Question

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see Fig. 10.14). Find the sides AB and AC.

Answer 1

Answer:

Let there is a circle having center O touches the sides AB and AC of the triangle at point E and F respectively.

Let the length of the line segment AE is x.

Now in △ABC,

CF=CD=6 (tangents on the circle from point C)

BE=BD=6 (tangents on the circle from point B)

AE=AF=x (tangents on the circle from point A)

Now AB=AE+EB

⟹AB=x+8=c

BC=BD+DC

⟹BC=8+6=14=a

CA=CF+FA

⟹CA=6+x=b

Answer 2

Answer:

Let there is a circle having center O touches the sides AB and AC of the triangle at point E and F respectively.

Let the length of the line segment AE is x.

Now in △ABC,

CF=CD=6 (tangents on the circle from point C)

BE=BD=6 (tangents on the circle from point B)

AE=AF=x (tangents on the circle from point A)

Now AB=AE+EB

⟹AB=x+8=c

BC=BD+DC

⟹BC=8+6=14=a

CA=CF+FA

⟹CA=6+x=b

Now

Semi-perimeter, s=

2

(AB+BC+CA)

s=

2

(x+8+14+6+x)

s=

2

(2x+28)

⟹s=x+14