# Find the value of $AD,BF$ and $CF$ in the given figure.

Last updated date: 27th Mar 2023

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Answer

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Hint: Use the rule of the tangents drawn from an external point to a circle subtended are equal.

Complete step-by-step answer:

We are going to use the rule that tangents drawn from an external point to a circle are equal.

Therefore,

$ \Rightarrow AD = AF,BD = BE$ and $CE = CF$

Now let us take,

$ \Rightarrow AD = AF = a$,

$ \Rightarrow BD = BE = b$ and,

$ \Rightarrow CE = CF = c$,

Add and, we get,

$AB = AD + DB = a + b = 8$ …..(1)

$BC = BE + EC = b + c = 10$ …..(2)

$AC = AF + FC = a + b = 12$ …..(3)

Add (1), (2) and (3), we get,

$2\left( {a + b + c} \right) = 30$

Taking the numbers on one side and variable on the other, we have,

$\left( {a + b + c} \right) = \frac{{30}}{2} = 15$…..(4)

Now, we are going to use (1), (2), (3) and (4), to find the value of the variables.

Subtracting (1) from (4), we get $c = 7$

Subtracting (2) from (4), we get $a = 5$

Subtracting (3) from (4), we get $b = 3$

Therefore,

$AD = a = 5cm$,

$BE = b = 3cm$

And, $CF = c = 7cm$

Note: We started by taking the rule of tangents drawn from external points to circle are equal, using this we formed equations and then solved it to get the values.

Complete step-by-step answer:

We are going to use the rule that tangents drawn from an external point to a circle are equal.

Therefore,

$ \Rightarrow AD = AF,BD = BE$ and $CE = CF$

Now let us take,

$ \Rightarrow AD = AF = a$,

$ \Rightarrow BD = BE = b$ and,

$ \Rightarrow CE = CF = c$,

Add and, we get,

$AB = AD + DB = a + b = 8$ …..(1)

$BC = BE + EC = b + c = 10$ …..(2)

$AC = AF + FC = a + b = 12$ …..(3)

Add (1), (2) and (3), we get,

$2\left( {a + b + c} \right) = 30$

Taking the numbers on one side and variable on the other, we have,

$\left( {a + b + c} \right) = \frac{{30}}{2} = 15$…..(4)

Now, we are going to use (1), (2), (3) and (4), to find the value of the variables.

Subtracting (1) from (4), we get $c = 7$

Subtracting (2) from (4), we get $a = 5$

Subtracting (3) from (4), we get $b = 3$

Therefore,

$AD = a = 5cm$,

$BE = b = 3cm$

And, $CF = c = 7cm$

Note: We started by taking the rule of tangents drawn from external points to circle are equal, using this we formed equations and then solved it to get the values.

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