The lengths of the diagonal of a rhombus are 18 cm and 24 cm. Find the perimeter and the height of the rhombus.
The diagonals of a rhombus are perpendicular bisectors of each other, dividing the rhombus into four congruent right triangles.
Let's call the diagonal bisecting the longer side of the rhombus as d1 with length 24 cm, and the diagonal bisecting the shorter side of the rhombus as d2 with length 18 cm.
The length of one side of the rhombus can be found using the Pythagorean theorem in one of the right triangles formed:
a^2 + b^2 = c^2
(a/2)^2 + (b/2)^2 = (d1/2)^2
a^2 + b^2 = d1^2/4 (1)
Similarly, we can use the Pythagorean theorem in another right triangle formed:
a^2 + b^2 = d2^2/4 (2)
Adding equation (1) and equation (2), we get:
2(a^2 + b^2) = (d1^2 + d2^2)/4
2(a^2 + b^2) = (24^2 + 18^2)/4
2(a^2 + b^2) = (576 + 324)/4
2(a^2 + b^2) = 900/4
a^2 + b^2 = 225/2
Since the diagonals of a rhombus bisect each other at right angles, the diagonals split the rhombus into four congruent right triangles. Therefore, we can find the height of the rhombus by taking the square root of half the sum of the squares of the diagonals:
height = √[(d1^2 + d2^2)/4]
height = √[(24^2 + 18^2)/4]
height = √[(576 + 324)/4]
height = √[900/4]
height = √(225/2)
height = √(225)/√(2)
height = 15/√(2) cm
To find the perimeter of the rhombus, we multiply the length of one side by 4:
perimeter = 4 * side
perimeter = 4 * √(225/2) cm
perimeter = 4 * (15/√(2)) cm
perimeter = (4 * 15)/√(2) cm
perimeter = 60/√(2) cm
To rationalize the denominator, we multiply by √(2)/√(2):
perimeter = (60/√(2)) * (√(2)/√(2))
perimeter = (60√(2))/2 cm
perimeter = 30√(2) cm
Therefore, the perimeter of the rhombus is 30√2 cm and the height is 15/√2 cm.