Question

ABC is a right angled triangle right angled at B such that BC = 6 cm and AB = 8 cm. A circle with center O is inscribed in triangle ABC. The radius of the circle is
A. 1cm
B. 2cm
C. 3cm
D. 4cm

Answer 1

Hint: In these types of questions remember the Pythagoras theorem i.e. \[{(hypotenuse)^2} = {(base)^2} + {(perpendicular)^2}\] also use the basic theorem of tangent to a circle to find the value of the radius of the circle.

Complete step-by-step answer:
Let’s use the given information and draw a rough diagram
Let’s draw a right angle triangle with a circle inside it with center O.
Then draw a line OE and OF which is perpendicular to BC and AB and a line OB

First let’s find the value of AC
By the Pythagoras theorem
$A{C^2} = A{B^2} + B{C^2}$
\[A{C^2} = {6^2} + {8^2}\] (Since the value of AB and BC are given in the question)
$A{C^2} = 100$
Applying both side by square root
$ \Rightarrow $$\sqrt {A{C^2}} = \sqrt {100} $
$ \Rightarrow $AC = 10cm
By the theorem of Tangent to a circle i.e. if two tangents are drawn from an external point of the circle, then they are of equal lengths therefore BE = BF, CP = CF and AP = AE.
By the above statement BE = BF = r
And CP = CF
CP = CF $ \Rightarrow $AC-AP = BC – BF (equation 1)
Now putting the value of BF and BE in the equation 1
AC – AE = BC – BF since AP=AE by the theorem of tangent to a circle
AC – AE = BC – BF (equation 2)
Putting the value of AC, AE, BC and BF in equation 2
10 – (6 – r) = 8 – r
$ \Rightarrow $10 – 6 + r = 8 – r
$ \Rightarrow $2r = 8 – 4
$ \Rightarrow $r = 4/2
$ \Rightarrow $r = 2cm
Therefore the radius of the circle inside the triangle is 2cm.

Note: In these types of questions first draw a rough diagram then find the unknown value of side of the triangle then construct lines from the center of the circle OF and OE that are perpendicular to the line AB and BC also a line OB which bisect the BE and BF then with the help theorem of tangent to a circle show that AP = AE, CP = CF and BE = BF then use the statement CP = CF and put the values of CP and CF and find the value of radius of circle.