A straight line is drawn through point M of side AB of triangle ABC, perpendicular to the height BD

| June 17, 2021 | education

A straight line is drawn through point M of side AB of triangle ABC, perpendicular to the height BD of the triangle and intersecting side BC to point K. It is known that BM = 7cm, BK = 9cm, BC = 27cm.Find: a) the length of the side AB b) the ratio of the areas of triangles ABC and MBK

According to the BD condition, the height of the triangle to the AC side, therefore, the CDB angle is straight, and the MK segment is perpendicular to the BD height, therefore, the MOD angle is also straight.

The angles CDB and MOD are crosswise and are equal at the intersection of the segments AC and MK of the secant DB, therefore, the AC is parallel to the MK.

Consider triangles ABC and MBK, in which the angle B of the common sides AC and MK are parallel, and the angles CAB = KMB, and ACB = MKB, as the corresponding angles. Therefore, triangles ABC and MBK are similar in three angles.

Then: AB / MB = CB / KB.

AB / 7 = 27/9.

AB = 27 * 7/9 = 21 cm.

The coefficient of similarity of triangles will be equal to: k = CB / KB = 3.

The ratio of the areas of similar triangles is equal to the square of the similarity coefficient.

Savs / Smvk = k2 = 9.

Answer: AB = 21 cm, Savs / Smvk = 9/1.



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