof is structured with a given statement, a to-do statement, and a solution that proves the chords are equal.

TYPE: NO_QUESTION

The text does not contain a mathematical question but rather a proof of a geometric statement. Therefore, it does not fit into the categories that involve questions or answers to questions.