1.Determine the area of a parallelogram in which 2 adjacent sides are 10cm and 13cm and the angle between them is 55 degrees?

2.If the area of triangle ABC is 5000m squared with a=150m and b=70m what are the two possible sizes of angle C?

To find the area of a parallelogram, you can use the formula:

Area = base x height

However, in this case, we do not have the height. Instead, we have the length of two adjacent sides and the angle between them. We can use the formula:

Area = a * b * sin(C)

Where "a" and "b" are the lengths of the adjacent sides and "C" is the angle between them.

1. Using the given information, where side a = 10 cm and side b = 13 cm, and the angle between them is 55 degrees, we can calculate the area of the parallelogram.

First, convert the angle from degrees to radians:

Angle (in radians) = Angle (in degrees) * (π / 180)
Angle (in radians) = 55 * (π / 180)

Now, substitute the values into the formula:

Area = a * b * sin(C)
Area = 10 cm * 13 cm * sin(55 * (π / 180))

Using a calculator, calculate the sine of the angle:

Area = 10 cm * 13 cm * sin(0.9599)
Area = 166.7 cm²

Hence, the area of the parallelogram is 166.7 cm².

2. In the second question, we are given that the area of triangle ABC is 5000 m², and side "a" is 150 m, and side "b" is 70 m.

We can use the formula:

Area = (1/2) * a * b * sin(C)

Given:

5000 = (1/2) * 150 * 70 * sin(C)

Now, solve for sin(C):

sin(C) = (2 * 5000) / (150 * 70)
sin(C) = 14.2857 / 105
sin(C) = 0.1362

Now, take the inverse sine (sin^-1) of 0.1362 to find the possible values of angle C. Using a calculator:

C = sin^-1(0.1362)
C = 7.82 degrees

Since sin(C) has a positive value, there is also an obtuse angle in the triangle. Since the sum of angles in a triangle is 180 degrees, the obtuse angle is:

Obtuse angle = 180 - C
Obtuse angle = 180 - 7.82
Obtuse angle = 172.18 degrees

Hence, the two possible sizes of angle C are approximately 7.82 degrees and 172.18 degrees.

1. To determine the area of a parallelogram, you need to know the length of the base and the height. In this case, the two adjacent sides of the parallelogram are given as 10cm and 13cm, and the angle between them is 55 degrees.

To find the height (h) of the parallelogram, we can use trigonometry. The height (h) is the perpendicular distance between the base and the opposite side.

First, find the sine of the angle between the sides:
sin(55 degrees) = h / 10cm

Rearrange the equation to solve for h:
h = 10cm * sin(55 degrees)

Calculate the sine of 55 degrees using a calculator or mathematical software. Let's assume it is approximately 0.819.

h = 10cm * 0.819
h ≈ 8.19cm

Now that you have the height (h) and the base (10cm), you can calculate the area of the parallelogram using the formula:

Area = base * height
Area = 10cm * 8.19cm
Area ≈ 81.9 square cm

Therefore, the area of the parallelogram is approximately 81.9 square cm.

2. The area of a triangle can be calculated using the formula: Area = (1/2) * base * height, where the base is one side of the triangle and the height is the perpendicular distance from the base to the opposite vertex.

In this case, the area of triangle ABC is given as 5000m squared, and the lengths of sides a and b are given as 150m and 70m, respectively.

Using the area formula, we can rearrange it to solve for the height (h):
Area = (1/2) * base * height
5000m^2 = (1/2) * 150m * h

Multiply both sides of the equation by 2 to eliminate the fraction:
10000m^2 = 150m * h

Now, solve for the height (h):
h = 10000m^2 / 150m

Simplify the expression:
h ≈ 66.67m

The height of the triangle is approximately 66.67m.

To find the two possible sizes of angle C, we can use the trigonometric sine formula:

sin(C) = h / b

Substitute the known values:
sin(C) = 66.67m / 70m

Calculate the sine inverse (sin^(-1)) of both sides to isolate angle C:
C ≈ sin^(-1)(66.67m / 70m)

Using a calculator or mathematical software, calculate the inverse sine of (66.67m / 70m) to find the possible sizes of angle C. Let's say the result is approximately 61 degrees.

Therefore, the two possible sizes of angle C are approximately 61 degrees.

#1.

Note that if you draw a broken line corresponding to the height of the parallelogram, you would form a right triangle in which the hypotenuse is 10 cm and one of the angles is 55 degrees.
We can actually solve for the height, which is equal to
10 * sin (55) = 8.19 cm
A of parallelogram = bh
A = 13 * 8.19
A = 106.5 cm^2

#2.
I'm not sure if this one requires a figure.

Anyway, hope this helps~ :3

Thank u so much