The course for a boat race starts at point A and p

Questions: The course for a boat race starts at point A and proceeds to the SW at 52° to point B. The boats turn to the SE to approach point C. Then, they travel 8 km due north to the finish line where they began the race. What is the m<C?

Transcript text: The course for a boat race starts at point A and proceeds to the SW at $52^{\circ}$ to point B. The boats turn to the SE to approach point C. Then, they travel 8 km due north to the finish line where they began the race. What is the $m

Solution

Solution Steps

Step 1: Find the measure of angle ABC.

The boat travels SW at 52° from point A to point B. Then it turns SE towards point C. Since SE is 180° from NW, and NW is 90° counterclockwise from SW, the boat must have turned (180 - 90 - 52) = 38° at point B. The bearing from B to C is 40° southeast, which means angle DBC is 40°. Since angle ABC is the supplement of angle DBC, angle ABC = 180° - 40° = 140°. Thus angle ABC = 140°.

Step 2: Find the measure of angle BAC.

The angles in triangle ABC add up to 180°. Since angle ABC = 140°, and angle CAB = 52°, then angle BCA = 180° - (140° + 52°) = 180° - 192° = -12° which is impossible. So there's an issue. The question says the boat turns SE at B. If this means that AB continues straight to form an obtuse angle with BC, then <ABC = 180 - (180-90-52-40) = 180 - (-2) = 182 degrees.

If the race path at B has a bearing of 40° SE from B to C, and if AB has a bearing of 52° SW from A to B, then the angle at B must be: B = 180 - (90-40) - (52) = 180 - 50 - 52 = 78°

Then, <BCA = 180 - 78 - 52 = 50°.