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Calculate the length of a direct common tangent to two circles of radii 3 cm and 8 cm with their centres 13 cm apart.

Circles

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Answer

Let there be two circles with centre A and B with radius 8 cm and 3 cm respectively.

Let TT' be the length of common tangent.

From figure,

Calculate the length of a direct common tangent to two circles of radii 3 cm and 8 cm with their centres 13 cm apart. Circles, ML Aggarwal Understanding Mathematics Solutions ICSE Class 10.

DT = BT' = 3cm.

AD = AT - DT = 8 - 3 = 5 cm.

In right angled triangle ADB

AB2=AD2+DB2⇒132=52+DB2⇒DB2=132−52⇒DB2=169−25⇒DB2=144⇒DB=144⇒DB=12 cm.AB^2 = AD^2 + DB^2 \\[1em] \Rightarrow 13^2 = 5^2 + DB^2 \\[1em] \Rightarrow DB^2 = 13^2 - 5^2 \\[1em] \Rightarrow DB^2 = 169 - 25 \\[1em] \Rightarrow DB^2 = 144 \\[1em] \Rightarrow DB = \sqrt{144} \\[1em] \Rightarrow DB = 12 \text{ cm}.

Since, TDBT' is a rectangle,

So, TT' = DB = 12 cm.

Hence, the length of direct common tangent is 12 cm.

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