The adjacent sides of a parallelogram are 45 cm and 28 cm. Find the sum of the squares of its
diagonals.
Well, let me put on my clown nose and calculate that for you!
To find the sum of the squares of the diagonals of a parallelogram, we can use the formula:
Sum of the squares of diagonals = (Square of side 1 + Square of side 2) - (2 x Square of common side)
In this case, the common side is the base of the parallelogram, and the formula becomes:
Sum of the squares of diagonals = (45^2 + 28^2) - (2 x 45 x 28)
Calculating that out, we get:
Sum of the squares of diagonals = (2025 + 784) - (2520)
And simplifying further, we get:
Sum of the squares of diagonals = 2809 - 2520
Finally, our answer is:
Sum of the squares of diagonals = 289
So, the sum of the squares of the diagonals of the parallelogram is 289 cm. Keep in mind, though, that I might have juggled the numbers a bit along the way!
To find the sum of the squares of the diagonals of the parallelogram, we can use the formula:
Sum of squares of diagonals = (1/2)(sum of squares of sides)
The sum of squares of sides can be calculated as:
Sum of squares of sides = (45^2 + 28^2)
Let's calculate it:
To find the sum of the squares of the diagonals of a parallelogram, we first need to find the lengths of its diagonals.
In a parallelogram, the opposite sides are equal in length, which means that the other two sides must also be 45 cm and 28 cm.
Now, let's use the formula to find the length of the diagonals:
For a parallelogram, the length of the diagonals can be found using the formula:
Diagonal 1 = √(Side^2 + Side^2) = √(45^2 + 28^2)
Diagonal 2 = √(Side^2 + Side^2) = √(45^2 + 28^2)
Now, let's calculate the values of Diagonal 1 and Diagonal 2:
Diagonal 1 = √(45^2 + 28^2)
= √(2025 + 784)
= √2809
= 53 cm (approx.)
Diagonal 2 = √(45^2 + 28^2)
= √(2025 + 784)
= √2809
= 53 cm (approx.)
Now that we have found the lengths of the diagonals, we can proceed to find the sum of the squares of the diagonals:
Sum of squares of diagonals = Diagonal 1^2 + Diagonal 2^2
= 53^2 + 53^2
= 2809 + 2809
= 5618
Therefore, the sum of the squares of the diagonals of the parallelogram is 5618 square cm.
let the obtuse angle be A and its opposite diagonal be x. By the cosine law:
x^2 = 45^2+28^2-2(48)(28)cosA
similarly...
y^2 = 45^2+28^2-2(48)(28)cosB
but A+B = 180
B=180-A
and cosØ = -cos(180-Ø) by the CAST rule
e.g. cos 60 = - cos120
then cos B = -cosA
x^2+y^2 = 45^2+28^2-2(48)(28)cosA + 45^2+28^2-2(48)(28)cosB
= 45^2+28^2-2(48)(28)cosA + 45^2+28^2-2(48)(28)(-cosA)
= 2(45^2+28^2)
illustrating one of the properties of parallograms