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Locating a point equidistant from three villages

Locating a point equidistant from three villages

IP 1
7 marks

\(A\), \(B\) and \(C\) are three points that are not on one straight line.

  1. The point \(T\) lies on the perpendicular bisector of \(AB\) and on the perpendicular bisector of \(BC\). Explain why \(TA=TC\).[2]
  2. Hence explain why the perpendicular bisectors of the three sides of any triangle meet at one point.[2]
  3. Three villages \(A\), \(B\) and \(C\) want a water tower the same distance from all three. Describe how its position can be found by a ruler-and-compasses construction on a scale map, and state the type of triangle \(ABC\) for which the tower lies outside the triangle.[2]
  4. Where does the tower lie if \(\angle ABC=90^\circ\)?[1]

Solution:

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Answer:(a) \(TA=TB\) and \(TB=TC\) from the two bisectors (b) \(T\) is equidistant from \(A\) and \(C\), so it lies on the third bisector (c) intersection of two constructed perpendicular bisectors; outside when \(ABC\) is obtuse-angled (d) the midpoint of \(AC\) (diagonals of rectangle \(ABCD\) are equal and bisect each other)

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