The sides of the rectangle are 3: 4, and the bisector of the right angle divides the diagonal into segments

| August 8, 2021 | education

The sides of the rectangle are 3: 4, and the bisector of the right angle divides the diagonal into segments, the difference of which is 5. Find the sides of the rectangle.

Let the large side of the rectangle be 4 * X cm, then its smaller side will be equal to 3 * X cm.

We denote the length of the segment AH by Y cm, then, by condition, CH = (Y + 5) cm.

By the property of the bisector of a triangle, it divides the opposite side into segments proportional to the adjacent sides.

Then AB / AN = BC / CH.

3 * X / Y = 4 * X / (Y + 5).

4 * Y = 3 * Y + 15.

Y = 15 cm.

Then CH = 15 + 5 = 20 cm.

By the Pythagorean theorem, AC ^ 2 = AB ^ 2 + BC ^ 2.

400 = 9 * X ^ 2 + 16 * X2 = 25 * X ^ 2.

X ^ 2 = 16.

X = 4.

Then AB = 3 * 4 = 12 cm, BC = 4 * 4 = 16 cm.

Answer: The sides of the rectangle are 12 cm and 16 cm.



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