The three vertices of a parallelogram abcd are A (3,-4) B(-1,-3) and C(-6,2) . Find the coordinates of vertex D?

Question

Answer

Answer

Answer

A(3,-4) B(-1,-3) C(-6,2) D(x,y)=?

diagonals of a ||gram bisect each other

let the diagonals of the ||gram intersect each other at o

such that AO=OC and BO=OD

------> they are divided in the ratio of 1:1

.'. by special case of section formula

O(x,y) =( 3+(-6)/2 , -4+2/2 )

=( 3-6/2 , -2/2 )

=( -3/2 , -1)

-----> x=-3/2 , y=-1

also.

O(x,y) =(-1+x/2 , -3+y/2)

O(-3/2,-1)=(-1+x/2 , -3+y/2)

compairing abcissa and ordinate on both sides,

-3/2=-1+x/2 -1=-3+y/2

-3/2*2=-1+x -1*2=-3+y

-3=-1+x -2=-3+y

x=-3+1 y=3-2

x=-2 y=1

.'. coordinates of D are (-2,1)

Answer

Given Information:
The three vertices of a parallelogram ABCD are A(3,-4), B(-1,-3), and C(-6,2).

Goal:
Find the coordinates of vertex D.

Approach:
To find the coordinates of vertex D, we will use the properties of a parallelogram. A parallelogram is a quadrilateral with opposite sides that are parallel and equal in length.

Step 1: Find the length and slope of AB.
Using the distance formula, we can find the length of AB:
Length of AB = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Substituting the given coordinates, we get:
Length of AB = sqrt((-1 - 3)^2 + (-3 - (-4))^2)
= sqrt((-4)^2 + (1)^2)
= sqrt(16 + 1)
= sqrt(17)

To find the slope of AB, we can use the formula:
Slope of AB = (y2 - y1) / (x2 - x1)

Substituting the given coordinates, we get:
Slope of AB = (-3 - (-4)) / (-1 - 3)
= (-3 + 4) / (-1 - 3)
= 1 / -4
= -1/4

Step 2: Find the coordinates of point D.
Since AB and CD are parallel, they must have the same slope. Therefore, the slope of CD is also -1/4.

We know that D is the opposite vertex of A. So, we can use the slope-intercept form of a line to find the equation of line CD:
y = mx + b

Using the slope (-1/4) and one of the known points (C: -6, 2), we can substitute the values to find the y-intercept (b):
2 = (-1/4)(-6) + b
2 = 3/2 + b
b = 2 - 3/2
b = 1/2

Therefore, the equation of line CD is:
y = (-1/4)x + 1/2

Now, we can substitute the x-coordinate of A (3) into the equation to find the y-coordinate of D:
y = (-1/4)(3) + 1/2
y = -3/4 + 1/2
y = -3/4 + 2/4
y = -1/4

Thus, the coordinates of vertex D are (3, -1).

Final Answer:
The coordinates of vertex D are (3, -1).

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