the acute angles of a rhombus are 76 degrees if the shorter diagonal is of length 8 cm calculate of the sides of the rhombus to the nearest mm
the diagonals of a rhombus right-bisect each other, so
make a sketch and look at the right-angled triangle with angle 38° (half the 76) and leg of 4 cm
The hypotenuse is the side of the rhombus, call it x
sin 38° = 4/x
x = 4/sin38 = appr. 6.487 cm
= appr 65 mm
Ms Sue pls help
To find the length of the sides of the rhombus, we can use the properties of a rhombus.
In a rhombus, all four sides are equal in length. Let's denote the length of each side as "s" (in cm).
Since the adjacent acute angles in a rhombus are supplementary (add up to 180 degrees), we can conclude that there is an isosceles triangle formed by half of the shorter diagonal and two sides of the rhombus.
Given that the acute angle in this isosceles triangle is 76 degrees, the base angles of the triangle are equal. Thus, each base angle in the triangle is (180 - 76) / 2 = 52 degrees.
Using the properties of an isosceles triangle, we can find the length of the sides using trigonometry.
Using the law of cosines, we have:
s^2 = (8/2)^2 + (8/2)^2 - 2 * (8/2) * (8/2) * cos(52)
s^2 = 16 + 16 - 2 * 16 * cos(52)
s^2 = 32 - 32 * cos(52)
s^2 = 32(1 - cos(52))
s^2 ≈ 32(1 - 0.6157)
s^2 ≈ 32 * 0.3843
s^2 ≈ 12.3
s ≈ √12.3
s ≈ 3.5 cm
Therefore, the length of each side of the rhombus is approximately 3.5 cm to the nearest mm.
To find the lengths of the sides of the rhombus, we can use the trigonometric properties of a rhombus.
1. First, let's start by drawing and labeling the given information. We know that the acute angles of the rhombus are 76 degrees and one of the diagonals has a length of 8 cm.
D1
_________
| 76° |
________| |_______
| |
| |
| |
| |
| 8 cm |
| |
| |
| |
|_______| |_______|
D2
2. Since a rhombus is a parallelogram, its diagonals bisect each other at right angles. Therefore, we can divide the rhombus into four congruent right triangles.
3. Let's focus on one of the right triangles and label the vertices as follows:
- The vertex where the shorter diagonal intersects the side of the rhombus as point A.
- The vertex where the longer diagonal intersects the side of the rhombus as point B.
- The midpoint of the shorter diagonal as point M.
A
|\
| \
| \
| \
| \
|________\B
M
4. In right triangle AMB, angle AMB is 76 degrees. We can apply trigonometry to find the length of side AB using the sine function.
sin(76°) = AB / 8 cm
5. Rearranging the equation, we can solve for AB:
AB = 8 cm * sin(76°)
6. Now, we can calculate the value of AB:
AB ≈ 8 cm * 0.97437 (rounded to 5 decimal places)
≈ 7.795 cm (rounded to 3 decimal places)
7. Since the rhombus has congruent sides, the length of each side is equal to AB.
Side length = 7.795 cm
8. To convert the length to the nearest millimeter, we can round the value:
Side length ≈ 7.8 cm (rounded to the nearest millimeter)
Therefore, the sides of the rhombus, to the nearest millimeter, are approximately 7.8 cm.