Find the type of the quadrilateral if points A(-4, -2), B(-3, -7) C(3, -2) and D(2, 3) are joined serially
18«. Find the type of the quadrilateral if points A(-4, -2), B(-3, -7) C(3, -2) and D(2, 3) are joined serially
Solution
The given points are A(–4, –2), B(–3, –7) C(3, –2) and D(2, 3).
If they are joined serially so,
Slope of AB = [frac{- 7 + 2}{- 3 + 4} = – 5]
Slope of BC = [frac{- 2 + 7}{3 + 3} = frac{5}{6}]
Slope of CD =[frac{3 + 2}{2 – 3} = – 5]
Slope of AD = [frac{3 + 2}{2 + 4} = frac{5}{6}]
Opposite sides are parallel.
AC = [sqrt{left( 3 + 4 right)^2 + left( – 2 + 2 right)^2} = sqrt{49} = 7]
BD = [sqrt{left( 3 + 7 right)^2 + left( 2 + 3 right)^2} = sqrt{125} = 5sqrt{5}]
Diagonals are not equal.
Hence, the given points form a parallelogram.
Explanation:-
The given points are A(-4, -2), B(-3, -7), C(3, -2) and D(2, 3). We need to determine whether these points form a parallelogram or not.
To do this, we will first calculate the slopes of all four sides of the figure:
- Slope of AB = (-7 – (-2))/(-3 – (-4)) = -5
- Slope of BC = (-2 – (-7))/(3 – (-3)) = 5/6
- Slope of CD = (3 – (-2))/(2 – 3) = -5
- Slope of AD = (3 – (-2))/(2 – (-4)) = 5/6
We can observe that the opposite sides AB and CD have the same slope (-5), as do the opposite sides BC and AD (5/6). This means that the opposite sides are parallel.
Next, we will calculate the lengths of both diagonals AC and BD:
- Length of AC = sqrt((3 – (-4))^2 + (-2 – (-2))^2) = sqrt(49) = 7
- Length of BD = sqrt((3 – (-3))^2 + (2 – 3)^2) = sqrt(25 + 1) = 5sqrt(5)
We can observe that the two diagonals are not equal in length. Since a parallelogram has opposite sides parallel and equal diagonals, we can conclude that the given points do not form a parallelogram.
Chapter 5. Co-ordinate Geometry – Problem set 5 (Page 122)