In the rectangle МРКН О – the point of intersection of the diagonals. Points A, B are the middle of the sides МР and МН

| May 4, 2021 | education

In the rectangle МРКН О – the point of intersection of the diagonals. Points A, B are the middle of the sides МР and МН, respectively. Point C divides the segment MK in a ratio of 1: 7, counting from point M. AC is perpendicular to MK. Find the ratio ВO: РН.

Let the length of the segment CM = X cm, then, by condition, CK = 7 * X cm.

Segment MK = CM + CK = X + 7 * X = 8 * X cm.

Since the diagonals, at the point of intersection, are divided in half, then OH = OK = OM = OP = 8 * X / 2 = 4 * X.

Segment OC = OM – MC = 4 * X – X = 3 * X.

Let us prove that right-angled triangles AMC and MOВ are similar.

Let the angle ОМВ be equal to X0, then the angle ОМВ = (90 – X) 0. Angle MOВ = OMA = CMA as criss-crossing angles at the intersection of parallel lines OB and AM secant MO, then the triangles AMC and MOВ are similar in acute angle.

Then MS / OB = AM / OM

X / OB = AM / 4 * X.

Since MAOB is a rectangle, then OB = AM, then 4 * X^2 = OB^2.

ОВ = 2 * X.

OB / РН= 2 * X / 8 * X = 1/4.

Answer: ВO / ВН = 1/4.



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