 UPSKILL MATH PLUS

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### Theory:

Result $$4$$:
If  two circles touch externally, the distance between their centres is equal to the sum of their radii.
Explanation: If  two circles touch externally at a point $$B$$, then the distance $$OA$$ is equal to the sum of the radii $$OB$$ and $$AB$$.

$$\Rightarrow$$ $$OA$$ $$=$$ $$OB$$ $$+$$ $$AB$$
Proof for the result:
Let the two circles with centres $$O$$ and $$A$$ intersect each other externally at the point $$B$$.

Let the radius $$OB$$ $$=$$ $$r_{1}$$ and $$AB$$ $$=$$ $$r_{2}$$ and $$r_{1}$$ $$>$$ $$r_{2}$$.

Let the distance between the centres be $$d$$.

$$\Rightarrow$$ $$OA$$ $$=$$ $$d$$

From the figure, we observe that $$OA$$ $$=$$ $$OB$$ $$+$$ $$AB$$.

$$\Rightarrow$$ $$d$$ $$=$$ $$r_{1}$$ $$+$$ $$r_{2}$$
Example:
Two circle with radii $$4$$ $$cm$$ and $$5$$ $$cm$$ intersect at a point $$O$$ externally. If so, find the distance between their centres.

Solution:

By the result, we know that:

Distance between the centres $$=$$ Sum of the radii.

Thus, the distance between the centres $$=$$ $$4$$ $$cm$$ $$+$$ $$5$$ $$cm$$

$$=$$ $$9$$ $$cm$$