# In trapezium ABCD (AD and BC of the base), the diagonals intersect at point O, SAOD = 32 cm2, SBos = 8 cm2.

In trapezium ABCD (AD and BC of the base), the diagonals intersect at point O, SAOD = 32 cm2, SBos = 8 cm2. Find the smaller base of the trapezoid if the larger one is 10 cm.

Let’s draw our trapezoid, suppose that the larger base is AD, because the area of ​​the triangle containing this segment is large.
Let’s draw the diagonals of the trapezoid.
It can be seen from the image that the angles are:
<OAD = <OCB – as internal contiguous
<CBO = <ODA – as internal overlapping
<AOD = <BOC – as vertical angles when two straight lines intersect.
For triangles AOD and BOC, the corresponding angles are equal, which means that these triangles are similar in terms of the equality of three angles.
The areas of similar triangles are referred to as the squares of the corresponding sides:
(AD / BC) ^ 2 = SAOD / SBCO

(AD / BC) ^ 2 = 32/8 = 4
AD / BC = 2, hence:
BC = AD / 2 = 10/2 = 5 cm

Answer: the smaller base is 5 cm.

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