Question

A circle is inscribed in a triangle ABC, having sides 8cm, 10cm and 12cm. Find AD, BE and CF (these 3 are altitudes of triangle ABC).

Answer 1

Hint: To find this first we will draw a figure of a circle inscribed in a triangle ABC where the sides of triangle can touch the edge of the circle by making points D, E, & F. From the figure we can easily understand that AD, BE & CF are tangents of a circle. Then with the help of the given sides of a triangle we will find the length of tangent made by external point of circle by substituting method

Complete step-by-step answer:

From the figure we get to know that the external points of the circle of circle A, B, & C are the tangents and $AB=12cm,BC=8cm,AC=10cm$ .
We know that the tangents drawn from the external points of the circle are equal.
Therefore, $AD=AF,CF=CE,BE=BD.$
Now let us consider –
$\begin{array}{rl}& AD=AF=x\\ & BD=BE=y\\ & CE=CF=z\end{array}$
It is given that $AB=12cm,BC=8cm,AC=10cm$.
Where,
$\Rightarrow AB=AD+BD$
$x+y=12cm$ ……………….. (1)
$\Rightarrow BC=BE+CE$
$y+z=8cm$ …………………… (2)
$\Rightarrow AC=AF+CF$
$x+z=10cm$ ………………… (3)
By adding equation (1), (2) and (3), we get –
$\begin{array}{rl}& (x+y)+(y+z)+(x+z)=12cm+8cm+10cm\\ & 2x+2y+2z=30cm\end{array}$
By taking 2 as common we get –
$2(x+y+z)=30cm$
By dividing both sides by 2, we get –
$x+y+z=15cm$ ………………………….. (4)
Now, we will substitute equation (1) in equation (4). So, we get –
$\begin{array}{rl}& (x+y)+z=15\\ & 12+z=15\end{array}$
By subtracting 12 from both the sides, we get –
$\begin{array}{rl}& z=15-12\\ & z=3\end{array}$
Now, we will substitute the value of ‘z’ in equation (3).
$\begin{array}{rl}& x+z=10\\ & x+3=10\end{array}$
By subtracting 3 from both sides, we get –
$\begin{array}{rl}& x=10-3\\ & x=7\end{array}$
By substituting the value of ‘z’ in equation (2), we get –
$\begin{array}{rl}& y+z=8\\ & y+3=8\end{array}$
By subtracting 3 from both sides, we get –
$\begin{array}{rl}& y=8-3\\ & y=5\end{array}$
Therefore,
$\begin{array}{rl}& AD=x=7cm\\ & BE=y=5cm\\ & CF=z=3cm.\end{array}$

Note: Generally students make mistakes while solving this problem by taking x, y & z as the sides of the triangle then divide it by 2 to get an answer which is completely wrong. Students should know that the external points of the circle which is forming a triangle are the tangents of a circle. The lengths of two tangents from an external point of a circle are equal.