APPLICATIONS OF CONCURRENCY 1. The first-aid centre for a cycle race need to be at a point equidistant from three cycle routes that intersect to form a triangle. Locate this point so that in an emergency medical personel will be able to get to any one of the points by the shortest route possible. Which point of concurrency is it. 2. Razzia wants to install a circular sink in her new triangular countertop. She wants to choose the largest sink that will fit. Which poit of concurrency must she locate? Explain. 3. Mbulelo wishes to centre a butcher block at a location equidistant from the refrigerator, till and sink. Which point of concurrency does Mbulelo need to locate? 4. Marlene is interested in stained-glass art. She wishes to circumscribe a circle about a triangle in her latest abstract design. Which point of concurrency does she need to locate? 5. Deodat is designing a large triangular hang-glider. He needs to locate the centre of gravity for his glider. Which point does he need to locate. Deodat wishes to decorate his glider with the largest possible circle within his triangular glider. Which point of concurrency does he need to locate? EXTENSION: If you have access to geometry software e.g. Geometer’s Sketchpad, you may wish to investigate the four points of concurrency relate to a special line called the Euler line. Draw a scalene triangle and construct the different poits of concurrency for the triangle. What do you notice? State your discovery as a conjecture. Another extension is to investigate: Is there more to the orthocentre. Draw a triangleABC and construct its orthocentre O. Drag a vertex around and observe the behaviour of the orthocentre. Where does the orthocentre lie in an acute triangle, an obtuse triangle, a right-angled triangle? Hide the altitudes. Draw segments from each vertex to the orthocentr., forming three triangles within the original triangle. Now find the orthocentre of each of the three triangles. What happened? What does this mean? Experiment dragging different points, and observe the relationship among the four orthocentres. Drag the orthocentre towards each vertex. What happens. Write a paragraph of your findings, concluding with a conjecture about the orthocentre. Share your findings with your group members or class mates.