A rhombus ADFE is inscribed in triangle ABC so that angle A at the bottom is common

A rhombus ADFE is inscribed in triangle ABC so that angle A at the bottom is common, and the opposite vertex F lies on the side of triangle BC. find the perimeter of the rhombus if AB = 3 cm, AC = 7 cm.

Since ADFE is a rhombus, DF is parallel to AE and AC.

Then triangles ABC and BDF are similar in two angles.

The angle B in triangles is common, the angle ВDF = BAC as the corresponding angles at the intersection of parallel straight lines АС and DF secant AB.

Let AD = DF = FE = AE = X cm.

Then BD = AB – AD = 3 – X cm.

From the similarity of triangles: AB / AC = BD / DF.

3/7 = (3 – X) / X.

21 – 7 * X = 3 * X.

10 * X = 21.

X = 2.1 cm.

Then Radfe = 4 * X = 4 * 2.1 = 8.4 cm.

Answer: The perimeter of the rhombus is 8.4 cm.

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