A parallelogram has diagonals of length 10 cm and 17 cm. An angle between them is 125°. Find:

a. the area of the parallelogram
b. the dimensions of the parallelogram.

I have found a. to be:
1/2 * 10 * 17 *sin(125)
= 69.63cm^2

Just need help with part b

The diagonals of a parallelogram cut it into 4 triangles of equal area,

so one of them = (1/2)(5)(8.5)sin125° = 17.407
so the whole parallelogram = 4(17.04) = 69.63 cm^2, (you had that.)
for the sides, I would use the cosine law on one of the triangles
longside^2 = 5^2 + 8.5^2 - 2(5)(8.5)cos125°
longside = appr 12.09 cm

shortside^2 = 5^2 + 8.5^2 - 2(5)(8.5)cos55°
.... take over

thank you!

To find the dimensions of the parallelogram, we can set up a system of equations using the given information.

Let's denote the length of one side of the parallelogram as "a" and the length of the adjacent side as "b".

From the given information, we can deduce the following:

1) The length of one diagonal is 10 cm, which is equal to the sum of the two adjacent sides (a + b).
2) The length of the other diagonal is 17 cm, which is equal to the difference between the two adjacent sides (|a - b|).
3) The angle between the diagonals is given as 125°.

Using these conditions, we can set up the following system of equations:

a + b = 10 (equation 1)
|a - b| = 17 (equation 2)

First, we will solve equation 2 to find the possible values of a and b by considering both the positive and negative difference values:

Case 1: a - b = 17
Case 2: b - a = 17

Solving case 1:
a - b = 17
Solving for b:
b = a - 17

Now, substitute this value of b into equation 1:
a + (a - 17) = 10
2a - 17 = 10
2a = 27
a = 13.5

Now, substitute this value of a back into equation 1 to find the corresponding value of b:
13.5 + b = 10
b = 10 - 13.5
b = -3.5

However, since lengths cannot be negative, this solution is not valid for our parallelogram.

Solving case 2:
b - a = 17
Solving for b:
b = a + 17

Now, substitute this value of b into equation 1:
a + (a + 17) = 10
2a + 17 = 10
2a = -7
a = -3.5

Again, since lengths cannot be negative, this solution is also not valid.

Therefore, the given conditions do not yield a valid parallelogram.

To find the dimensions of the parallelogram, we can use the lengths of its diagonals and the angle between them.

Let's label the diagonals of the parallelogram as AC and BD, with AC having a length of 10 cm and BD having a length of 17 cm. The angle formed between them is 125°.

First, let's find the length of side AB. Since opposite sides of a parallelogram are equal in length, AB is also equal to 10 cm.

To find the length of side BC, we can use the Cosine Law, which states that in a triangle, the square of one side equals the sum of the squares of the other two sides minus twice the product of the lengths of those two sides multiplied by the cosine of the included angle.

In the parallelogram, triangle ABC is formed by sides AB, BC, and AC.

Applying the Cosine Law to triangle ABC, we have:
BC^2 = AB^2 + AC^2 - 2 * AB * AC * cos(angle_ABC)

angle_ABC = 180° - angle_BAC,
where angle_BAC is the angle between sides AB and AC (which is 125°).

Substituting the values into the equation, we get:
BC^2 = 10^2 + 17^2 - 2 * 10 * 17 * cos(180° - 125°)

Calculating the expression inside the cosine function:
cos(180° - 125°) = cos(55°)

Substituting the value back into the equation and evaluating it, we find:
BC^2 = 389 - 340cos(55°)
BC ≈ √(389 - 340cos(55°))

Finally, to find the length of side CD, we can again apply the properties of a parallelogram. Since opposite sides are equal, CD is also equal to BC.

Therefore, the dimensions of the parallelogram are:
AB = 10 cm
BC ≈ √(389 - 340cos(55°)) cm
CD ≈ √(389 - 340cos(55°)) cm
AD = 17 cm

These values represent the approximate dimensions of the parallelogram.