The sum of the diagonal of the rhombus is 70 cm, the side of 25 cm find the height of the rhombus.

Consider a right-angled triangle in which the side of the rhombus is the hypotenuse, half of the diagonals of the rhombus are the legs. The sum of the squares of the legs is equal to the square of the hypotenuse, which means:
(d1 / 2) ^ 2 + (d2 / 2) ^ 2 = a ^ 2;
(d1 ^ 2) / 4 + (d2 ^ 2) / 4 = 25 ^ 2 = 625;
(d1 ^ 2 + d2 ^ 2) / 4 = 625;
d1 ^ 2 + d2 ^ 2 = 625 * 4 = 2500.
According to the problem statement, the sum of the diagonals of the rhombus is 70 cm:
d1 + d2 = 70;
We square both sides of the equality, we get:
(d1 + d2) ^ 2 = 70 ^ 2;
d1 ^ 2 + d2 ^ 2 + 2 * d1 * d2 = 4900.
Substituting the value found above for d1 ^ 2 + d2 ^ 2, we get:
2500 + 2 * d1 * d2 = 4900;
2 * d1 * d2 = 4900-2500 = 2400;
d1 * d2 = 2400/2 = 1200.
The area of ​​the rhombus is equal to half the product of the diagonals: S = (d1 * d2) / 2 = 1200/2 = 600 cm2.
On the other hand, the area of ​​a rhombus is equal to the product of its side by the height: S = a * h.
Hence h = S / a = 600/25 = 24.
The desired height of the rhombus is 24 cm.