Point B is 12 km north of point A. Point C is 9 km from point B on a bearing of 060°. Find the distance from A to C.
Looks like the angle at B is 150° and we can just use the cosine law:
AC^2 = 12^2 + 9^2 - 2(12)(9)cos 150°
= 225 - 216( -√3/2)
= ....
AC = + √ .....
I got AC = appr 20.3
To find the distance from point A to point C, we can break this problem down into two steps:
Step 1: Find the coordinates of point C.
Step 2: Use the distance formula to calculate the distance from point A to point C.
Step 1: Finding the coordinates of point C
To determine the position of point C, we start from point B and move 9 km on a bearing of 060°. A bearing of 060° means moving in a direction 60° clockwise from north.
Since point B is 12 km north of point A, we can use this information along with the bearing to determine the coordinates of point C.
Using trigonometry, we can break down the motion into horizontal and vertical components. The horizontal component can be found using the cosine function, and the vertical component can be found using the sine function.
Horizontal Component = 9 km * cos(60°)
Vertical Component = 9 km * sin(60°)
We can now find the coordinates of point C by summing up the horizontal and vertical components:
Coordinates of point C = (Coordinates of point B) + (Horizontal Component, Vertical Component)
Step 2: Calculating the distance from point A to point C
Now that we have the coordinates of point C, we can use the distance formula to calculate the distance between point A and point C.
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Where (x1, y1) are the coordinates of point A and (x2, y2) are the coordinates of point C.
With this information and the values you have provided (12 km for point B north of point A, 9 km distance from point B on a bearing of 060°), you can apply these steps to solve for the distance from point A to point C.