# 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.

To Find: AB and AC

Let the given circle touch the sides AB and AC of the triangle at point E and F respectively and the length of the line segment AF be x.

In ∆ABC,

From the theorem which states that the lengths of two tangents drawn from an external point to a circle are equal.
So,

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

BE = BD = 8cm (Tangents on the circle from point B)

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

AB = AE + EB = x + 8

BC = BD + DC = 8 + 6 = 14

CA = CF + FA = 6 + x

By heron's formula,

where, a, b and c are sides of triangle and s is semi perimeter.

2S = AB + BC + CA

= x + 8 + 14 + 6 + x

= 28 + 2x

S

=√(14 + x)(x)(8)(6)

Also, area of triangle =
And from the given figure it is clear that
Area of ΔABC = Area of ΔOBC + Area of ΔOCA + Area of ΔOAB

Area of ΔOBC =

Area of ΔOCA =

Area of ΔOAB =

Area of ΔABC = Area of ΔOBC + Area of ΔOCA + Area of ΔOAB

⇒

⇒

⇒

⇒ 3x = 14 + x

⇒ 2x = 14

⇒ x = 7

Therefore, x = 7

Hence,
AB = x + 8 = 7 + 8 = 15 cm

CA = 6 + x = 6 + 7 = 13 cm
Hence, AB = 15 cm and AC = 13 cm

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Quiz | Imp. Qs. on Circles37 mins
Quiz | Testing Your Knowledge on Circles32 mins
Short Cut Trick to Find Area of Triangle43 mins
Quiz | Areas Related to Circles43 mins
RD Sharma | Area of Sector and Segments25 mins
Quiz | Area Related with Circles47 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses