Please help!!!!!
The points A,B,C,D have coordinates (3,3), (8,0), (-1,1), (-6,4) respectively.
a) Find the gradients of the lines AB and CD
b) Show that ABCD is a parallelogram
c) Find the coordinates of the point of intersection of the diagnols AC and BD.
Please help me by showing all methods of working out answer.
I assume that by gradient you mean slope, m
m AB = (0-3)/(8-3) = -3/5
m CD = (4-1)/(-6+1) = -35
so AB and CD are parallel (same slope)
if DA and CB are also parallel, it is a parallelogram
m DA = (4-3)/(-6-3) = -1/9
m CB = (0-1)/(8+1) = -1/9
sure enough, parallelogram
where is the middle?
halfway between A and C (or D and B)
do A and C
average x = (-1 + 3)/2 = 1
average y = (1+3)/2 = 2
so (1,2)
a) To find the gradient of a line, you need to use the formula:
Gradient = (change in y-coordinates) / (change in x-coordinates)
For line AB:
Coordinates of A: (3, 3)
Coordinates of B: (8, 0)
Change in y-coordinates: 0 - 3 = -3
Change in x-coordinates: 8 - 3 = 5
Gradient of AB = (-3) / 5 = -0.6
For line CD:
Coordinates of C: (-1, 1)
Coordinates of D: (-6, 4)
Change in y-coordinates: 4 - 1 = 3
Change in x-coordinates: -6 - (-1) = -5
Gradient of CD = 3 / -5 = -0.6
b) To show that ABCD is a parallelogram, we need to check if the opposite sides are parallel.
The gradient of AB is -0.6, and the gradient of CD is also -0.6. Since both gradients are equal, we can conclude that AB and CD are parallel.
Next, let's check if the opposite sides AC and BD are also parallel. To do this, we need to find the gradients of AC and BD:
For line AC:
Coordinates of A: (3, 3)
Coordinates of C: (-1, 1)
Change in y-coordinates: 1 - 3 = -2
Change in x-coordinates: -1 - 3 = -4
Gradient of AC = (-2) / (-4) = 0.5
For line BD:
Coordinates of B: (8, 0)
Coordinates of D: (-6, 4)
Change in y-coordinates: 4 - 0 = 4
Change in x-coordinates: -6 - 8 = -14
Gradient of BD = 4 / (-14) ≈ -0.286
Since the gradient of AC is 0.5 and the gradient of BD is approximately -0.286, we can see that the opposite sides AC and BD are not parallel.
Therefore, ABCD is not a parallelogram.
c) To find the coordinates of the point of intersection of the diagonals AC and BD, we can set the equations of the two lines equal to each other and solve for the point of intersection.
The equation of line AC:
y = mx + c
Substituting the coordinates of A (3, 3):
3 = 0.5(3) + c
3 = 1.5 + c
c = 3 - 1.5 = 1.5
So, the equation of line AC is y = 0.5x + 1.5.
The equation of line BD:
y = mx + c
Substituting the coordinates of B (8, 0):
0 = -0.286(8) + c
0 = -2.288 + c
c = 2.288
So, the equation of line BD is y = -0.286x + 2.288.
Now, we can set these equations equal to each other:
0.5x + 1.5 = -0.286x + 2.288
Simplifying the equation:
0.5x + 0.286x = 2.288 - 1.5
0.786x = 0.788
x = 0.788 / 0.786 ≈ 1
Substituting the value of x into either equation (let's use the equation of line AC):
y = 0.5(1) + 1.5
y = 0.5 + 1.5
y = 2
Therefore, the point of intersection of the diagonals AC and BD is approximately (1, 2).