Theory question 109
ConstructionsThree towns X, Y and Z are such that Y is 20 km from X and 22 km from Z. Town X is 18 km from Z. A Health Centre is to be built by the government to serve the three towns. The centre is to be located such that patients from X and Y will always travel equal distance to access the Health Centre while patients from Z will travel exactly 10 km to reach the Health Centre. (a) Using a scale of 1 cm to 2 km, find by construction, using a pair of compasses and ruler only, the possible positions the Health Centre can be located. (b) In how many possible locations can the Health Centre be built? (c) Measure and record the distances of the locations from town X. (d) Which of these locations would be convenient for all the three towns?
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(a) Draw triangle XYZ with XY = 10 cm, YZ = 11 cm, XZ = 9 cm. Construct the perpendicular bisector of XY. With centre Z and radius 5 cm (10 km) draw a circle. The Health Centre is where they meet.
(b) Two possible locations.
(c) About 12.7 km and 28.0 km from X (6.3 cm and 14.0 cm on the drawing).
(d) The location about 12.7 km from X (inside the triangle), since it is closest to all three towns.
Points equidistant from X and Y lie on the perpendicular bisector of XY; points 10 km from Z lie on a circle.