The perimeter of a rhombus is 40 cm and the distance between a pair of parallel side is 5.6 cm. If the length of one of its diagonals is 7 cm, find the length of the other diagonal.
Since the perimter is 40, each side is 10.
So, the area is 5.6*10 = 56.
The area is also half the product of the diagonals. So, to get the other diagonal d,
56 = 1/2 * 7 * d
d = 16
Yes, I agree with Reiny that I did the math wrong when computing the other diagonal. However, the statement that the set of dimensions are incompatible with a rhombus remains valid.
Well, let's put on our clown shoes and solve this puzzle! Now, let's start by picturing a rhombus with its parallel sides and its diagonal. It's like a fancy diamond, but without all the glamor!
Since the perimeter of the rhombus is 40 cm, and a rhombus has four equal sides, each side must be 10 cm long. Now, here comes the circus trick: if we draw the perpendicular bisectors of those sides, they will intersect right in the middle, forming a right-angled triangle.
This right-angled triangle has a hypotenuse (represented by the diagonal of the rhombus), a base (represented by half the distance between the parallel sides), and a height (represented by half the length of one side).
In this case, the base is 5.6 cm (half the distance between the parallel sides), and the height is 10 cm (half the length of one side).
Now we can apply some clown math magic! The Pythagorean theorem says that the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, we can use it to find the length of the diagonal we're looking for.
7 cm squared (diagonal squared) is equal to 5.6 cm squared (base squared) plus 10 cm squared (height squared).
Okay, let's add a little circus magic here: 7 squared is 49. And if we subtract 10 squared (which is 100) from both sides, we get 49 minus 100, which is -51.
Uh-oh! We seem to have made a mistake here! It looks like we stumbled into a trapdoor. But no worries, my trusty clown calculator says that a negative length doesn't make sense in this context.
So, sorry to disappoint you, but it seems like something went wrong along the way. Maybe we made a wrong assumption or took a somersault when we weren't supposed to. Let's double-check our calculations and try again!
Remember, kids, even the best clowns occasionally slip on banana peels. Keep smiling, and let's figure this out together!
To find the length of the other diagonal, we can use the properties of a rhombus.
A rhombus has the following properties:
1. All sides are equal in length.
2. Opposite angles are equal.
3. Diagonals bisect each other at a right angle.
In this case, the perimeter of the rhombus is given as 40 cm. Since a rhombus has all sides equal in length, we divide the perimeter by 4 to get the length of each side:
Side length = Perimeter / 4 = 40 cm / 4 = 10 cm.
The distance between a pair of parallel sides is given as 5.6 cm. This distance represents the height of the rhombus. In a rhombus, the diagonals bisect each other at a right angle, so we can now form a right-angled triangle.
Let's consider one half of the diagonal as the base of this right-angled triangle, and the given height as the height of this triangle.
We have:
Base = 7 cm / 2 = 3.5 cm
Height = 5.6 cm
Using Pythagoras' theorem, we can find the length of the other half of the diagonal.
Applying the theorem, we have:
(Base)^2 + (Height)^2 = (Diagonal)^2
Rearranging the equation to solve for the diagonal, we get:
(Diagonal)^2 = (Base)^2 + (Height)^2
Substituting the known values, we have:
(Diagonal)^2 = (3.5 cm)^2 + (5.6 cm)^2
Calculating the right side of the equation, we get:
(Diagonal)^2 = 12.25 cm^2 + 31.36 cm^2
Adding the terms on the right side, we have:
(Diagonal)^2 = 43.61 cm^2
To find the length of the other diagonal, we need to take the square root of both sides of the equation:
Diagonal = √43.61 cm^2
Diagonal ≈ 6.60 cm
Therefore, the length of the other diagonal is approximately 6.60 cm.
The data given for the problem contradicts each other and does not make sense.
drwls pointed out correctly that we would get 4 congruent right-angled triangles,
the hypotenuse would be 10, one side is 3.5, let the other side be x
x^2 + 3.5^2 = 10^2
x = √87.75 or appr. 9.367
making the other diagonal 18.734
and then the area of the rhombus would be
(1/2)(7(18.734) or 65.57
So far I have ignored the information that the distance between one set of parallel sides is 5.6
As Steve has done, that way the area would have to be (10(5.6) or 56 , which contradicts my earlier calculation of the area.
So we have a bogus question.
Each side length is the same (since it is a rhombus) and therefore 10 cm. The diagonals intersect at right angles, forming four congruent right trangles.
The sides of those triangles are
5, 3.5 and sqrt[5^2 - 3.5^2] = 3.571, making the other diagonal length 2*3.571 = 7.141
You have been given more information than necessary to solve the problem. The distance between pairs of parallel sides is inconsistent with the lengths of diagonals.
The distance from the centroid to one side, d, must be such that
(1/2)*10*d = (1/2)*7*7.141
Therefore d = 4.999 and the distance between parallel faces is 9.997
Your provided data is inconsistent with a rhombus.