Explain angle on same segment. ( i.e., angle subtended on the chord is equal to angle formed by the chord and tangent )
 
Prove it . 

Asked by sucharitasahoo1 | 14th Oct, 2017, 08:50: PM

Expert Answer:

 
begin mathsize 16px style GIven colon space straight A space circle space with space centre space straight O. space Tangent space PQ thin space touches space the space circle space at space point space straight A. space
Through space straight A comma space the space point space of space contact comma space straight a space chord space AB space is space drawn.

To space prove colon space angle BAQ equals angle ACB space space and space space angle BAP equals angle ADB

Proof colon
In space triangle ABR comma
angle ABR equals 90 degree space space space space space left parenthesis Angle space of space straight a space semi minus circle right parenthesis
rightwards double arrow angle ARB plus angle RAB equals 90 degree space space space space space space space.... left parenthesis straight i right parenthesis
angle OAQ equals 90 degree space space space space space space space space space space left parenthesis Angle space between space the space radius space and space the space tangent right parenthesis
rightwards double arrow angle RAB plus angle BAQ equals 90 degree space space space space space space.... left parenthesis ii right parenthesis
rightwards double arrow angle ARB plus angle RAB equals angle RAB plus angle BAQ space space space space space space space.... left square bracket From space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis right square bracket
rightwards double arrow angle ARB equals angle BAQ space space space
But comma space angle angle ARB equals angle ACB space space space space space space left parenthesis Angles space of space the space same space segment right parenthesis space space
rightwards double arrow angle BAQ equals angle ACB space space space space space space left parenthesis proved right parenthesis

Now comma space angle BAP plus angle BAQ equals 180 degree
and space angle ACB plus angle ADB equals 180 degree space space space space space left parenthesis opposite space angles space of space cyclic space quadrilateral right parenthesis
rightwards double arrow angle BAP plus angle BAQ equals angle ACB plus angle ADB
rightwards double arrow angle BAP equals angle ADB space space end style

Answered by Rashmi Khot | 4th Dec, 2017, 03:17: PM