the sides of a quadrilateral touch a circle, prove that the sum of a pair of opposite side is equal to the sum of the other pair.
Answers
Step-by-step explanation:
Given: the sides of a quadrilateral touch a circle
To prove: the sum of a pair of opposite sides is equal to the sum of the other pair.
Proof:
From the theoram which states that the lengths of the two tangents drawn from an external point to a circle are equalFrom points A the tangents drawn are AP and AS,AP = AS .... (1)From points B the tangents drawn are BP and BQ,BP = BQ ..... (2)From points D the tangents drawn are DR and DS,DR = DS ....(3)From points C the tangents drawn are CR and CQ,CR = CQ ..... (4)Add 1,2,3 and 4 to getAP+BP+DR+CR = AS+BQ+DS+CQ(AP+BP)+(DR+CR )= (AS+DS)+ (BQ+CQ)AB+ DC = AD + BCHence proved
Answer:
as quadrilateral touches the circle
it is cyclic quadrilateral
therefore oppo sides are supplementary i.e 180degrees by theorem of cyclic quadrilateral
let the quadrilateral ABCD therefore A + C is equal to 180 degree by cyclic quadrilateral theorem and b + d is equals to 180 degree by cyclic quadrilateral theorem therefore from eq 1 and eq 2 therefore from one end to a + b is equals to b + D
hence proved