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Describe two different ways of showing, using co-ordinate geometry techniques, that the points form a parallelogram ABCD - Leaving Cert Mathematics - Question b & e - 2012
Question b & e
Describe two different ways of showing, using co-ordinate geometry techniques, that the points form a parallelogram ABCD. Any TWO of: - Show that opposite sides ar... show full transcript
Worked Solution & Example Answer:Describe two different ways of showing, using co-ordinate geometry techniques, that the points form a parallelogram ABCD - Leaving Cert Mathematics - Question b & e - 2012
Step 1
Show that opposite sides are parallel by showing the slopes of opposite lines are equal.
Answer
To find the slopes of sides AB and CD:
- Points A and B are A(6, -1) and B(12, -3).
- The slope of AB is calculated as follows:
[ \text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - (-1)}{12 - 6} = \frac{-2}{6} = -\frac{1}{3} ]
- Points C and D are C(8, 5) and D(2, 7).
- The slope of CD is:
[ \text{slope}_{CD} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 5}{2 - 8} = \frac{2}{-6} = -\frac{1}{3} ]
Since the slopes of AB and CD are both -\frac{1}{3}, the opposite sides AB and CD are parallel.
Step 2
Show that the diagonals bisect each other.
Answer
First, we find the midpoints of the diagonals AC and BD.
- The midpoint of AC is found using the formula:
[ \text{Midpoint}_{AC} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{6 + 8}{2}, \frac{-1 + 5}{2} \right) = (7, 2) ]
- For diagonal BD, we find:
[ \text{Midpoint}_{BD} = \left( \frac{12 + 2}{2}, \frac{-3 + 7}{2} \right) = \left( 7, 2 \right) ]
Since both midpoints of AC and BD are (7, 2), the diagonals bisect each other.