Find to the area to the triangle:?

Question

Point C has a bearing of N28E from point A and a bearing of N12W from point B. What is the area of the triangle ABC if B is due east of A and 23km from A?

Solve and explain how you go the answer please.

N 28degrees E-----N 12 degrees W ... incase you didnt understand that part

Answer 1

The angles are not hard to find (draw a sketch):


It is also known that AB = 23.

The Law of Cosines gives us the other sides:

AB / (sin BC = X * (sin CA = X * (sin
The area can be calculated as
1/2 * (side 1) * (side 2) * sin (angle 3)

in this case for instace
1/2 * BC * CA * sin ... = 1/2 * 31.95 * 35.00 * sin 40 = 355.4 km2.

Answer 2

Try to draw in a paper triangle ABC, where AB is the base, A is to the left, B is to the right, and C is above AB. To simplify, we let the segments:
AB = c
AC = b
BC = a

We have:
AB = 23 km

or
c = 23 km

Since C is N 28 E from A, then angle A measures 90 - 28 = 62. (since AB is horizontal, it is perpendicular to the N-S line).

Since C is N 12 W from B, then angle B measures 90 - 12 = 78. (since AB is horizontal, it is perpendicular to the N-S line).

Since the sum of the angles of a triangle is 180, then angle C measures 180 - 62 - 78 = 40.

We can now use the Sine Law:
a/sin A = b/sin B = c/sin C

Here we can omit a/sin A (since it will not be used)
b/sin B = c/sin C

solve for b.
b = c sin B/sin C

We have the area formula:
Area = bc sin A

Substitute the expression for b
Area = (c sin B/sin C) (c sin A)

Thus, another formula for the area is:
Area = c² · sin A sin B/sin C

We can now substitute the values:
Area = (23 km)² · sin 62 sin 78/sin 40

Simplifying
Area = 529 sin 62 sin 78/sin 40 km²

Or approximately
Area ≈ 710.76739 km²

^_^

Answer 3

Draw an imaginary line due north to make two right triangles ending at point D - use proportions to figure out the length of one of the new bases AD or BD -use trig to get the length of your imaginary line - use the standard rule for the area of a triangle to solve.

That's how - now you should be able to do the rest yourself.

Answer 4

My answer is kinda like Jeopardy, in the form of a question. Why don't you do your OWN homework?