AB is a chord of a circle, with centre O, such that AB = 16 cm and radius of the circle is 10 cm. Tangents at A and B intersect each other at P. Find the length of PA.
Answers
AB is a chord of a circle, with centre O, such that AB = 16 cm and radius of the circle is 10 cm. Tangents at A and B intersect each other at P. Find the length of PA.
bisects {AB}
AL = BL = 8 cm
OL = √( 10^2 - 8^2)
OL = √36
OL = 6 cm
Tangent is perpendicular to the radius through the point of contact.
∠AOL + ∠PAL = 90°
∠PAL + ∠APL = 90°
( Because sum of angles in ∆APL is 180° )
∠PLA = ∠OLA = 90 ( Each )
∠APL = ∠OLA = 90° ( Proved )
∆APL and ∆OAL are similar triangles
=> PA/OA = AL/OL
( Corresponding Sides are proportional ).
=> PA/10 = 8/6
=> PA =( 8 × 10 )/6
=> PA = 80/6
Or PA = 40/3
Tangent is perpendicular to the radius through the point of contact.
∠AOL + ∠PAL = 90°
∠PAL + ∠APL = 90°
( Because sum of angles in ∆APL is 180° )
∠PLA = ∠OLA = 90 ( Each )
∠APL = ∠OLA = 90° ( Proved )
∆APL and ∆OAL are similar triangles
=> PA/OA = AL/OL
( Corresponding Sides are proportional ).
=> PA/10 = 8/6
=> PA =( 8 × 10 )/6
=> PA = 80/6
Or PA = 40/3
THEREFORE,
The length of PA = 40/3 cm