UPSKILL MATH PLUS

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### Theory:

Result $$6$$:
The two direct common tangents drawn to the circles are equal in length.
Explanation:

The two direct common tangents $$AC$$ and $$BD$$ from $$P$$ drawn to the circles are equal in length.

$$\Rightarrow$$ $$AC$$ $$=$$ $$BD$$
Proof for the result:
By the result $$3$$, we have:
The lengths of the two tangents drawn from an exterior point to a circle are equal.
$$PA$$ $$=$$ $$PB$$ and $$PC$$ $$=$$ $$PD$$.

Subtract the above two equations.

$$PA$$ $$-$$ $$PC$$ $$=$$ $$PB$$ $$-$$ $$PD$$

$$AC$$ $$=$$ $$BD$$
Example:
In the above given figure if $$PB$$ $$=$$ $$9$$ $$cm$$ and $$AC$$ $$=$$ $$6$$ $$cm$$, find the length of the tangent $$PD$$.

Solution:

By the result, we know that $$AC$$ $$=$$ $$BD$$.

So, $$BD$$ $$=$$ $$6$$ $$cm$$.

From the figure it is observed that, $$PD$$ $$=$$ $$PB$$ $$-$$ $$BD$$.

$$PD$$ $$=$$ $$9$$ $$cm$$ $$-$$ $$6$$ $$cm$$

$$PD$$ $$=$$ $$3$$ $$cm$$

Therefore, the length of the tangent $$PD$$ $$=$$ $$3$$ $$cm$$.