In a parallelogram a diagonal of the length 20 cm is perpendicular to one of the sides. Find the longer side of parallelogram if its perimeter is 80 cm.
let long side = x
short side = s and is perpendicular to diag
s^2 + 400 = x^2
so s^2 = x^2 - 400
2 s + 2 x = 80
s = 40 - x
1600 - 80 x + x^2 = x^2 - 400
80 x = 2000
x = 200/8 = 100/4 = 25
length=x
width=80-2x, 40-x
so using pythagorean theorem you get
(40-x)^2+(20)^2=x^2
1600-80x+x^2+400=x^2
-x^2. -x^2
1600+400-80x=0
2000-80x=0
-80x=-2000
x=25
so 25 is the width and 15 is the length
I'm sorta Still cunflufled. /:
I need a picture. How is a diagonal perpendicular to a side?
Why did the parallelogram get a job as a construction worker? Because it had good angles! Now, let's solve this shape-tastic problem together!
Let's call the longer side of the parallelogram "x" cm. Since the perimeter of the parallelogram is 80 cm, this means that the sum of all the sides is 80 cm. Now, we know that opposite sides of a parallelogram are equal in length, so we have two sides of length x cm each.
The diagonal of length 20 cm divides the parallelogram into two congruent right-angled triangles. So, we have a right-angled triangle with one side of length x cm, another side of 20 cm, and the hypotenuse of the parallelogram as the diagonal.
Using the Pythagorean theorem, we can find the length of the shorter side of the parallelogram. According to the theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
So, x^2 + 20^2 = (20 cm)^2
x^2 + 400 = 400
x^2 = 0
Oops, it seems like there's an error in our calculations!
Let's try a different approach. Since the diagonal of the parallelogram is perpendicular to one of the sides, we have a right-angled triangle. The diagonal of the parallelogram is the hypotenuse, and one of the sides of length x cm is the altitude.
Using the Pythagorean theorem again, we can find the length of the shorter side of the parallelogram. So, we have x^2 + 20^2 = (diagonal)^2 = (20 cm)^2
x^2 + 400 = 400
x^2 = 0
Uh-oh, another math malfunction! It seems that something's not right.
I think there might be a mistake in the problem statement or my calculations! Sorry about that. Do you have any other questions, or can I help you with something else?
To solve this problem, we need to use the properties of a parallelogram and set up equations based on the given information.
Let's start by labeling the sides and angles of the parallelogram:
Let the longer side of the parallelogram be represented by "a" and the shorter side be represented by "b".
Given: Diagonal length = 20 cm, Perimeter = 80 cm
Since the diagonal is perpendicular to one of the sides, it bisects the parallelogram into two congruent right-angled triangles. Let's call the point where the diagonal intersects the side "P".
Now, using the Pythagorean theorem, we can find the relationship between the sides of the right-angled triangle:
a^2 = b^2 + P^2 ----(1)
a + b = 40 ----(2) since the perimeter is the sum of all four sides.
From equation (2), we can solve for "a" in terms of "b":
a = 40 - b
Now substitute this value of "a" in equation (1):
(40 - b)^2 = b^2 + P^2
Expanding and simplifying the equation:
1600 - 80b + b^2 = b^2 + P^2
Simplifying further:
1600 - 80b = P^2
From this equation, we can find the value of P, which is the distance from the point P to the nearest side.
Given: P = 20 cm (as it is one of the diagonals)
Substituting P = 20 in the equation:
1600 - 80b = 20^2
1600 - 80b = 400
-80b = -1200
b = 15 cm
Now we can find the longer side of the parallelogram "a" using equation (2):
a + b = 40
a = 40 - 15
a = 25 cm
Therefore, the longer side of the parallelogram is 25 cm.