prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact .
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see diagram.
This point can be proved by the powerpoint of a circle wrt a point theorem.
Tangent is a line which touches a circle exactly at one point. All other lines intersect the circle at two points. Take a secant PQOR. Tangent be PT.
Let the radius be R. Let PQ = x.
powerpoint of P wrt circle:
= PQ * PR
= x ( x + 2R)
= x² + 2 R x
= (x + R)² - R²
= PO² - R²
So powerpoint of P wrt circle depends only on distance PO and Radius.
Powerpoint of P calculated along PT, (that is Q and R merge at T):
= PT * PT
Hence, we have : PT² = PO² - R²
In the triangle PTO, this means Pythagoras theorem is valid. So the triangle is right angled.
Hence PT ⊥ OT.
This point can be proved by the powerpoint of a circle wrt a point theorem.
Tangent is a line which touches a circle exactly at one point. All other lines intersect the circle at two points. Take a secant PQOR. Tangent be PT.
Let the radius be R. Let PQ = x.
powerpoint of P wrt circle:
= PQ * PR
= x ( x + 2R)
= x² + 2 R x
= (x + R)² - R²
= PO² - R²
So powerpoint of P wrt circle depends only on distance PO and Radius.
Powerpoint of P calculated along PT, (that is Q and R merge at T):
= PT * PT
Hence, we have : PT² = PO² - R²
In the triangle PTO, this means Pythagoras theorem is valid. So the triangle is right angled.
Hence PT ⊥ OT.
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