In a figure circle inscribed in a triangle ABC touches its side AB, BC, and AC at points D, E, and F respectively. If AB=12cm, BC=8cm, and AC=10cm, then find the length of AD, BE, and CF.

Question

Vedantu Content Team · Accepted Answer

Hint: We know that the tangents drawn from an external to a circle are equal in length. We can see that AD and AF are tangents drawn from point A. Similarly, BE and BD are tangents drawn from point B and CF and CE are tangent drawn from point C. We have the length of the sides AB, AC, and BC. We will suppose the length of a tangent as x, y, and z. We can see from the figure that, AB=AD+DB, BC=BE+EC, and AC=AF+FC. So, we can find the value of AD, BE, and CF.Complete step-by-step solution:

We have figure as given above,We know that the tangents drawn from an external to a circle are equal in length. So from the figure we can see that,$\begin{align}  & \Rightarrow AF=AD \\  & \Rightarrow CF=CE \\  & \Rightarrow BE=BD \\ \end{align}$ We will assume that,$\begin{align}  & \Rightarrow AF=AD=x \\  & \Rightarrow CF=CE=y \\  & \Rightarrow BE=BD=z \\ \end{align}$We can clearly see from the figure that,$\begin{align}  & \Rightarrow AB=AD+DB \\  & \Rightarrow BC=BE+EC \\  & \Rightarrow AC=AF+FC \\ \end{align}$ We know the values of AB, BC and AC, we can write the above equation as,$\begin{align}  & \Rightarrow 12=x+z.......(i) \\  & \Rightarrow 8=z+y..........(ii) \\  & \Rightarrow 10=x+y........(iii) \\ \end{align}$We will add equation (i), (ii) and (iii), we will get,$\begin{align}  & \Rightarrow 2(x+y+z)=30 \\  & \Rightarrow x+y+z=15.......(iv) \\ \end{align}$ We will substitute equation(iii) in equation (iv), we will get,$\begin{align}  & \Rightarrow x+y+z=15 \\  & \Rightarrow 10+z=15 \\  & \Rightarrow z=5 \\ \end{align}$We can calculate value of x and y using above result, we will get,$\begin{align}  & \Rightarrow x+z=12\Rightarrow x+5=12 \\  & \Rightarrow x=7 \\  & \Rightarrow x+y=10\Rightarrow 7+y=10 \\  & \Rightarrow y=3 \\ \end{align}$ We know that,$\begin{align}  & \Rightarrow AF=AD=x \\  & \Rightarrow CF=CE=y \\  & \Rightarrow BE=BD=z \\ \end{align}$ So, AD=7cm, BE=5cm and CF=3cmNote:  Before attending this question, students must know the concepts of geometry of the circle. It is important for the student to analyze the figure properly, before jumping to a conclusion, half of the question can be solved using figure.