The longer diagonal of a parallelogram measures 62 cm and makes an angle of 30 degrees with the base. Find the area of the parallelogram if the diagonals intersect at angle of 70 degrees.

Since the angle of intersection of the diagonals is 70 degrees then the other angle of intersection of the diagonals is 110 degrees.

180 = 70 + θ
where θ = the other angle of intersection
θ = 110 degrees

A parallelogram makes 4 oblique triangles. One triangle has angles : 30, 40, and 110 degrees.
Since the diagonals intersect at their midpoints, the length of a side of one oblique triangle will measure 31 degrees (which is from 62/2).
By the given values you can solve for half the length of the shorter diagonal using the law of sines.
31/sin40 = x/sin30
where x = the length of one side of the oblique triangle and also half the length of the shorter diagonal.
x = 24.11371932 cm
2(24.11371932) = 48.23 --> length of shorter diagonal

use the formula:
Area of parallelogram = (lenth of longer diagonal x lenth of shorter diagonal x sinθ)/2
where θ = angle of intersection of the two diagonals (either 70 degrees or 110 degrees)

A = ((62 cm)(48.23)sin70)/2
A = 1404.962628 or 1405 cm^2

1405

Well, calculating the area of a parallelogram can sometimes feel like trying to unravel a mysterious tangle of geometry secrets. But fear not, I am here to make it a little less tangled!

Let's break this down. We know that the longer diagonal measures 62 cm and forms an angle of 30 degrees with the base. We also know that the diagonals intersect at an angle of 70 degrees. Hmm, quite the predicament!

To get started, let's draw ourselves a little diagram. Imagine a parallelogram with sides and diagonals, and let this process be compensated with your imagination. Now, we have a 70-degree interior angle, and since opposite angles in a parallelogram are congruent, the other 70-degree angle will be at the opposite corner.

With me so far? Great! Now imagine drawing perpendiculars from the points where the diagonals intersect to the opposite sides of the parallelogram. This creates four right-angled triangles, with two triangles on each side of the longer diagonal.

Oh, the suspense! But fear not, my friend. Since the longer diagonal forms an angle of 30 degrees with the base, we can split these triangles into two 15-75-90 triangles.

Using our knowledge of trigonometry (hey, it finally comes in handy!), we can determine that the length of the shorter diagonal is 62 cm * sin(70°) / sin(30°). In simpler terms, it's like "a sin of 70° away, the shorter diagonal goes". And who said geometry couldn't be poetic?

Now that we have both diagonals' lengths, we can finally calculate the area of the parallelogram. The area of any parallelogram can be determined by multiplying the length of one of its diagonals by the length of the perpendicular from that diagonal to the opposite side. In our case, that would be the longer diagonal times the perpendicular from the longer diagonal to the base.

So, area = 62 cm * (62 cm * sin(70°) / sin(30°)).

But don't fret! Grab your trusty calculator, follow these steps, and you'll be able to unwrap the geometry secrets to find the area of the parallelogram. Good luck, brave mathematician!

To find the area of the parallelogram, we need to know the lengths of the base and height.

First, let's find the length of the base. We are given that the longer diagonal of the parallelogram measures 62 cm. Since the diagonals of a parallelogram bisect each other, the longer diagonal is divided into two equal parts. Therefore, each half of the longer diagonal is 62 cm ÷ 2 = 31 cm.

Now, let's find the length of the height. We know that the longer diagonal makes an angle of 30 degrees with the base. Since opposite angles in a parallelogram are congruent, the shorter diagonal also makes an angle of 30 degrees with the base.

Using trigonometry, we can determine the length of the height. In a right triangle formed by the base, the height, and half of the longer diagonal, the angle opposite the height is 30 degrees. We can use the trigonometric function tangent to find the height. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height, and the adjacent side is half of the longer diagonal.

So, the height can be calculated as follows:

height = (1/2) * longer diagonal * tangent(angle)
= (1/2) * 62 cm * tan(30 degrees)
= (1/2) * 62 cm * 0.577 (approximately, using the tangent of 30 degrees)
= 17.767 cm (approximately)

Now that we have the base (31 cm) and the height (17.767 cm), we can calculate the area of the parallelogram using the formula:

area = base * height
= 31 cm * 17.767 cm
= 549.477 cm² (approximately)

Therefore, the area of the parallelogram is approximately 549.477 cm².