Given that a right angled trapezium has an inscribed circle. Prove that the length of the right angled leg is the Harmonic mean of the lengths of bases

Options

A. Harmonic mean of length of bases
B. Geometric mean of length of bases
C. Arithmetic mean of length of bases
D. None of these

Answer

A

Explanation

Let length of right angled leg `= 2r = OA`
`:.` Equation of circle is
`(x-r)^(2) + (y+r)^(2) = r^(2) "….."(i)`
`:.` Length of tangent
`BP = BQ = sqrt((x_(1) - r)^(2) + (2r-r)^(2)-r^(2))`
`=x_(1)-r`
Simliarly length of tangent `PC = x_(2) -r`
Now `BC = BP + PC`
`sqrt((x_(2)-x_(1))^(2) + 4r^(2)) = x_(1) + x_(2) - 2r`
Square
`(x_(2) - x_(1))^(2) + 4r^(2) = (x_(1) + x_(2))^(2) + 4r^(2) - 4r(x_(1) + x_(2))`
`rArr x_(1)x_(2) = r(x_(1) + x_(2)) rarr 1/r = 1/(x_(2)) + (1)/(x_(1))`
`rArr r = (x_(1)x_(2))/(x_(1)x_(2)) rArr 2r = (2x_(1)x_(2))/(x_(1)+x_(2))`