In a convex quadrilateral, ABCDs have different lengths. The dioganals of the quadrilateral intersect

| March 2, 2021 | education

In a convex quadrilateral, ABCDs have different lengths. The dioganals of the quadrilateral intersect at the 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 the triangles AOD and BOC.

Let us prove the similarity of the triangles AOD and ВOС.

The ВOС angle is equal to the AOD angle as the vertical angles at the intersection of straight lines AC and ВD.

ВO / DO = 6/18 = 1/3.

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

The two sides of the triangles are proportional. Then the triangles AOD and ВOС are similar in two proportional sides and the angle between them.

In such triangles, the angles between similar sides are equal, then the angle BCO = OAD.

Since these are criss-crossing angles at the intersection of two secant lines, then the BC is parallel to the ABP, and therefore the quadrilateral AВСD is a trapezoid, which was required to be proved.

The ratio of the areas of similar triangles is equal to the square of the similarity coefficient, then:

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

Answer: The area ratio is 1/9.



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