In the figure, the circles with centeres A and B touch each other at E. Line l is a common tangent which touches the circles at C and D respectively. Find the length of seg CD, if the radii of the circle are 4 cm, 6 cm.

Answer

`CD = 4sqrt(6) cm`

Explanation

Draw seg AC, seg BD seg AB. Draw seg `AM bot "seg BD and B-M-D"`.

A-E-B …(By theorem of touching circles)
`{:(AE = AC = 4cm),(Be=BE=6cm):}}" ...(Radiiof the same circle)"`
`/_ACD = /_BDC = 90^2` ...(Tangent theorem) ....(1)
`/_ACD+/_CDM+/_AMD+/_CAM = 360^@` ...(Sum of all angles of quedrilateral is `360^@`)
`:." "90^@ + 90^@ +90^@ /_CAM = 360^@` ...[From consturction, (1) and B-M-D]
`:." "270+/_CAM = 360^@`
`:." "CAM = 360^@-270^@`
`:." "CAM = 90^@`
`BM+MD = BD" ...(B-M-D)"`
`:. BM = 6-4`
`:." "AB=10 cm`
`:." "AB = 4+6`
`AB =AE +EB" ...(A-E-B)"`
`:." "AB = 10 cm.`
` :." "AB=10cm`
In `Delta AMB, /_ AMB = 90^@" ...(Construction)"`
` :." "` by Pythagoras therem,
`AB^2 = AM^2 +BM^2`
`:." "10^2 = AM^2 +2^2` ltbr `:." "AM^2 = 100-4`
` :." "AM^2 = 96`
`:." "AM = 4sqrt6" ":." "AM^2 = 96`
`:." "AM = 4sqrt6 cm` ...(Taking square roots of both the sides ) `CD=AM`
` :." "CD = AM` ...(Opposite sides of rectangle are equal )
`:." "CD = 4sqrt4` cm
`CD = 4sqrt(6) cm`