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

The diagonals of the rhombus intersect at right angles and the intersection point is halved:

AO = OC = AC / 2;

BO = OD = BD / 2.

Consider the ABO triangle. Behind the Pythagorean theorem, you can find the segments BO and AO:

AB ^ 2 = AO ^ 2 + BO ^ 2.

Since the diagonals of the rhombus are related as 3: 4, the segments AO and BO will be equal to half of this length and are related as: (3/2) 🙁 4/2) = 1.5: 2. Thus, we express the segments AO and BO as follows:

1.5x is the length of the AO segment;

2x – length of the BO segment;

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

2.25x ^ 2 + 4x ^ 2 = 2500;

6.25x ^ 2 = 2500;

x ^ 2 = 2500 / 6.25 = 400;

x = √400 = 20;

AO = 1.5 20 = 30 cm;

BO = 2 * 20 = 40 cm.

AC = AO · 2;

AC = 30 2 = 60 cm;

BD = BO · 2;

ВD = 40 2 = 80 cm.

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