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Dudeney: Choosing a Site in a Right Estate

For the context of this applet, check out Choosing a Site in an Isosceles Estate first. Here we generalize HED’s problem to the case of the man who wanted to build his house on his right-angled triangular estate. How can he locate it so that a straight carriage-drive made from the front of his house to each of the three roads bordering his estate is most economical? Initially, the applet shows a right-angled triangle with sides a = 5 (units), b = 7 and the hypotenuse, c, is determined from these. Two options (green and orange) are shown for the location of the house, each requiring roads having a total length of 5. (Normally only one of the green and orange will work, with the other ‘undefined’. This will be seen by exploration.) In addition to the location of the house, the red points will allow you to change the lengths of the sides of the triangle (but in this introduction, we leave them as they are). At the left end of the base (a), there are two movable points, red & blue. Moving the blue point to the right or left allows you to consider different ‘contours’ (the lines with the large dashes) for the road length. Explore moving the green, orange & blue points … What about the circle? First restore the diagram to its initial state. Let p be the length of the vertical green road (part of the side b). It can be shown that . The radius of the circle (centred at the left end of side a) is . The line (with small dashes) is drawn from the right end of a to where the circle intersects the hypotenuse, and depends only on a, b & c. It is perpendicular to the contours mentioned above. This construction is what is required make the contours. There is plenty to explore here … Be sure to look at triangles with sides of various lengths (by moving the red points). You may like to derive the formulas for p & r