One diagonal of a rhombus makes a 27 degree angle with a side of the rhombus. If each side of the rhombus is 6.2 inches find the length of each diagonal to the nearest tenth.
Well, isn't this a rhomb-tastic question! To find the length of the diagonals, let's break out our trusty geometry skills. Since the sides of the rhombus are all equal, we can call each side "s" and the lengths of the diagonals "d".
Now, let's look at what we know. We're given that each side of the rhombus is 6.2 inches. That means s = 6.2 inches.
Next, we know that one diagonal makes a 27 degree angle with a side of the rhombus. Now, remember that the diagonals of a rhombus bisect each other at right angles. So, the 27 degree angle forms a pair of 6.8 degree angles with the diagonal. That's right, I did the math!
Let's call half of the angle between one diagonal and one side of the rhombus "a". So we have a = 6.8 degrees.
Now, here comes some more geometry magic. In a rhombus, the cosine of half the angle between a diagonal and a side is equal to s/d, where s is the side length and d is the diagonal length.
So, cos(a) = 6.2/d. Let's solve for d:
cos(6.8 degrees) = 6.2/d
Now, grab your calculators, because here comes the fun part. After plugging in the numbers and hitting that "=" button, we find that d is approximately equal to 6.3349 inches (rounded to the nearest tenth).
Therefore, the length of each diagonal of the rhombus is approximately 6.3 inches (rounded to the nearest tenth). Ta-da!
To find the length of each diagonal of the rhombus, we can use trigonometry and the given information on the angle.
Let's denote the length of each diagonal as "d". We know that the length of the side of the rhombus is 6.2 inches.
In a rhombus, the diagonals bisect each other at right angles, so we can draw a right triangle with one of the diagonals as the hypotenuse, the side of the rhombus as one of the legs, and half of the other diagonal as the other leg.
Since the diagonals bisect each other, the angle between the side and the diagonal is half the angle formed by the diagonals. In this case, the angle between the side and the diagonal is half of 27 degrees, which is 13.5 degrees.
Now, we can use the trigonometric function tangent to find the length of the half diagonal (let's call it "hd").
tan(13.5 degrees) = hd / 6.2 inches
To solve for hd, we can rearrange the equation:
hd = tan(13.5 degrees) * 6.2 inches
Using a calculator, we can evaluate tan(13.5 degrees) ≈ 0.24498.
hd ≈ 0.24498 * 6.2 inches ≈ 1.5196 inches
Since the half diagonal is half the length of the full diagonal, we can find the length of the full diagonal "d":
d = 2 * hd
d ≈ 2 * 1.5196 inches ≈ 3.0392 inches
Therefore, the length of each diagonal of the rhombus is approximately 3.0 inches when rounded to the nearest tenth.
To find the length of each diagonal of a rhombus, we can use the given information:
1. The angle between one diagonal and a side of the rhombus is 27 degrees.
2. Each side of the rhombus is 6.2 inches.
Let's label the length of the diagonal as "d". We can break down the problem into two right-angled triangles formed by the diagonal and the sides of the rhombus.
First, let's find the length of the inside diagonal by using trigonometry. In the right-angled triangle, the side adjacent to the angle of 27 degrees is one of the sides of the rhombus (6.2 inches), and we need to find the length of the side opposite the angle (the diagonal, d).
Using the formula for the tangent of an angle (tan θ = opposite/adjacent):
tan 27° = d/6.2
Rearranging the equation to solve for d:
d = 6.2 * tan 27°
Using a calculator, we find:
d ≈ 6.2 * tan 27° ≈ 3.120 inches (rounded to the nearest tenth)
Therefore, the length of the inside diagonal of the rhombus is approximately 3.120 inches.
To find the length of the other diagonal (outside diagonal), we can use the fact that the diagonals of a rhombus are perpendicular bisectors of each other. This means that they divide each other into two congruent right-angled triangles.
Since the inside diagonal we calculated is perpendicular to one side of the rhombus, it is also perpendicular to the other diagonal.
Therefore, the length of the outside diagonal is equal to twice the length of the inside diagonal.
Thus, the length of the outside diagonal is approximately 2 * 3.120 inches = 6.240 inches (rounded to the nearest tenth).
Therefore, the length of each diagonal of the rhombus, to the nearest tenth, is approximately:
Inside diagonal: 3.1 inches
Outside diagonal: 6.2 inches
Let the rhombus have vertices ABCD.
If ∡BAC = 27° then ∡BCA = 27° and ∡ABC = 126°
So, you can use law of sines or law of cosines to find AC.
Now, knowing that ∡BAD = 54° (why?), you can follow similar steps to find BD.