in the given figure a triangle abc is drawn to circumscribe a circle of radius 3 cm such that the segment bd and dc are respectively of length 6 and 9 cm if the area of traingle abc is 54cm2 then find side ab and ac

Asked by jennyy | 26th Mar, 2022, 08:41: PM

Expert Answer:

Let O be the centre of the circumscribed circle.
Join OA, OB, OC
 
OE perpendicular AB space and space OF perpendicular AC
As space tangents space drawn space from space an space external space point space are space equal comma space we space have
AE equals AF equals straight x space cm comma space BD equals BE equals 6 space cm comma space CD equals CF equals 9 space cm
Now comma space straight A left parenthesis increment ABC right parenthesis equals straight A open parentheses increment BOC close parentheses plus straight A open parentheses increment AOB close parentheses plus straight A open parentheses increment AOC close parentheses
rightwards double arrow 54 equals 1 half cross times BC cross times OD plus 1 half cross times AB cross times OE plus 1 half cross times AC cross times OF
rightwards double arrow 108 equals 15 cross times 3 plus open parentheses 6 plus straight x close parentheses cross times 3 plus open parentheses 9 plus straight x close parentheses cross times 3
rightwards double arrow straight x equals 3 space cm
Therefore comma space AB equals 9 space cm space and space AC equals 12 space cmOE perpendicular AB space and space OF perpendicular AC
As space tangents space drawn space from space an space external space point space are space equal comma space we space have
AE equals AF equals straight x space cm comma space BD equals BE equals 6 space cm comma space CD equals CF equals 9 space cm
Now comma space straight A left parenthesis increment ABC right parenthesis equals straight A open parentheses increment BOC close parentheses plus straight A open parentheses increment AOB close parentheses plus straight A open parentheses increment AOC close parentheses
rightwards double arrow 54 equals 1 half cross times BC cross times OD plus 1 half cross times AB cross times OE plus 1 half cross times AC cross times OF
rightwards double arrow 108 equals 15 cross times 3 plus open parentheses 6 plus straight x close parentheses cross times 3 plus open parentheses 9 plus straight x close parentheses cross times 3
rightwards double arrow straight x equals 3 space cm
Therefore comma space AB equals 9 space cm space and space AC equals 12 space cm

Answered by Renu Varma | 28th Mar, 2022, 12:10: PM