In triangle ABC it is known that BC = 72 cm AD- height, AD = 24 cm. Rectangle MNKP is inscribed in this triangle

| September 18, 2021 | education

In triangle ABC it is known that BC = 72 cm AD- height, AD = 24 cm. Rectangle MNKP is inscribed in this triangle so that vertices M and P belong to side BC, and vertices N and K to sides AB and AC, respectively. Find the sides of the rectangle if MP: MN = 9: 5 with explanation

Let the lengths of the sides of the rectangle MNKP, MN = KP = 5 * X cm, then the lengths of the sides MP = NK = 9 * X cm.

Since the points M and P lie on the segment BC, and the quadrilateral MNKP is a rectangle, then BC is parallel to NK.

Then triangles ABC and ANK are similar in two angles.

The height AO of triangle ANK is equal to: AO = AD = NM = (24 – 5 * X).

Then BC / NK = AD / AO.

72/9 * X = 24 / (24 – 5 * X).

216 * X = 1728 – 360 * X.

576 * X = 1728.

X = 3.

Then MN = 5 * 3 = 15 cm.

NK = 9 * 3 = 27 cm.

Answer: The sides of the rectangle are 15 cm and 27 cm.



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