Diagonal of a rhombus are in the ratio 3:4. If its perimeter is 40cm, find the length of the side and diagonals of rhombus
In a rhombus all 4 sides are equal, so each side = 10 cm
Since the diagonals bisect each other, the "half-diagonals" are also in the ration of 3 : 4
Let those half-diagonals be 3x and 4x
We have 4 congruent right-angled triangle, in each one:
(3x)^2 + (4x)^2 = 10^2
25x^2 = 100
x^2 = 4
x = 2
So the half-diagonals are 6 and 8 , making the diagonals equal to 12 and 16
Let's denote the lengths of the diagonals of the rhombus as 3x and 4x respectively.
We know that the perimeter of the rhombus is 40 cm, which means that the sum of all four sides is 40 cm. As the rhombus has equal sides, each side will have a length of 40 cm divided by 4, which is 10 cm.
Since the diagonals of a rhombus bisect each other at right angles, we can use the Pythagorean theorem to find the lengths of the diagonals.
Using the Pythagorean theorem, we have:
(3x)^2 = (4x)^2 - (10/2)^2
9x^2 = 16x^2 - 25
16x^2 - 9x^2 = 25
7x^2 = 25
x^2 = 25/7
x = √(25/7)
x ≈ 2.23 cm
Therefore, the lengths of the diagonals are approximately 3x ≈ 6.69 cm and 4x ≈ 8.92 cm.
Similarly, the length of each side is 10 cm.
To find the length of the side and diagonals of a rhombus, we can start by using the given information about the ratio of the diagonals.
Let's assume the length of the shorter diagonal is 3x and the length of the longer diagonal is 4x. Since the diagonals of a rhombus bisect each other at right angles, the sides of the rhombus can be obtained using the Pythagorean theorem.
Let's denote the length of the side of the rhombus as 's'. Now, we can see that the side 's' becomes the hypotenuse of two right triangles formed by the diagonals.
Using the Pythagorean theorem, we can determine the relation between the diagonals and the side.
For the first right triangle, with the shorter diagonal:
(1) (s/2)^2 + (3x)^2 = s^2
For the second right triangle, with the longer diagonal:
(2) (s/2)^2 + (4x)^2 = s^2
Next, we can find the value of 'x' by using the ratio of the diagonals. Given that the diagonals are in the ratio 3:4, we can set up the equation:
(3) 3x/4x = 3/4
Simplifying equation (3), we get:
(4) 3x = 4(3) = 12
Solving equation (4), we find:
(5) x = 12/3 = 4
Now, we substitute the value of 'x' into equations (1) and (2) to find the value of 's', the length of the side of the rhombus.
From equation (1), we have:
(s/2)^2 + (3x)^2 = s^2
Substituting the value of 'x' from equation (5), we get:
(s/2)^2 + (3 * 4)^2 = s^2
(s/2)^2 + 12^2 = s^2
(s/2)^2 + 144 = s^2
s^2/4 + 144 = s^2
s^2 + 576 = 4s^2
3s^2 = 576
s^2 = 576/3
s^2 = 192
s = √192
So, the length of the side of the rhombus is √192 cm.
To find the length of the diagonals, we can substitute the value of 'x' into equations (1) and (2).
For the shorter diagonal:
(√192/2)^2 + (3 * 4)^2 = s^2
(√192/2)^2 + 12^2 = s^2
(√192/2)^2 + 144 = s^2
s^2/4 + 144 = s^2
3s^2 = 576
s^2 = 576/3
s^2 = 192
s = √192
For the longer diagonal:
(√192/2)^2 + (4 * 4)^2 = s^2
(√192/2)^2 + 16^2 = s^2
(√192/2)^2 + 256 = s^2
s^2/4 + 256 = s^2
3s^2 = 1024
s^2 = 1024/3
s^2 = 341.33
s = √341.33
Therefore, the length of the side of the rhombus is approximately √192 cm, and the lengths of the diagonals are approximately √192 cm and √341.33 cm.