The diagonals of the rhombus are in a ratio of 3: 4. The perimeter of the rhombus is 300. Find the height of the rhombus.
Find the side length of a given rhombus
The problem statement says that the perimeter of this rhombus is 300.
Since the perimeter of any rhombus is equal to the side length of this rhombus multiplied by 4, the length of the side length of any rhombus is equal to the perimeter of this rhombus divided by 4.
Therefore, the length of the side of this rhombus is:
a = 300/4 = 75.
Find the lengths of the diagonals of a given rhombus
Let’s denote by x one third of the length of the smaller diagonal of the rhombus.
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 a rhombus are related as 3: 4, therefore, the length of the larger diagonal of this rhombus will be 4x 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.
Consider one of the four right-angled triangles into which the rhombus is divided by the diagonals.
The legs of such a triangle are equal to half the diagonals of the rhombus, and the hypotenuse is equal to the side of the rhombus.
Using the Pythagorean theorem, we get the following equation:
(1.5x) ^ 2 + (2x) ^ 2 = 75 ^ 2.
We solve the resulting equation:
2.25x ^ 2 + 4x ^ 2 = 5625;
6.25x ^ 2 = 5625;
x ^ 2 = 5625 / 6.23;
x ^ 2 = 900;
x ^ 2 = 30 ^ 2;
x = 30.
Find the lengths of the diagonals of the rhombus:
3x = 3 * 30 = 90;
4x = 4 * 30 = 120.
Find the height of the given rhombus
Let’s find the area of one triangle into which the rhombus is divided by its diagonals:
(1/2) * (90/2) * (120/2) = (1/2) * 45 * 60 = 45 * 30 = 1350.
Find the area of the rhombus:
S = 3 * 1350 = 5400.
Find the height of the rhombus:
h = S / a = 5400/75 = 72.
Answer: The height of the rhombus is 72.
