Question

The centres of two circles are 41cm apart. The smaller circle has a radius of 4 cm and the larger one has a radius of 5cm. The length of the common internal tangent is:
A 41cm
B 39cm
C 39.8cm
D 40cm

Answer 1

Hint: In this problem, first we need to draw the figure of the given situation. Now, use the property of the congruent triangle and Pythagoras theorem to obtain the length of the internal tangent.

Complete step-by-step answer:

From the above figure, it can be observed that \[\Delta {C_1}AC\] and \[\Delta {C_2}BC\] are congruent. Therefore,
\[\begin{gathered}
  \,\,\,\,\,\,\dfrac{{A{C_1}}}{{B{C_2}}} = \dfrac{{{C_1}C}}{{{C_2}C}} \\
   \Rightarrow \dfrac{4}{5} = \dfrac{x}{{41 - x}} \\
   \Rightarrow 164 - 4x = 5x \\
   \Rightarrow 9x = 164 \\
   \Rightarrow x = \dfrac{{164}}{9}cm \\
\end{gathered}\]
Now, apply the Pythagoras theorem in \[\Delta AC{C_1}\] as shown below.
\[\begin{gathered}
  \,\,\,\,\,AC = \sqrt {{C_1}{C^2} - {C_1}{A^2}} \\
   \Rightarrow AC = \sqrt {{{\left( {\dfrac{{164}}{9}} \right)}^2} - {4^2}} \\
   \Rightarrow AC = \sqrt {\dfrac{{25600}}{{81}}} \\
   \Rightarrow AC = \dfrac{{160}}{9}cm \\
\end{gathered}\]
Again, In \[\Delta {C_1}AC\] and \[\Delta {C_2}BC\],
\[\begin{gathered}
  \,\,\,\,\,\,\dfrac{{A{C_1}}}{{B{C_2}}} = \dfrac{{AC}}{{BC}} \\
   \Rightarrow \dfrac{4}{5} = \dfrac{{\dfrac{{160}}{9}}}{{BC}} \\
   \Rightarrow BC = \dfrac{{160}}{9} \times \dfrac{5}{4} \\
   \Rightarrow BC = \dfrac{{200}}{9}cm \\
\end{gathered}\]
Thus, the length of the internal tangent AB is calculated as follows:
\[\begin{gathered}
  \,\,\,\,\,AB = AC + BC \\
   \Rightarrow AB = \dfrac{{160}}{9} + \dfrac{{200}}{9} \\
   \Rightarrow AB = \dfrac{{360}}{9} \\
   \Rightarrow AB = 40cm \\
\end{gathered}\]
Thus, the length of the internal tangent is 40 cm, hence, option (D) is the correct answer.
Note: Two triangles are said to be congruent if, two sides and included angle of one triangle is congruent with the second triangle under SAS configuration. When two triangles are similar or congruent, the ratios of the corresponding sides are also equal.