Q. 65.0( 3 Votes )

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.

Answer :

Formula used.

Perpendicular to tangent pass through centre of circle

Mid-point of chord is perpendicular line passes through centre


As we join the figure

If P is mid-point of chord YZ


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°

In Quadrilateral XAOP

A + P = 90° + 90° = 180°

A + P + O + X = 360°

O + X = 360° - 180° = 180°

Sum of both opposite angles are 180°

Quadrilateral XAOP is cyclic quadrilateral

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