Question

Please solve q 38?

Answer 1

`r=5/7` units

Explanation:

As `S and T` are points of tangency of the two circles, `PS and RT` are perpendicular to `AC, => PR` // `AC, => ST=PR=2r`
`DeltaPQR and DeltaABC` are similar,
`=> (QR)/(PR)=(BC)/(AC)`,
`=> QR=2r*3/5=(6r)/5`
Similarly, `(PQ)/(PR)=(AB)/(AC)`,
`=> PQ=2r*4/5=(8r)/5`
Now, `AS+ST+TC=AC, => x+2r+y=5`,
`=> color(red)(x=5-2r-y)`
SImilarly, `UP+QR+VC=BC, => r+(6r)/5+y=3`,
`=> color(red)((11r)/5+y=3 ----- Eq(1))`
Similarly, `AU+PQ+RV=AB, => x+(8r)/5+r=4`,
`=> (5-2r-y)+(8r)/5+r=4`
`=> color(red)((3r)/5-y=-1 ----- Eq(2))`
`Eq(1)+Eq(2), => (14r)/5=2`
`=> r=(2xx5)/14=5/7` units