Question

A scout troop is practicing its orienteering skills with map and compass. First they walk due east for \(1.2 \mathrm{km}\) Next, they walk \(45^{\circ}\) west of north for \(2.7 \mathrm{km} .\) In what direction must they walk to go directly back to their starting point? How far will they have to walk? Use graph paper, ruler, and protractor to find a geometrical solution.

Short answer

Answer: To find the direction and distance for the scout troop to return to their starting point, follow these steps: 1. Draw the vectors on graph paper, creating a triangle with points A, B, and C. 2. Measure the angle from point C to point A using a protractor, starting from due east and moving counter-clockwise. 3. Calculate the distance by measuring the length of the line segment between points C and A and multiplying it by the chosen scale (0.6 km/cm). 4. The final answer will consist of the measured angle as the direction and the calculated distance in kilometers.

Step-by-step solution

Draw the vectors on graph paper

Choose a suitable scale (such as 1 cm = 0.6 km) and draw the two movements on the graph paper. Begin at a designated starting point, A, and draw an arrow 2 cm long pointing due east (toward the right). Label the end of the arrow as point B. From point B, draw an arrow that is 4.5 cm long, representing 2.7 km, at an angle of 45 degrees west of north (135 degrees from due east). Label the end of this arrow as point C. Points A, B, and C should form a triangle.

Find the required direction

Using the protractor, measure the angle from point C to point A (the direction the scouts need to walk). Start measuring the angle from due east and move counter-clockwise towards point A.

Calculate the distance

Use the ruler to measure the length of the line segment between points C and A. Multiply the length by the chosen scale (0.6 km/cm) to obtain the distance in kilometers.

Final answer

The angle measured in step 2 represents the required direction to return to the starting point, and the distance calculated in step 3 gives the distance they need to walk.