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The diagram below shows a quadrilateral WXYZ, as well as its diagonals [WY] and [XZ] - Junior Cycle Mathematics - Question 6 - 2021
Question 6
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The diagram below shows a quadrilateral WXYZ, as well as its diagonals [WY] and [XZ]. Without measuring, perform constructions on the diagram below to show that the ... show full transcript
Worked Solution & Example Answer:The diagram below shows a quadrilateral WXYZ, as well as its diagonals [WY] and [XZ] - Junior Cycle Mathematics - Question 6 - 2021
Step 1
Perform constructions to show diagonals bisect each other
Answer
- Draw the diagonals WY and XZ.
- Label the intersection point of the diagonals as O.
- Using a compass, measure the distance from W to O and from Y to O.
- Show that WO = OY.
- Likewise, measure the distance from X to O and from Z to O.
- Show that XO = OZ.
- Since WO = OY and XO = OZ, this proves that the diagonals bisect each other.
Step 2
Prove that the triangle ABO is isosceles
Answer
- In triangle DAB, we have |DAB| = |ABC| because opposite angles of a parallelogram are equal.
- Therefore, let |DAB| = x and |ABC| = x.
- The angles at point O, |AOB| and |DOB| also create two triangles ABO and DBO that share the same height from point O to line AB.
- Since |DAB| = |CBA|, triangles DAB and CBA are congruent by the Angle-Angle criterion.
- Consequently, triangle ABO is isosceles since two of its angles are equal, specifically |OAB| = |OBA|.