Line EF intersects sides AB and BC of triangle ABC at points E and F, respectively, so that the sum of angle A

| February 6, 2021 | education

Line EF intersects sides AB and BC of triangle ABC at points E and F, respectively, so that the sum of angle A and angle EFC is 180 °, and the area of the quadrilateral AEFC refers to the area of triangle EBF as 16: 9. Prove that triangle BFE is similar to triangle BAC and find the similarity factor of these triangles.

By condition, the sum of the angles (BAC + EFC) = 1800, and the sum of the angles (EFC + EFB) = 180, as the sum of the vertical angles, then the angle BAC = BFE.

In triangles ABC and BEF, the angle B is common, the angle BAC = BFE, then these triangles are similar in two angles, which was required to be proved.

Let the area of ​​the triangle EBF = 9 * X cm2, then the area of ​​the quadrangle AEFC is 16 * X cm2.

Then the area of ​​the triangle ABC = 9 * X + 16 * X = 25 * X cm2.

The ratio of the areas of similar triangles BFE and ABC is: Svfe / Savs = 9 * X / 25 * X = 9/25.

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

Then K ^ 2 = 9/25.

K = 3/5.

Answer: The coefficient of similarity is 3/5.



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