The perimeter of the triangle ABC is 36 cm. The sum of the lengths of the sides AB and BC is 24 cm. The ratio of the length of the side AB to the length of the side BC is 7: 5. Construct triangle A1CB1 symmetrical to triangle ABC about its vertex C. Find the lengths of sides A1C; A1B1 and B1C.
By condition, the perimeter of the triangle ABC is
P = AB + BC + AC = 36.
AB + BC = 24,
AB = (7/5) * BC.
From these data, we compose a system of equations and find each side of the ABC triangle:
AC = 36 – 24 = 12 cm.
(7/5) * BC + BC + 12 = 36, which means (12/5) * BC = 24, therefore BC = 10 cm.
Then AB = (7/5) * BC = (7/5) * 10 = 14 cm.
Consider triangle A1CB1. Symmetry about point C is constructed as follows: side A1C continues side AC, C is the apex of triangle A1CB1, and B1C continues side BC, AB is parallel to A1B1.
A1C = AC = 12 cm.,
В1С = ВС = 10 cm,
A1B1 = AB = 14 cm.
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