From 2 points 400 yards apart on the bank of a straight river flowing due south, the bearings of a point on the opposite bank are 148 degrees and 28 degrees.

How wide is the river?

I am having a hard time understanding how this is set up.
Help please?

my diagram looked like this :

two vertical lines, points A and B 400 yards apart on the left line, with A above B.
Point P on the other line, so that angle PBA = 28º and angle PAB = 32º (the supplementary angle to 180)

now find the perpendicular distance from P to AB.
Does that help?

how do you find that? I have the angle measurments, but no side lengths.

Except for 400..

To understand how this problem is set up, it's helpful to visualize the situation. Let's imagine a river flowing from north to south, and we are standing on the bank of the river on the northern side. We have two points, A and B, on this bank, which are 400 yards apart. We are trying to determine the width of the river, which is the distance between the opposite bank and the line connecting points A and B.

The problem provides us with the bearings of a point on the opposite bank from points A and B. Bearings are angles measured in degrees clockwise from a reference line. In this case, the bearings are given as 148 degrees and 28 degrees.

To solve this problem, we can use trigonometry, specifically the tangent function. Here's how we can do it step by step:

Step 1: Draw the Diagram
Start by drawing a diagram that represents the given information. Draw a line to represent the river, and label the two points on the bank as A and B. Mark the bearings of the point on the opposite bank from points A and B.

Step 2: Identify Right Angles
Since the river is flowing due south, we know that it forms a right angle (90 degrees) with the bank where points A and B are located. Identify this right angle in the diagram.

Step 3: Calculate Angles
Next, calculate the angles formed between the line connecting points A and B and the bearings to the opposite bank. Subtract each given bearing from 90 degrees to find these angles. In this case, we have:

Angle A = 90 - 148 = -58 degrees (anticlockwise from the line)
Angle B = 90 - 28 = 62 degrees (clockwise from the line)

Note: The negative value for Angle A indicates that it lies on the opposite side of the line connecting points A and B.

Step 4: Apply Trigonometry
Now that we have the angles, we can use the tangent function to find the width of the river. The tangent of an angle is calculated by dividing the length of the side opposite the angle by the length of the side adjacent to the angle.

In this case, the tangent of Angle A can be used to calculate the width of the river. From the diagram, we can see that the side opposite Angle A is the width of the river, and the side adjacent to Angle A is the distance between points A and B, which is 400 yards.

Using the tangent function, we have:

tan(Angle A) = (width of the river) / (400 yards)

Step 5: Solve for the Width
Rearrange the equation to solve for the width of the river:

(width of the river) = tan(Angle A) * (400 yards)

Plug in the value for Angle A in the equation and calculate the width of the river.

Keep in mind that if the resulting width is negative, it means the point on the opposite bank is on the same side as point A, rather than point B.

By following these steps, you should be able to calculate the width of the river using the given bearings.