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A circle is inscribed in a trapezium in which one of the non-parallel sides is perpendicular to the two parallel sides. Then

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The diameter of the inscribed circle is the geometric mean of the lengths of the parallel sides

The diameter of the inscribed circle is the harmonic mean of the lengths of the parallel sides

The area of the trapezium is the area of the rectangle having lengths of its sides as the lengths of the parallel sides of the trapezium

The area of the trapezium is half the area of the rectangle having lengths of its sides as the lengths of the parallel sides of the trapezium

answer is B, C.

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Detailed Solution

From the above fig, since the length of the tangent

Drawn from an external point to the circle is the same

$A B = A S, B P = B Q, C R = C Q, D R = D S$

$\begin{array}{l} A B + D C = A D + C B \\ C B = a + b - 2 r \end{array}$

$in Δ C N B,$ ${(2 r)}^{2} + {(b - a)}^{2} = {(a + b - 2 r)}^{2}$

$4 r a + 4 r b = 4 a b r a + r b = a b r (a + b) = a b r = \frac{a b}{a + b}$

Area is $\frac{1}{2} (a + b) 2 r = a b$

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