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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 that the diagonals bisect each other
Answer
- Draw the diagonals WY and XZ on the quadrilateral WXYZ.
- Mark the intersection point of the diagonals as point O.
- Use a compass and straightedge to show that line segment WO is equal to line segment OY and line segment XO is equal to line segment OZ. Indicate these equal lengths on the diagram.
- Therefore, conclude that the diagonals bisect each other since WO = OY and XO = OZ.
Step 2
Prove that the triangle ABO is isosceles
Answer
- In parallelogram ABCD, since opposite angles are equal, we have |DAB| = |ABC|.
- Since OA and OB are opposite sides of the diagonals and they bisect each other at O, triangles DAB and CBA share side AB, which is common.
- Therefore, using the Side-Angle-Side (SAS) congruence criterion, triangles DAB and CBA are congruent. Thus, |DAB| = |CBA|.
- Since angles DAB and CBA are equal, triangle ABO is isosceles with OA = OB.