plz give answer fast a triangle is drawn to circumscribe a circle of radius 4cm such that the segments bd and dc into which BC is divided by the point of contact d are of the lengths 8cm and 6 cm respectively find the sides ab and Ac it finds who is brainliest in you
Answers
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Answer:
AB = 15 cm and AC = 13 cm
Step-by-step explanation:
In ΔABC,
Length of two tangents drawn from the same point to the circle are equal,
∴ CF = CD = 6cm
∴ BE = BD = 8cm
∴ AE = AF = x
We observed that,
AB = AE + EB = x + 8
BC = BD + DC = 8 + 6 = 14
CA = CF + FA = 6 + x
Now semi perimeter of circles,
⇒ s = (AB + BC + CA
)/2
= (x + 8 + 14 + 6 + x
)/2
= (28 + 2x
)/2
= 14 + x
Area of ΔABC = √s (s - a)(s - b)(s - c)
= √(14 + x) (14 + x - 14)(14 + x - x - 6)(14 + x - x - 8)
= √(14 + x) (x)(8)(6)
= √(14 + x) 48x ------- (1)
also,
Area of ΔABC = 2 * area of (ΔAOF + ΔCOD + ΔDOB)
= 2 * [(1/2 * OF * AF) + (1/2 * CD * OD) + (1/2 * DB * OD)]
= 2 * 1/2 (4x + 24 + 32)
= 56 + 4x ----------- (2)
Equating equation (1) and (2) we get,
√(14 + x) 48 x = 56 + 4x
Squaring both sides,
48x (14 + x) = (56 + 4x)2
⇒ 48x = [4(14 + x)]2/(14 + x)
⇒ 48x = 16 (14 + x)
⇒ 48x = 224 + 16x
⇒ 32x = 224
⇒ x = 7 cm
Hence,
AB = x + 8 = 7 + 8 = 15 cm
CA = 6 + x = 6 + 7 = 13 cm
Therefore,
AB = 15 cm and AC = 13 cm
Hope it helps!
