If the two sides of a pair of opposite sides of a cyclic quadrilateral are equal, prove that is diagonals are equal.
Given:
The two sides of a pair of opposite sides of a cyclic quadrilateral are equal.
To do:
We have to prove that is diagonals are equal.
Solution:
Let in a cyclic quadrilateral $ABCD, AB = CD, AC$ and $BD$ are the diagonals.
$AB = CD$
arc $AB =$ arc $CD$
Adding arc $BC$ to both sides, we get,
arc $AB +$ arc $BC =$ arc $BC +$ arc $CD$
arc $AC =$ arc $BD$
Therefore,
$AC = BD$
Hence the diagonals of the cyclic quadrilateral are equal.
Related Articles
- If the sides of a quadrilateral touch a circle, prove that the sum of a pair of opposite sides is equal to the sum of the other pair.
- If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
- Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.
- Which of the following statements are true (T) and which are false (F)?In any quadrilateral, if a pair of opposite sides is equal, it is a parallelogram.
- Prove that, in a parallelogram1)opposite sides are equal 2) opposite angles are equal 3) Each diagonal will divide the parallelogram into two congruent triangles
- A parallelogram whose all sides are equal opposite angles are equal opposite sides are parallel is called a rhombus. Why? Why not square?
- Say True or False :(a) Each angle of a rectangle is a right angle.(b) The opposite sides of a rectangle are equal in length.(c) The diagonals of a square are perpendicular to one another.(d) All the sides of a rhombus are of equal length.(e) All the sides of a parallelogram are of equal length.(f) The opposite sides of a trapezium are parallel.
- Which of the following statements are true (T) and which are false (F)?If the opposite sides of a quadrilateral are equal, it is a parallelogram.
- If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
- Which of the following statements are true (T) and which are false (F):Angles opposite to equal sides of a triangle are equal.
- The opposite sides of a quadrilateral are parallel. If one angle of the quadrilateral is $60^o$, find the other angles.
- Prove that in a quadrilateral the sum of all the sides is greater than the sum of its diagonals.
- Prove that the sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.
- The figure formed by joining the mid-pointsof the sides of a quadrilateral ABCD, taken inorder, is a square only, ifA. Diagonals of ABCD are equalB. Diagonals of ABCD are equal and perpendicular
- Which of the following statements are true (T) and which are false (F)?If all the sides of a quadrilateral are equal it is a parallelogram.
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies