a rhombus has an area of 50 cm^2 and an internal angle of size 63 . find the lenght of its sides.
Rhombus area = a^2 sin A
where a is the side length and A is an internal angle at a corner. (Either the acute or the obtuse angle could be used, since that are supplementary)
Solve a^2 sin 63 = 50 to get since length a.
a^2 = 50/0.891
a = ____ cm
thank u
:)
To find the length of the sides of a rhombus, we can use the formula for the area of a rhombus:
Area = (diagonal₁ * diagonal₂) / 2
Since a rhombus has diagonals that intersect at right angles, we can divide the rhombus into four congruent right-angled triangles.
Let's denote the length of one side of the rhombus as "s", and the length of one of its diagonals as "d". The other diagonal will also have a length of "d".
Since the diagonals bisect each other at right angles, each of the right-angled triangles formed will have a base of length "d/2" and height "s/2".
The area of each right-angled triangle is given by:
Area = (base * height) / 2
Substituting the values, we have:
50 cm² = [(d/2) * (s/2)] / 2
Simplifying the equation:
100 cm² = ds/8
Since we know the area (50 cm²), we can substitute it into the equation:
100 cm² = d*s/8
Cross multiplying, we get:
800 cm² = ds
Now, we need to find the length of the sides "s". To do this, we need to know the length of one of the diagonals "d".
Unfortunately, the information about the size of the internal angle (63°) is not sufficient to calculate the length of the diagonals or sides of the rhombus.
To find the lengths of the sides, we would need either the length of one of the diagonals or the length of one side.
To find the lengths of the sides of a rhombus given its area and an internal angle, you can use the formulas:
Area of a rhombus = 1/2 * (product of diagonals)
Internal angle = arcsin(square root of (1/2 * (1 - (diagonal1/diagonal2)^2)))
Since we have the area and an internal angle, we can solve for the lengths of the sides using the following steps:
Step 1: Substitute the given area into the formula for the area of a rhombus:
50 cm^2 = 1/2 * (product of diagonals)
Step 2: Solve for the product of the diagonals:
Product of diagonals = 2 * 50 cm^2 = 100 cm^2
Step 3: Substitute the given internal angle into the formula for the internal angle of a rhombus:
63° = arcsin(square root of (1/2 * (1 - (diagonal1/diagonal2)^2)))
Step 4: Rearrange the formula to solve for (diagonal1/diagonal2)^2:
(diagonal1/diagonal2)^2 = 1 - (2 * sin^2(63°))
Step 5: Calculate (diagonal1/diagonal2)^2:
(diagonal1/diagonal2)^2 = 1 - (2 * (sin(63°))^2)
Step 6: Calculate the values of (diagonal1/diagonal2)^2:
(diagonal1/diagonal2)^2 = 1 - (2 * (0.890)^2) (rounding sin(63°) to three decimal places)
Step 7: Take the square root of (diagonal1/diagonal2)^2:
diagonal1/diagonal2 = square root of (1 - (2 * (0.890)^2)) (rounding to three decimal places)
Step 8: Calculate the lengths of the sides of the rhombus:
Since a rhombus has equal side lengths, the lengths of the sides are equal to the lengths of the diagonals.
sides = diagonal1 = diagonal2 = square root of (100 cm^2 / diagonal1/diagonal2)
By following these steps, you can find the lengths of the sides of the rhombus. Please use a scientific calculator to perform the necessary calculations.