the length of the diagonals of a rhombus are in the ratio 3:4 if one side of the rhombus is 10 cm the length of both the diagonals of the rhombus
By the definition of a rhombus , all sides are equal, so if one side is 10, then all sides are equal to 10.
Also, the diagonals bisect each other at right angles creating 4 congruent right-angled triangles.
Looking at one of these triangles, we have sides
3x, 4x, and hypotenuse 10
9x^2 + 16x^2 = 100
25x^2 = 100
x^2 = 4
x = 2
Each triangle is 6 cm by 8 cm by 10 cm
According to my definitions, the diagonals would be 6x and 8x, or
12 cm and 16 cm.
Ab=62
Ad=48
To find the length of the diagonals of the rhombus, we need to use the given ratio and the length of one side of the rhombus.
First, let's denote the length of the shorter diagonal as 3x (since the ratio is 3:4) and the length of the longer diagonal as 4x.
Since the diagonals of a rhombus bisect each other at right angles, we can use the Pythagorean theorem to find the relationship between the length of the sides and the length of the diagonals.
In a rhombus, the diagonals create four congruent right triangles. Since each side of the rhombus has a length of 10 cm, we can use half of a side (5 cm) as one leg of the right triangle formed by the diagonals.
Using the Pythagorean theorem, we have:
(3x)^2 = (4x)^2 + (5 cm)^2, since the diagonals bisect each other at right angles.
Simplifying the equation:
9x^2 = 16x^2 + 25 cm^2
Subtracting 16x^2 from both sides:
7x^2 = 25 cm^2
Divide both sides by 7:
x^2 = 25/7
Taking the square root of both sides:
x ≈ √(25/7)
x ≈ √(25)/√(7)
x ≈ 5/√(7)
Therefore, the length of the shorter diagonal (3x) is approximately 3 * (5/√(7)) cm, and the length of the longer diagonal (4x) is approximately 4 * (5/√(7)) cm.