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Given the co-ordinates of the vertices of a quadrilateral ABCD, describe three different ways to determine, using co-ordinate geometry techniques, whether the quadrilateral is a parallelogram - Leaving Cert Mathematics - Question 1 - 2012
Question 1
Given the co-ordinates of the vertices of a quadrilateral ABCD, describe three different ways to determine, using co-ordinate geometry techniques, whether the quadri... show full transcript
Worked Solution & Example Answer:Given the co-ordinates of the vertices of a quadrilateral ABCD, describe three different ways to determine, using co-ordinate geometry techniques, whether the quadrilateral is a parallelogram - Leaving Cert Mathematics - Question 1 - 2012
Step 1
method 1: Check whether both pairs of opposite sides have the same slope (slope formula).
Answer
To determine if the quadrilateral is a parallelogram, calculate the slopes of the sides AB, BC, CD, and DA using the formula:
Calculate each:
- Slope of AB: ( m_{AB} = \frac{-2 - (-5)}{-4 - 21} = \frac{3}{-25} \
- Slope of CD: ( m_{CD} = \frac{10 - 7}{-17 - 8} = \frac{3}{-25}
Since ( m_{AB} = m_{CD} ), they are parallel.
Now calculate slopes of BC and DA:
- Slope of BC: ( m_{BC} = \frac{7 - (-5)}{8 - 21} = \frac{12}{-13} \
- Slope of DA: ( m_{DA} = \frac{-2 - 10}{-4 - (-17)} = \frac{-12}{13}
Since ( m_{BC} = m_{DA} ), they are also parallel.
Step 2
method 2: Check whether both pairs of opposite sides are equal in length (distance formula).
Answer
To determine if opposite sides are of equal length, use the distance formula:
Calculate each:
-
Length of AB: ( d_{AB} = \sqrt{(-4 - 21)^2 + (-2 - (-5))^2} = \sqrt{(-25)^2 + 3^2} = 25.09 \
-
Length of CD: ( d_{CD} = \sqrt{(-17 - 8)^2 + (10 - 7)^2} = \sqrt{(-25)^2 + 3^2} = 25.09
-
Length of BC: ( d_{BC} = \sqrt{(8 - 21)^2 + (7 - (-5))^2} = \sqrt{(-13)^2 + 12^2} = 17.48 \
-
Length of DA: ( d_{DA} = \sqrt{(-4 - (-17))^2 + (-2 - 10)^2} = \sqrt{(13)^2 + (-12)^2} = 17.48
Since ( d_{AB} = d_{CD} ) and ( d_{BC} = d_{DA} ), the quadrilateral is a parallelogram.
Step 3
method 3: Check whether the midpoints of the diagonals coincide (diagonals bisecting each other).
Answer
To check if the diagonals bisect each other, calculate the midpoints of the diagonals AC and BD:
Use the midpoint formula:
- Midpoint of AC: ( M_{AC} = \left( \frac{-4 + 8}{2}, \frac{-2 + 7}{2} \right) = \left( 2, 2.5 \right) \
- Midpoint of BD: ( M_{BD} = \left( \frac{21 + (-17)}{2}, \frac{-5 + 10}{2} \right) = \left( 2, 2.5 \right)
Since both midpoints are equal, the diagonals bisect each other, confirming that the quadrilateral is a parallelogram.
Step 4
Using method 1: Check whether both pairs of opposite sides have the same slope.
Answer
I have already calculated the slopes above, confirming that both pairs of opposite sides have equal slopes. Therefore, the quadrilateral with vertices (–4, –2), (21, –5), (8, 7) and (–17, 10) is indeed a parallelogram.