9 Find lengths for all line segments in the figure below ABC

9 Find lengths for all line segments in the figure below. ABCD is a quadrilateral and the circle is inscribed in ABCD. You are given ABI = 10. IARI = 7. DC-411 = 5 (BFI =

Solution

The following segment lengths are equal as they are tangents to the circle from outside

AE = AH ---- (1)

EB = BF ----(2)

FC = CG -----(3)

GD = DH -----(4)

Since AH = 7, from (1) AE = 7 --- (5)

Given AB = 10 and from (5) AE = 7. Therefore BE = 10 - 7 = 3 --- (6)

From (2) BE = BF = 3 --- (7)

Given BC = 5 and from (7), BF = 3, therefore FC = 5 - 3 = 2---(8)

From (3) FC = CG = 2 --- (9)

Given DG = 4, therefore DC = DG + CG = 4 + 2 = 6

From (4) DG = DH = 4 --- (10)

Therefore AD = AH + HD = 7 + 4 = 11

The values are as follows

AE = 7

BF = 3

CF = 2

AD = 11

CG = 2

DH = 4

BE = 3

CD = 6