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TMUA 2018 · Paper 2 · Question 5 of 20

TMUA 2018 Paper 2 Question 5

Coordinate geometry — Quadrilaterals with perpendicular diagonals. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 2Coordinate geometryQuadrilaterals with perpendicular diagonals4 options
The two diagonals of the quadrilateral Q are perpendicular.

Consider the following statements:

I   One of the diagonals of Q is a line of symmetry of Q.
II  The midpoints of the sides of Q are the vertices of a square.

Which of these statements is/are necessarily true for the quadrilateral Q?

  1. Aneither of them
  2. BI only
  3. CII only
  4. DI and II
Show the answer and worked solution
answer · A
  1. Aneither of them
  2. BI only
  3. CII only
  4. DI and II
Build a quadrilateral with perpendicular diagonals that is as lopsided as possible: take the vertices (0, 3), (−1, 0), (0, −2), (4, 0), so one diagonal is vertical and the other horizontal. Neither diagonal is a line of symmetry, because neither cuts the other in half, so I fails. For II, the midpoints of the sides always form a parallelogram whose sides are parallel to the diagonals and half their lengths, so perpendicular diagonals make it a rectangle — but it is a square only when the diagonals are also equal in length, which they are not here. Neither statement is forced.