Let $s$ denote the semi-perimeter of a triangle $A B C$ in which $B C = a, C A = b, A B = c .$ If a circle touches the sides $B C, C A, A B$ at $D, E, F$ , respectively, prove that $B D = s - b .$

Open in App

Solution

Prove the statement:
Let's draw the figure,
Given: Semi Perimeter $= s$
Therefore, Perimeter $= 2 s$
$2 s = A B + B C + A C$
We know that tangents drawn from an external point to a circle are equal.
Hence,
$A F = A E B F = B D C D = C E$
Now, add all together .
$A F + B F + C D = A E + B D + C E \Rightarrow A B + C D = A C + B D$ $[∵ AF + BF = AB ∵ AE + CE = AC]$
Add $B D$ both side,
$\Rightarrow$ $A B + C D + B D = A C + B D + B D$
$\Rightarrow$ $A B + B C - A C = 2 B D$
$\Rightarrow$ $A B + B C + A C - A C - A C = 2 B D$
$\Rightarrow$ $2 s - 2 A C = 2 B D$ $[2 s = A B + B C + A C]$
$\Rightarrow$ $2 B D = 2 s - 2 b$ $[A C = b]$
$\Rightarrow$ $B D = s - b$
Hence,it is proved that $BD = s - b$ .

Suggest Corrections

Similar questions

Q. If a, b, c are the lengths of sides, BC, CA and AB of a triangle ABC, prove that

\vec{B C} + \vec{C A} + \vec{A B} = \vec{0}

and deduce that

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} .

In the given figure, a circle inscribed in a triangle ABC touches the sides AB, BC and CA at points D, E and F respectively. If AB = 14 cm, BC = 8 cm and CA = 12 cm. Find the lengths AD, BE and CF.

Q. Let

s

denote the semi-perimeter of a triangle

A B C

in which

B C = a, C A = b, A B = c

. If a circle touches the side

B C

C A

A B

D, E, F

respectively, prove that

B D = s - b

Q. Let s denotes the semi – perimeter of a ∆ ABC, in which BC=a, CA=b and AB=c , if a circle touches the sides BC, CA , AB at D,E,F respectively prove that BD = s - b.

Q. Question 2
Let s denotes the semi – perimeter of a ∆ ABC, in which BC=a, CA=b and AB=c , if a circle touches the sides BC, CA , AB at D,E,F respectively prove that BD = s - b.

Join BYJU'S Learning Program

Let s denote the semi-perimeter of a triangle ABC in which BC=a,CA=b,AB=c. If a circle touches the sides BC,CA,ABat D,E,F, respectively, prove that BD=s-b.

Let $s$ denote the semi-perimeter of a triangle $A B C$ in which $B C = a, C A = b, A B = c .$ If a circle touches the sides $B C, C A, A B$ at $D, E, F$ , respectively, prove that $B D = s - b .$