Vertex C of parallelogram ABCD is connected to point H on side AB. Line segment CH intersects diagonal

| May 4, 2021 | education

Vertex C of parallelogram ABCD is connected to point H on side AB. Line segment CH intersects diagonal BD at point P. The area of triangle BHP is 18 and the area of triangle BCP is 24. Find the area of the parallelogram.

Consider triangles BHP and HРВ, the area of ​​which is known. Both triangles have a common height ВK, then the ratio of the areas of these triangles is equal to the ratio of their bases.

Svnr / Svrs = НР / СР.

18/24 = HP / CP = 3/4.

The triangles ВНР and СРD are similar, since they have a common angle P, and the angle ВНР = DСР as crosswise. The coefficient of similarity of triangles is HP / CP = 3/4.

Then the ratio of the areas of these triangles is equal to the square of the similarity coefficient.

Svnr / Ssrd = 9/16.

18 / Ssrd = 9/16.

Ssrd = 18 * 16/9 = 32 cm2.

The area of ​​the triangle ВСD will be equal to: Svsd = Svsd + Svrs = 32 + 24 = 56 cm2.

Since the diagonal of the parallelogram divides it into two equal triangles, then Savsd = 2 * Svsd = 2 * 56 = 112 cm2.

Answer: The area of ​​the parallelogram is 112 cm2.



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