In a rhombus, one diagonal has a length of 10 cm and the area is 120 cm2. Find the length of one side.
a.)10cm
b.)12cm
c.)13cm
A rhombus has equal side lengths. The two intersecting diagonals create four congruent right triangles of area 30 cm^2 each.
Let side length = a
30 = (1/2)*5* sqrt(a^2 -5^2)
12 = sqrt(a^2 -25)
144 = a^2 -25
a^2 = 169
a = 13
The length of the other diagonal is
2*sqrt[(13^2 - 5^2)] = 24 cm
Interesting problem!
This is an interesting problem:) Thanks!!
To find the length of one side of the rhombus, we need to use the formula for the area of a rhombus:
Area = (diagonal1 * diagonal2) / 2
Given that one diagonal has a length of 10 cm and the area is 120 cm², we can substitute these values into the formula:
120 = (10 * diagonal2) / 2
To isolate the variable, we can multiply both sides of the equation by 2:
240 = 10 * diagonal2
Now, divide both sides of the equation by 10:
24 = diagonal2
So, the length of the other diagonal (diagonal2) is 24 cm.
In a rhombus, the diagonals are perpendicular bisectors of each other and divide the rhombus into four congruent right-angled triangles. Since the diagonals of a rhombus bisect each other at right angles, each triangle has two sides that are equal in length.
Let's use the Pythagorean theorem to find the length of one side of the rhombus using one of these right-angled triangles:
a² + b² = c²
In this case, one side of the rhombus is the hypotenuse (c), and the other two sides are equal (a and b).
Let's assume that the length of one side is x cm.
Using the Pythagorean theorem, we can set up the equation:
(x/2)² + (x/2)² = 24²
Simplifying the equation:
(x²/4) + (x²/4) = 576
Multiplying both sides of the equation by 4 to eliminate the fractions:
x² + x² = 2304
Combine like terms:
2x² = 2304
Now, divide both sides of the equation by 2:
x² = 1152
To find the length of one side (x), we need to take the square root of both sides of the equation:
x = √1152
Using a calculator, the square root of 1152 is approximately 33.94.
So, the length of one side of the rhombus is approximately 33.94 cm.
Therefore, none of the given answer options (a, b, c) are correct.