Result 5 on circles and tangents — lesson. Mathematics State Board, Class 10.

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Result $5$ :

If two circles touch internally, the distance between their centres equals the difference in their radii.

Explanation:

If two circles touch internally at a point

$B$ , then the distance

$OA$ is equal to the difference between the radii

$OB$ and

$AB$ .

$\Rightarrow$

$OA$

$=$

$OB$

$-$

$AB$

Proof of the result:

Let the two circles with centres

$O$ and

$A$ intersect each other internally at 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

$O$ internally. If so, find the distance between their centres.

Solution:

By the result, we know that:

Distance between the centres

$=$ Difference of the radii.

Thus, the distance between the centres

$=$

$5$

$cm$

$-$

$4$

$cm$

$=$

$1$

$cm$

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6. Result 5 on circles and tangents

Theory: