How do you find the height of a scalene triangle when given 2 angles and 1 side: side c = 1200, Angle A = 72, Angle B = 77?

To find the height of a scalene triangle given two angles and one side, we can follow these steps:

Step 1: Find the third angle.

The sum of the angles in any triangle is 180°. Given that we know angles A and B, we can find angle C as follows:

Angle C = 180° - Angle A - Angle B
Angle C = 180° - 72° - 77°
Angle C = 31°

Step 2: Use the Law of Sines to find the other two sides.

The Law of Sines states that:

(sin(A) / a) = (sin(B) / b) = (sin(C) / c)

We are given side c = 1200, and we need to find the other sides (a and b). First, we will use the Law of Sines to find side b.

(sin(A) / a) = (sin(C) / c)

Rearranging for a:

a = (c * sin(A)) / sin(C)

Substitute the values:

a = (1200 * sin(72°)) / sin(31°)

Now calculate the values of the sine functions:

a = (1200 * 0.95106) / 0.51504
a ≈ 2205.36

Next, we will find side b using the Law of Sines again:

(sin(B) / b) = (sin(C) / c)

Rearranging for b:

b = (c * sin(B)) / sin(C)

Substitute the values:

b = (1200 * sin(77°)) / sin(31°)

Now calculate the values of the sine functions:

b = (1200 * 0.97493) / 0.51504
b ≈ 2272.33

Step 3: Use the area formula to find the height.

Now that we have sides a and b, we can use the area formula of a triangle to find the height. The area (A) of a triangle can be found using:

A = 0.5 * base * height

We can use side a or side b as the base. Let's choose side b = 2272.33 as the base. The area of the triangle can also be found using:

A = 0.5 * a * b * sin(C)

Substitute the values:

A = 0.5 * 2205.36 * 2272.33 * sin(31°)

A = 0.5 * 2205.36 * 2272.33 * 0.51504
A ≈ 1204034.23

Now, using the area formula with base b, we can find the height (h):

A = 0.5 * b * h

Substitute the values:

1204034.23 = 0.5 * 2272.33 * h

Now solve for h:

h = 1204034.23 / (0.5 * 2272.33)
h ≈ 1054.51

Answer: The height of the triangle is approximately 1054.51 units.