In a rectangular trapezoid, the base and the smaller side are equal to a, b, and c, respectively. Find the distances from the intersection of the diagonals to the bases and the smaller side.
Let OK = X cm, OM = Y cm, OH = Z cm.
Rectangular triangles ABD and OKD are similar in acute angle ADB.
Then x / c = (b – y) / b (1).
Right-angled triangles ABC and AMO are similar in acute angle BAC.
Then y / a = x / c (2).
Let’s solve equations 1 and 2 with two unknowns.
y / a = (b – y) / b.
y * b = a * b – a * y.
y * (b + a) = a * b.
y = a * b / (b + a).
(a * b / (b + a)) / a = x / c.
x * a = c * a * b / (b + a).
x = b * c / (b + a).
Then Z = AB – OK = c – x = c – (b * c / (b + a)).
Answer: The distance to the side is a * b / (b + a), to the larger base b * c / (b + a), to the smaller base (c – (b * c / (b + a))).
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