The diagonals of a rhombus are in the ratio 3:4. If its perimeter is 40 cm,
find the lengths of the sides and diagonals of the rhombus
The diagonals are perpendicular, and bisect each other.
Draw the diagram. Each triangle formed is a 3:4:5 right triangle.
To find the lengths of the sides and diagonals of the rhombus, we can use the given information about the ratio of the diagonals and the perimeter.
Let's assume that the lengths of the diagonals are 3x and 4x, where x is a common factor.
We know that the sum of all sides of a rhombus is equal to its perimeter. Since a rhombus has four sides of equal length, each side will have a length of 40 cm divided by 4, which is 10 cm.
Using the Pythagorean theorem, we can find the value of x:
Let's consider half of the diagonals as right-angled triangles. The length of each side is x, and the length of the hypotenuse is 3x/2 or 4x/2.
Applying Pythagoras theorem, we have:
(x^2) + (x^2) = (3x/2)^2
Simplifying the equation:
2x^2 = 9x^2/4
Multiplying both sides by 4 to eliminate the fraction:
8x^2 = 9x^2
Subtracting 9x^2 from both sides:
8x^2 - 9x^2 = 0
- x^2 = 0
We have x^2 = 0, which implies x = 0.
But since lengths cannot be negative or zero, x cannot be zero.
Hence, there is no solution for this given information.