The diagonals of the rhombus are 3: 4 and the side is 50cm. Find the diagonals of the rhombus.

In order to simplify the solution of this problem, let us choose as the unknown x one third of the length of the smaller diagonal of the rhombus, expressed in centimeters.

Then the length of the smaller diagonal of this rhombus will be 3 cm.

Let us express in terms of x the length of the greater diagonal of this rhombus.

In the problem statement, it is said that the lengths of the diagonals of the rhombus are related as 3: 4, therefore, the length of the larger diagonal of this rhombus will be 4 cm.

Then half of the length of the smaller diagonal of this rhombus will be equal to 3x / 2 = 1.5x cm, and half of the length of the larger diagonal of this rhombus will be equal to 4x / 2 = 2x cm.

By its diagonals, the rhombus is divided into 4 equal right-angled triangles, in each of which the legs are equal to half the diagonals of the rhombus and the hypotenuse is equal to the side of the rhombus.

Consider one of these triangles. Using the Pythagorean theorem, we get the following equation:

(1.5x) ^ 2 + (2x) ^ 2 = 50 ^ 2.

Solve the equation and find the diagonals of the rhombus
2.25x ^ 2 + 4x ^ 2 = 2500.

We present similar terms on the left side of the resulting equation:

6.25x ^ 2 = 2500.

Divide both sides of the equation by 6.25:

6.25x ^ 2 / 6.25 = 2500 / 6.25;

x ^ 2 = 400;

x ^ 2 = 20 ^ 2;

x = 20 cm.

Knowing x, we find the lengths of the smaller and larger diagonals of the given rhombus:

3x = 3 * 20 = 60 cm;

4x = 4 * 20 = 80 cm.

Answer: the lengths of the diagonals of this rhombus are 60 cm and 80 cm.