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 figure). Find the sides $AB$ and $AC$. "

Given:

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

To do:

We have to find the lengths of sides $AB$ and $AC$.

Solution:

Let the given circle touch the sides AB and AC of the triangle at points $E$ and $F$ respectively and let the

length of line segment $AF$ be $x$.

$AE = AF = x\ cm$ In $∆ABC$,

$a = 6 + 8$

$= 14\ cm$

$b = (x + 6)\ cm$

$c = (x + 8)\ cm$

Therefore,

$s=\frac{a+b+c}{2}$

$=\frac{14+x+6+x+8}{2}$

$=\frac{2 x+28}{2}$

$=(x+14) \mathrm{cm}$

$\operatorname{ar}(\Delta \mathrm{ABC})=\sqrt{s(s-a)(s-b)(s-c)}$

$=\sqrt{(x+14) \times x \times 8 \times 6}$

$=\sqrt{48 x \times(x+14)} \mathrm{cm}^{2}$.........(i)

$\operatorname{ar}(\Delta \mathrm{ABC})=\operatorname{ar}(\Delta \mathrm{OBC})+\operatorname{ar}(\Delta \mathrm{OCA})+\operatorname{ar}(\Delta \mathrm{OAB})$

$=\frac{1}{2} \times 4 \times a+\frac{1}{2} \times 4 \times b+\frac{1}{2} \times 4 \times c$

$=2 a+2 b+2 c$

$=2(a+b+c)$

$=2 \times 2(x+14)$............(ii)

From (i) and (ii), we get,

$\sqrt{48 x(x+14)}=4(x+14)$

$48 x(x+14)=4^{2}(x+14)^{2}$

$48 x(x+14) =16(x+14)^{2}$

$3 x(x+14)=(x+14)^{2}$

$3 x =x+14$

$2 x=14$

$x=7$

$\mathrm{AB}=x+8$

$=7+8$

$=15 \mathrm{~cm}$

$\mathrm{AC} =x+6$

$=7+6$

$=13 \mathrm{~cm}$

Related Articles In Figure 4, a $\vartriangle ABC$ is drawn to circumscribe a circle of radius $3\ cm$, such that the segments BD and DC are respectively of lengths $6\ cm$ and $9\ cm$. If the area of $\vartriangle ABC$ is $54\ cm^{2}$, then find the lengths of sides AB and AC."\n
In figure 5, a triangle PQR is drawn to circumscribe a circle of radius 6 cm such the segment QT into which QR is divided by the point of contact T, are of lengths 12 cm and 9 cm respectively. If the area of $\vartriangle PQR=189\ cm^{2}$, then find the lengths of sides PQ and PR."\n
In the figure, a \( \triangle A B C \) is drawn to circumscribe a circle of radius \( 4 \mathrm{~cm} \) such that the segments \( B D \) and \( D C \) are of lengths \( 8 \mathrm{~cm} \) and \( 6 \mathrm{~cm} \) respectively. Find the lengths of sides \( A B \) and \( A C \), when area of \( \triangle A B C \) is \( 84 \mathrm{~cm}^{2} \). "\n
A triangle \( P Q R \) is drawn to circumscribe a circle of radius \( 8 \mathrm{~cm} \) such that the segments \( Q T \) and \( T R \), into which \( Q R \) is divided by the point of contact \( T \), are of lengths \( 14 \mathrm{~cm} \) and \( 16 \mathrm{~cm} \) respectively. If area of \( \Delta P Q R \) is \( 336 \mathrm{~cm}^{2} \), find the sides \( P Q \) and \( P R \).
In a $\triangle ABC, D, E$ and $F$ are respectively, the mid-points of $BC, CA$ and $AB$. If the lengths of sides $AB, BC$ and $CA$ are $7\ cm, 8\ cm$ and $9\ cm$, respectively, find the perimeter of $\triangle DEF$.
$D$ and $E$ are the points on the sides $AB$ and $AC$ respectively of a $\triangle ABC$ such that: $AD = 8\ cm, DB = 12\ cm, AE = 6\ cm$ and $CE = 9\ cm$. Prove that $BC = \frac{5}{2}DE$.
In a $Δ\ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE\ ||\ BC$. If $AD\ =\ 4\ cm$, $DB\ =\ 4.5\ cm$ and $AE\ =\ 8\ cm$, find $AC$. "\n
Construct a triangle $ABC$ such that $BC = 6\ cm, AB = 6\ cm$ and median $AD = 4\ cm$.
In a $Δ\ ABC$, $D$ and $E$ are points on the sides $AB$ and $AC$ respectively such that $DE\ ||\ BC$. If $AD\ =\ 2\ cm$, $AB\ =\ 6\ cm$ and $AC\ =\ 9\ cm$, find $AE$. "\n
In a $Δ\ ABC$, $AD$ is the bisector of $∠\ A$, meeting side $BC$ at $D$. If $AB\ =\ 10\ cm$, $AC\ =\ 14\ cm$, and $BC\ =\ 6\ cm$, find $BD$ and $DC$. "\n
In a $Δ\ ABC$, $AD$ is the bisector of $∠\ A$, meeting side $BC$ at $D$.If $AB\ =\ 10\ cm$, $AC\ =\ 6\ cm$, and $BC\ =\ 12\ cm$, find $BD$ and $DC$."\n
In figure below, check whether AD is the bisector $\angle A$ of $\triangle ABC$ in each of the following:$AB=8\ cm, AC=24\ cm, BD=6\ cm$ and $BC=24\ cm$"\n
In a $Δ$ ABC, D and E are points on the sides AB and AC respectively such that DE $||$ BC.If AD $=$ 6 cm, DB $=$ 9 cm and AE $=$ 8 cm, find AC."\n
$D$ and $E$ are respectively the midpoints on the sides $AB$ and $AC$ of a $\vartriangle ABC$ and $BC = 6\ cm$. If $DE || BC$, then find the length of $DE ( in\ cm)$.
In the figure triangle $ABC$ is right-angled at $B$. Given that $AB = 9\ cm, AC = 15\ cm$ and $D, E$ are the mid points of the sides $AB$ and $AC$ respectively, calculate the length of $BC$."\n
Kickstart Your Career
Get certified by completing the course

Get Started