A parallelogram has adjacent sides 7 cm and 15 cm. If the shorter diagonal is 10 cm long, find the length of the longer diagonal.
A parallelogram has adjacent sides 8 cm and 13cm. If the shorter diagonal is 11 cm long, find the length of the longer diagonal.
To find the length of the longer diagonal of the parallelogram, you can use the formula for the length of a diagonal in a parallelogram.
The formula is:
Diagonal = √(a^2 + b^2 + 2ab cosθ)
Where a and b are the lengths of the adjacent sides, and θ is the angle between them.
In this case, the lengths of the adjacent sides are given as 7 cm and 15 cm. The shorter diagonal is given as 10 cm.
Let's call the longer diagonal x cm.
Using the formula, we can substitute the given values:
10 = √(7^2 + 15^2 + 2*7*15 cosθ)
To solve for x, we need to find the angle θ. We can use the Law of Cosines to find θ:
c^2 = a^2 + b^2 - 2ab cosθ
Substituting the given values:
10^2 = 7^2 + 15^2 - 2*7*15 cosθ
Simplifying:
100 = 49 + 225 - 210 cosθ
Rearranging:
210 cosθ = 49 + 225 - 100
210 cosθ = 174
Dividing both sides by 210:
cosθ = 174/210
cosθ = 0.82857
Now, we can substitute this value back into the formula for the longer diagonal:
x = √(7^2 + 15^2 + 2*7*15 *(0.82857))
Calculating:
x = √(49 + 225 + 210 *(0.82857))
x = √(49 + 225 + 173.9973)
x = √448.9973
x ≈ 21.2 cm
Therefore, the length of the longer diagonal is approximately 21.2 cm.
To find the length of the longer diagonal of the parallelogram, we can use the fact that the diagonals of a parallelogram bisect each other.
Step 1: Draw the parallelogram and label the given information.
Let's draw a parallelogram. Label one side as 7 cm and the adjacent side as 15 cm. The shorter diagonal is given as 10 cm.
7 cm
---------------
| |
15 cm | |
| |
---------------
10 cm
Step 2: Use the fact that diagonals of a parallelogram bisect each other.
The diagonals of a parallelogram bisect each other, which means they divide each other into two equal parts. So, the shorter diagonal of 10 cm divides the longer diagonal into two equal parts.
Step 3: Calculate the length of the longer diagonal.
Since the shorter diagonal divides the longer diagonal into two equal parts, each part is half the length of the longer diagonal. Therefore, the length of the longer diagonal is twice the length of the shorter diagonal.
Length of longer diagonal = 2 * length of shorter diagonal
= 2 * 10 cm
= 20 cm.
The length of the longer diagonal of the parallelogram is 20 cm.
using the law of cosines, we know that
7^2+15^2-2*7*15 cosθ = 10^2
cosθ = (7^2+15^2-10^2)/(2*7*15) = 29/35
So, that means the other diagonal, x is
x^2 = 7^2+15^2+2*7*15*29/35 = 448
x = 8√7