In a convex quadrilateral ABCD, all sides have different lengths. The diagonals of the quadrilateral intersect

| August 2, 2021 | education

In a convex quadrilateral ABCD, all sides have different lengths. The diagonals of the quadrilateral intersect at point O, OC = 5cm, OB = 6cm, OA = 15cm, OD = 18cm. a) Prove that the quadrilateral ABCD is a trapezoid b) Find the ratio of the areas of triangles AOD and BOC.

Consider triangles BOC and AOD.

For triangles, the angle BOC = AOD as vertical angles.

BO / DO ratio = 6/18 = 1/3.

CO / AO ratio = 5/15 = 1/3.

Then the triangles BOC and AOD are similar in two proportional sides and the angle between them.

In such triangles, similar angles are equal, the angle ADB = CBD, and since these are cross-lying angles with straight lines BC and AD and secant AC, then straight lines AD and BC are parallel, therefore ABCD is a trapezoid, in which AD and BC are bases, which was required to prove …

The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.

Svos / Saod = (1/3) 2 = 1/9.

Answer: The ratio of the areas of the triangles is 1/9.



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