# X is a point on the tangent at the point A lies on a circle with center O. A secant drawn from a point X intersects the circle at the points Y and z. If P is a mid-point of YZ, let us prove that XAPO or XAOP is a cyclic quadrilateral.

Formula used.

Perpendicular to tangent pass through centre of circle

Mid-point of chord is perpendicular line passes through centre

Solution.

As we join the figure

If P is mid-point of chord YZ

Then;

Line passing through centre to mid-point of line is perpendicular

Therefore OP is perpendicular to YZ

P = 90°

As there is tangent from point A on circle

Line passing through centre and point of contact is perpendicular to tangent

A = 90°

A + P = 90° + 90° = 180°

A + P + O + X = 360°

O + X = 360° - 180° = 180°

Sum of both opposite angles are 180°

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Champ Quiz | Area Related with the Circle45 mins
Quiz | Area Related with Circles47 mins
Quiz | A Letter to God39 mins
A Letter to God45 mins
A Letter to God50 mins
How to build a strong Vocabulary?53 mins
Time-Management: A key to Success37 mins
Let's Calculate - A Guide to an Economist's Dictionary54 mins
Tricks to MemoriseFREE Class
Smart and Effective study is the Key of success45 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses