- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# 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.

Given:

Two equal chords of a circle intersect within the circle

To do:

We have to prove that the segments of one chord are equal to corresponding segments of the other chord.

Solution:

Let $AB$ and $CD$ be two equal cords which intersect at point $R$.

From the centre of the circle, draw a perpendicular to $AB$ and another one perpendicular to $CD$

$OP \perp AB$

$OQ \perp CD$.

Join $OR$.

From the diagram,

$OP$ bisects $AB$ and $OP \perp AB$

$OQ$ bisects $CD$ and $OQ \perp CD$

$AB = CD$

This implies,

$AP = QD$.........(i)

$PB = CQ$.........(ii)

In triangles $OPR$ and $OQR$,

$\angle OPR = \angle OQR$

$OR = OR$ (Common side)

$OP = OQ$ ($AB$ and $CD$ are equal and they are equidistant from the centre)

Therefore, by RHS congruency,

$\triangle OPR \cong \triangle OQR$

This implies,

$PR = QR$..........(iii) (CPCT)

From (i) and (ii), we get,

$AP+PR = QD+QR$

$AR = RD$

From (ii) and (iii), we get,

$PB-PR = CQ-QR$

$BR = CR$

Hence proved.

- Related Questions & Answers
- Check if two line segments intersect
- Program to make the XOR of all segments equal to zero in Python
- Find the Diameter or Longest chord of a Circle in C++
- Check if a large number can be divided into two or more segments of equal sum in C++
- Count of strings that become equal to one of the two strings after one removal in C++
- Check if values of two arrays are the same/equal in JavaScript
- Area of circle inscribed within rhombus?
- Area of decagon inscribed within the circle in C Program?
- C Program for area of decagon inscribed within the circle?
- Maximum number of segments that can contain the given points in C++
- How do I find the intersection of two line segments in Matplotlib?
- What are the three segments of the default route, that is present in ASP .Net MVC C#?
- Finding two numbers that produce equal to the sum of rest in JavaScript
- Return True if two Numpy arrays are element-wise equal within a tolerance
- Java program to find the area of a circle
- Java program to find the circumference of a circle