Congruent Chords in the Same Circle are Equidistant from the Centre

Two congruent chords of a circle are equidistant from its centre.
NOTE: Two chords are said to be congruent if they are equal in lengths.


congruent-chord-circle

Consider the equation of the circle

{x^2}  + {y^2} = {r^2}\,\,\,\,{\text{ -  -  - }}\left( {\text{i}} \right)


Suppose that AB and CD be congruent chords with A\left( {{x_1},{y_1}} \right), B\left( {{x_2},{y_2}} \right), C\left( {{x_3},{y_3}} \right) and D\left( {{x_4},{y_4}} \right) as shown in the given diagram. Since the circle passes through the points A,\,B,\,C and D, so equation of circle becomes

\begin{gathered} {x_1}^2 + {y_1}^2 = {r^2}\,\,\,\,{\text{  -  -   - }}\left( {{\text{ii}}} \right) \\ {x_2}^2 + {y_2}^2 = {r^2}\,\,\,\,{\text{  -  -   - }}\left( {{\text{iii}}} \right) \\ {x_3}^2 + {y_3}^2 = {r^2}\,\,\,\,{\text{  -  -   - }}\left( {{\text{iv}}} \right) \\ {x_4}^2 + {y_4}^2 = {r^2}\,\,\,\,{\text{  -  -   - }}\left( {\text{v}} \right) \\ \end{gathered}


Since M be the midpoint of the chord AB, so

M\left(  {\frac{{{x_1} + {x_2}}}{2},\frac{{{y_1} + {y_2}}}{2}} \right)


Since N be the midpoint of the chord CD, so

N\left(  {\frac{{{x_3} + {x_4}}}{2},\frac{{{y_3} + {y_4}}}{2}} \right)


Now we shall find distance between O and M, as follows

\begin{gathered} {\left| {OM} \right|^2} = {\left(  {\frac{{{x_1} + {x_2}}}{2} - 0} \right)^2} + {\left( {\frac{{{y_1} + {y_2}}}{2}  - 0} \right)^2} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left(  {\frac{{{x_1} + {x_2}}}{2}} \right)^2} + {\left( {\frac{{{y_1} + {y_2}}}{2}}  \right)^2} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =  \frac{{{{\left( {{x_1} + {x_2}} \right)}^2}}}{4} + \frac{{{{\left( {{y_1} +  {y_2}} \right)}^2}}}{4} = \frac{{{{\left( {{x_1} + {x_2}} \right)}^2} +  {{\left( {{y_1} + {y_2}} \right)}^2}}}{4} \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =  \frac{{{x_1}^2 + {x_2}^2 + 2{x_1}{x_2} + {y_1}^2 + {y_2}^2 + 2{y_1}{y_2}}}{4}   \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, =  \frac{{{x_1}^2 + {y_1}^2 + {x_2}^2 + {y_2}^2 + 2{x_1}{x_2} + 2{y_1}{y_2}}}{4}   \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{r^2} +  {r^2} + 2{x_1}{x_2} + 2{y_1}{y_2}}}{4} = \frac{{2\left( {{r^2} + {x_1}{x_2} +  {y_1}{y_2}} \right)}}{4} \\ \Rightarrow {\left| {OM} \right|^2} =  \frac{{{r^2} + {x_1}{x_2} + {y_1}{y_2}}}{2}\,\,\,\,{\text{ -  -  -  }}\left( {{\text{vi}}} \right) \\ \end{gathered}


Similarly, we can show that

  \Rightarrow {\left| {ON} \right|^2} = \frac{{{r^2} + {x_3}{x_4} +  {y_3}{y_4}}}{2}\,\,\,\,{\text{ -  -  - }}\left( {{\text{vii}}} \right)


Since AB and CD are congruent chords, so {\left| {AB} \right|^2} = {\left| {CD} \right|^2}

\begin{gathered} \Rightarrow {\left( {{x_2} - {x_1}}  \right)^2} + {\left( {{y_2} - {y_1}} \right)^2} = {\left( {{x_4} - {x_3}}  \right)^2} + {\left( {{y_4} - {y_3}} \right)^2} \\ \Rightarrow {x_1}^2 + {x_2}^2 - 2{x_1}{x_2}  + {y_1}^2 + {y_2}^2 - 2{y_1}{y_2} = \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{x_3}^2  + {x_4}^2 - 2{x_3}{x_4} + {y_3}^2 + {y_4}^2 - 2{y_3}{y_4} \\ \end{gathered}

Using equation (2) and (3), we get the following result

\begin{gathered} \Rightarrow {r^2} + {r^2} - 2{x_1}{x_2} -  2{y_1}{y_2} = {r^2} + {r^2} - 2{x_3}{x_4} - 2{y_3}{y_4} \\ \Rightarrow   - 2{x_1}{x_2} - 2{y_1}{y_2} =  -  2{x_3}{x_4} - 2{y_3}{y_4} \\ \Rightarrow {x_1}{x_2} + {y_1}{y_2} =  {x_3}{x_4} + {y_3}{y_4} \\ \end{gathered}

Adding {r^2} on both sides of the above equation, we have

\begin{gathered} \Rightarrow {r^2} + {x_1}{x_2} + {y_1}{y_2}  = {r^2} + {x_3}{x_4} + {y_3}{y_4} \\ \Rightarrow \frac{{{r^2} + {x_1}{x_2} +  {y_1}{y_2}}}{2} = \frac{{{r^2} + {x_3}{x_4} + {y_3}{y_4}}}{2}\,\,\,{\text{  -  -   - }}\left( {{\text{viii}}} \right) \\ \end{gathered}


Using equation (vi) and (vii) in equation (viii), we get

\begin{gathered} {\left| {OM} \right|^2} = {\left| {ON}  \right|^2} \\ \Rightarrow \left| {OM} \right| = \left|  {ON} \right| \\ \end{gathered}


This shows that the congruent chords of a circle are equidistant from its centre.

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