In triangle ABC: side AB = 25 cm, BC = 20 cm, AC = 30 cm. On the side AB from point B, there is a segment

In triangle ABC: side AB = 25 cm, BC = 20 cm, AC = 30 cm. On the side AB from point B, there is a segment ВD = 4 cm, and on the side BC, point E is taken so that angle BDE = angle C, Find the perimeter of triangle BDE (looking for similar triangles)

Consider triangles ABC and BDE, in them:
∠ B – general;
∠ BDE = ∠ ACB.
We conclude that the triangles are similar in two corners.
Let’s write down the ratio of the parties:
DE / AC = BE / AB = BD / BC = 4/20 = 1/5 – coefficient of similarity of triangles.
Find the unknown sides of the triangle BDE:
DE = AC / 5 = 30/5 = 6 (cm);
BE = AB / 5 = 25/5 = 5 (cm).
Find the perimeter of the triangle BDE:
P BDE = BD + DE + BE = 4 + 6 + 5 = 15 (cm).
The second option for finding the perimeter of the smaller triangle is to multiply the perimeter of the larger triangle by the similarity factor:
R ABC = 25 + 20 + 30 = 75 (cm).
P BDE = P ABC * 1/5 = 75 * 1/5 = 15 (cm).
Answer: The perimeter of the triangle BDE is 15 cm.