In aright triangele ABC a circle with AB as diameter is drawn to intersect the hypotenuse AC in P .prove that the tangent to the circle at P bisects the base BC

Asked by lovemaan5500 | 24th Dec, 2017, 03:42: PM

Expert Answer:

 
begin mathsize 16px style Construction colon space Join space PB
Since space angle space of space semi minus circle equals 90 degree comma
angle APB equals 90 degree
rightwards double arrow angle BPC equals 90 degree space space space space space space left parenthesis APC space is space straight a space straight space line right parenthesis
rightwards double arrow angle BPQ plus angle CPQ equals 90 degree space space space space space space space.... left parenthesis straight i right parenthesis
In space right minus angled space triangle space ABC comma
angle straight A plus angle straight C equals 90 degree space space space space space space.... left parenthesis ii right parenthesis left parenthesis since space angle straight B equals 90 degree right parenthesis
Also comma space angle BPQ equals angle straight A space space space space space.... left parenthesis iii right parenthesis space left parenthesis Angles space in space the space alternate space segment right parenthesis
rightwards double arrow angle straight A plus angle CPQ equals 90 degree space space space space space space space.... left parenthesis iv right parenthesis space space space space space space left square bracket From space left parenthesis straight i right parenthesis space and space left parenthesis iii right parenthesis right square bracket
rightwards double arrow angle straight A plus angle CPQ equals angle straight A plus angle straight C
rightwards double arrow angle CPQ equals angle straight C
rightwards double arrow PQ equals CQ
But comma space PQ equals BQ space space left parenthesis Tangents space from space an space external space point space are space equal right parenthesis
rightwards double arrow CQ equals BQ
Thus comma space the space tangent space to space the space circle space at space straight P space bisects space the space base space BC. end style

Answered by Rashmi Khot | 26th Dec, 2017, 09:33: AM