A rectangle is inscribed in a square, the area of which is 81 cm square, so that the vertices of the rectangle

| February 4, 2021 | education

A rectangle is inscribed in a square, the area of which is 81 cm square, so that the vertices of the rectangle divide the sides of the square in a ratio of 2: 1. Find the area of the rectangle.

Knowing the area of ​​the square, we determine the length of its side.

AB = BC = CD = AD= √Savsd = √81 = 9 cm.

By condition, the vertices of the CMHР rectangle divide the sides of the square in a ratio of 2/1.

Let ВK = 2 * X cm, and СK = X cm.

ВK + СK = BC = 9 cm.

2 * X + X = 9 cm.

3 * X = 9.

X = СK = 3 cm, ВK = 2 * 3 = 6 cm.

Rectangular triangles ВKР, СMK, DНM, AHР are equal in two legs.

By the Pythagorean theorem РK ^ 2 = BP ^ 2 + BK ^ 2 = 9 + 36 = 45.

РK = 3 * √5 cm.

Then S = PK2 = 9 * 5 = 45 cm2.

The rectangle can be positioned differently, EKLN.

Then EK ^ 2 = BE ^ 2 + BK ^ 2 = 36 + 36 = 72.

EK = 6 * √2 cm.

EH ^ 2 = AE ^ 2 + AH ^ 2 = 9 + 9 = 18.

EH = 3 * √2 cm.

Then S = 6 * √2 * 3 * √2 = 36 cm2.

Answer: The area of ​​the rectangle is 45 cm2 or 36 cm2.



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