In the parallelogram ABCD AC = 15 cm. The midpoint M of the side AB is connected by a line segment to the vertex D.

| February 10, 2021 | education

In the parallelogram ABCD AC = 15 cm. The midpoint M of the side AB is connected by a line segment to the vertex D. Find the segments into which DM divides the diagonal AC.

Let us prove that the AOM triangle is similar to the СOD triangle.

Angle AOM = СOD as vertical angles. Angle OAM = OСD as lying crosswise at the intersection of parallel lines AB and СD secant AC, then the triangles AOM and СOD are similar in two angles.

By condition, point M is the middle of AB, AM = AB / 2, and since SD = AB, then AM = СD / 2.

Then the coefficient of similarity is: K = AM / СD = (СD / 2) / СD = 1/2.

Then AO / OС= 1/2.

OС = 2 * AO.

AO + OС = AC = 15 cm.

3 * AO = 15.

AO = 15/3 = 5 cm.

OС = AC – AO = 15 – 5 = 10 cm.

Answer: The AC diagonal is divided into 5 cm and 10 cm segments.



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