In trapezoid ABCD, the bases of BC and AD are 2 cm and 8 cm, and the diagonal AC

| September 20, 2021 | education

In trapezoid ABCD, the bases of BC and AD are 2 cm and 8 cm, and the diagonal AC is 4 cm. In what ratio does the diagonal AC divide the area of the trapezoid?

Consider triangles BCA and CAD. Since ABCD is a trapezoid, its bases are parallel, i.e. BC || AD. For straight lines BC, AD and secant AC, angles BCA and CAD are crosswise, and therefore equal. BC / CA = CA / AD, 2/4 = 4/8, we can reduce the fractions and get 1/2 = 1/2. According to the third criterion of similarity, we have: the two sides of one triangle are proportional to the two sides of the other triangle, and the angles between these sides are equal. Hence, triangles BCA and CAD are similar. And in such triangles, the areas are proportional to the squares of the similar sides. S BCA / S CAD = 1 ^ 2/2 ^ 2 = 1/4.
Answer: the diagonal AC divides the area of ​​the trapezoid in a ratio of 1: 4



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