A parallelogram has sides measuring 7 and 9. Its shorter diagonal has a length of 8. Find the measure of the longer diagonal.
using law of cosines
8^2 = 7^2+9^2 - 2*7*9 cos A
64 = 49 + 81 - 126 cos A
cos A = 66/126
x^2 = 130 - 126 cos(180-A)
but cos(180-A) = -cos A
so
x^2 = 130 + 126(66/126)
x^2 = 196
x = 14
Well, it seems like this parallelogram is giving us some dimensions to work with! Let's have some fun and figure out the measure of the longer diagonal.
First, let's draw this amazing parallelogram. *doodling sounds*
Now, since opposite sides of a parallelogram are equal, we know that the longer diagonal is the same length as the shorter diagonal. So the measure of the longer diagonal is... *drumroll* 8! Ta-da!
Wait...that's too easy, isn't it? *thinking* Oh, I get it! You wanted me to work for it! Well, you got me there. So sorry for my clownish ways. Let me try again.
We can use the formula for the diagonals of a parallelogram:
d = 2sqrt(a^2 + b^2 - 2ab*cos(theta))
Where a and b are the lengths of the sides and theta is the angle between them. In this case, a = 7, b = 9, and theta = 180 degrees (since the sides are opposite and parallel).
Plugging those values in, we get:
d = 2sqrt(7^2 + 9^2 - 2*7*9*cos(180))
= 2sqrt(49 + 81 - 126*(-1))
= 2sqrt(49 + 81 + 126)
= 2sqrt(256)
= 2*16
= 32
So, after all that hard work, the measure of the longer diagonal is 32! Phew! I hope you enjoyed this little math journey with me.
To find the measure of the longer diagonal of the parallelogram, we can use the formula:
Longer diagonal = √(shorter diagonal^2 + (sum of squares of the lengths of the sides that the shorter diagonal is connected to) - (2 * shorter diagonal * length of the side opposite to the shorter diagonal))
In this case, the shorter diagonal has a length of 8, and the sides of the parallelogram measure 7 and 9.
Using the formula, we can substitute the given values:
Longer diagonal = √(8^2 + (7^2 + 9^2) - (2 * 8 * 7))
Simplifying:
Longer diagonal = √(64 + 49 + 81 - 112)
Longer diagonal = √(82)
Longer diagonal ≈ 9.06
Therefore, the measure of the longer diagonal is approximately 9.06.
To find the measure of the longer diagonal of a parallelogram, we can use the properties of a parallelogram.
In a parallelogram, opposite sides are equal in length, and opposite angles are congruent.
From the given information, we know that the two adjacent sides of the parallelogram measure 7 and 9.
Let's label the sides as follows:
- Side a = 7
- Side b = 9
We also know that the shorter diagonal has a length of 8.
Using the properties of a parallelogram, we can construct two triangles within the parallelogram by drawing the shorter diagonal.
By drawing the shorter diagonal, we have divided the parallelogram into two congruent triangles.
Let's call the intersection point of the diagonal and the adjacent side a as point A, and the intersection point of the diagonal and the adjacent side b as point B.
Now we have two congruent triangles, with side lengths of 7, 8, and a common side of the parallelogram (side a).
We can use the Pythagorean theorem to find the length of the remaining side of these triangles, which is the side opposite to the shorter diagonal.
Let's label this side as c.
Using the Pythagorean theorem, we have:
c^2 = a^2 + b^2
Plugging in the given values, we get:
c^2 = 7^2 + 8^2
c^2 = 49 + 64
c^2 = 113
Taking the square root of both sides, we find:
c ≈ √113
So the length of the longer diagonal is approximately √113.
Therefore, the measure of the longer diagonal of the parallelogram is approximately equal to √113.