the length of the diagonals of a rhombus are 24cm and 32cm. the perimeter of the rhombus is
(A) 9cm
(B) 128cm
(C) 80cm
(D) 56cm
plz do explain how to find.
thank you
C:80 cm
80 cm
To find the perimeter of a rhombus, you need to know the length of at least one of its sides. However, we are given the lengths of the diagonals instead.
Let's break down the problem into steps:
Step 1: Recall the properties of a rhombus. In a rhombus, the diagonals bisect each other at 90 degrees, and they divide the rhombus into four congruent right-angled triangles.
Step 2: Use the Pythagorean theorem to find the length of each side. Since we have a right-angled triangle, we can use the Pythagorean theorem, which states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the diagonals of the rhombus are the hypotenuses of these right-angled triangles. We have the following lengths:
Diagonal 1 (d1) = 24 cm
Diagonal 2 (d2) = 32 cm
Let's label the sides of one of the congruent triangles formed by the diagonals:
a: half the length of one side of the rhombus
b: half the length of the other side of the rhombus
Using the Pythagorean theorem, we can write the following equation for the first right-angled triangle:
a^2 + b^2 = (d1/2)^2
Similarly, for the second right-angled triangle:
a^2 + b^2 = (d2/2)^2
Step 3: Solve the system of equations.
We can solve this system of equations simultaneously to find the values of 'a' and 'b'.
Let's substitute the given values:
a^2 + b^2 = (24/2)^2 --> a^2 + b^2 = 144
a^2 + b^2 = (32/2)^2 --> a^2 + b^2 = 256
Now, we have the following system of equations:
a^2 + b^2 = 144 ------(1)
a^2 + b^2 = 256 ------(2)
Subtracting equation (1) from equation (2), we get:
a^2 + b^2 - a^2 - b^2 = 256 - 144
0 = 112
Since this is not possible, it means that the system of equations is inconsistent, and there are no solutions. This suggests that there is an error in the given information, as the given lengths of the diagonals do not form a rhombus.
Therefore, we cannot determine the perimeter of the rhombus based on the given information. None of the provided options (A, B, C, D) can be correct.
A rhombus is made up of four right angle triangles.
http://www.coolmath.com/reference/rhombus.html
The two sides of each triangle are 12 cm and 16 cm.
Use the Pythagorean Theorem to find the hypotenuse of each triangle, which is also the side of the rhumbus.
a^2 + b^2 = c^2
12^2 + 16^2 = c^2
144 + 256 = c^2
400 = c^2
20 = c
4 * 20 = perimeter of rhombus