A secondary school, Beterekwa, is to be built to service three primary schools: Tugwi, Shinga, and Hlanga. Shinga is 16 km from Tugwi on a bearing of $150^\circ$. Hlanga is 15 km Northeast of Shinga. We need to: a) Construct a diagram using a scale of 1 cm to represent 2 km, showing the positions of the three primary schools. b) Beterekwa is 8 km from Shinga and equidistant from Tugwi and Hlanga. i) Construct the locus of points 8 km from Shinga and the locus of points equidistant from Tugwi and Hlanga. ii) Mark and label $B_1$ and $B_2$, the possible positions of Beterekwa. c) Use the diagram to find the shorter distance between Beterekwa and Tugwi.

1. Problem Description

A secondary school, Beterekwa, is to be built to service three primary schools: Tugwi, Shinga, and Hlanga. Shinga is 16 km from Tugwi on a bearing of $150^\circ$. Hlanga is 15 km Northeast of Shinga. We need to:

a) Construct a diagram using a scale of 1 cm to represent 2 km, showing the positions of the three primary schools.

b) Beterekwa is 8 km from Shinga and equidistant from Tugwi and Hlanga.

i) Construct the locus of points 8 km from Shinga and the locus of points equidistant from Tugwi and Hlanga.

ii) Mark and label $B_1$ and $B_2$, the possible positions of Beterekwa.

c) Use the diagram to find the shorter distance between Beterekwa and Tugwi.

2. Solution Steps

a) Constructing the diagram:

Scale: 1 cm = 2 km.

1. Position of Shinga relative to Tugwi: Shinga is 16 km from Tugwi on a bearing of $150^\circ$. On the diagram, this is 16 km / 2 km/cm = 8 cm. Draw a line 8 cm long from Tugwi at a bearing of $150^\circ$ to represent Shinga.

2. Position of Hlanga relative to Shinga: Hlanga is 15 km Northeast of Shinga. Northeast means a bearing of $045^\circ$. On the diagram, this is 15 km / 2 km/cm = 7.5 cm. Draw a line 7.5 cm long from Shinga at a bearing of $045^\circ$ to represent Hlanga.

b) Finding Beterekwa:

1. Beterekwa is 8 km from Shinga, which is 8 km / 2 km/cm = 4 cm on the diagram. Construct a circle with a radius of 4 cm centered at Shinga. This is the locus of points 8 km from Shinga.

2. Beterekwa is equidistant from Tugwi and Hlanga. Construct the perpendicular bisector of the line segment joining Tugwi and Hlanga. This is the locus of points equidistant from Tugwi and Hlanga.

3. The intersections of the circle (locus of points 8 km from Shinga) and the perpendicular bisector (locus of points equidistant from Tugwi and Hlanga) are the possible positions of Beterekwa. Label these points $B_1$ and $B_2$.

c) Measuring the distance:

1. Using the diagram, measure the distance between each of $B_1$ and $B_2$ from Tugwi.

2. Multiply the measured distance in cm by 2 to get the actual distance in km.

3. Identify the shorter distance.

Without the ability to draw the diagram, I can only provide the method. Let's assume $B_1$ is the closer point to Tugwi, and the distance between them measures 3.5 cm.

3. Final Answer

The shorter distance between Beterekwa and Tugwi is approximately 7 km. (3.5 cm * 2 km/cm = 7 km). Note: This answer is based on an assumption about the diagram measurement, as I am unable to create the diagram and measure the distance myself.