Question

If (2,1), (3,4) & (0,1) are three vertices taken in order of a parallelogram. Find the fourth vertex. ​

Answer 1

Given,

The three vertices of the parallelogram are (2,1) , (3,4) and (0,1).

To find,

The fourth vertex of the parallelogram.

Solution,

Let,the four vertices of the parallelogram = A,B,C,D

Now,

A = (2,1)  B = (3,4)  C = (0,1)

Let,the D = (x,y)

[Assume,x and y as variables to do the further mathematical calculations.]

Now, according to the geometry,the two diagonals of a parallelogram bisects each other. So, midpoint of AC diagonal and midpoint BD diagonal will be the same point.

The midpoint of AC = (2+0)/2 , (1+1)/2 = (1,1)

The midpoint of BD = (3+x)/2 , (4+y)/2

Now as they are same points,

(3+x)/2 , (4+y)/2 = (1,1)

By complying we get that,

(3+x)/2 = 1

3+x = 2

x = -1

And,

(4+y)/2 = 1

4+y = 2

y = -2

Hence, the fourth vertex is (-1,-2).

Answer 2

Answer:

(-1,2)

Step-by-step explanation:

Let the vertices be as follows,

A(2,1) ; B(3,4) ; C(0,1); D(x, y)

The line segments that make up our parallelogram are AD is parallel to BC, and CD is parallel to AB.  Since the definition of a parallelogram is a quadrilateral with 2 pairs of parallel sides, we can use this fact to find the coordinate to the 4 vertexes.  A pair of parallel sides also means that these 2 lines have the same slope.  So if we do this on both pairs of parallel lines, we should have 2 equations with 2 variables, which should be simple to solve.

A(2,1) ; B(3,4) ; C(0,1); D(x, y)

The slope of CD =  Slope of AB

(y - 1)/(x - 0) = (4 - 1)/(3 - 2)

(y - 1)/(x) = 3/1

(y - 1) = 3x

-1 = 3x - y

equation #1:  3x - y = -1

A(2,1) ; B(3,4) ; C(0,1); D(x, y)

The slope of AD = Slope of BC

(y - 1)/(x - 2) = (1 - 4)/(0 - 3)

(y - 1)/(x - 2) = -3/-3

(y - 1)/(x - 2) = 1/1

y-1=x-2

equation #2: x - y = 1

We now have 2 equations with 2 variables, and we can use any method (substitution, linear combination, matrix) to solve for the x and y coordinates.  

equation #1:  3x - y = -1

equation #2:  x - y = 1

Subtract equation 2 from equation 1

equation #1:  3x - y = -1

equation #2:   x - y = 1

eq 1 - eq 2:   2x = -2

x = -1

Using equation#2 to solve for y:

x - y = 1

-1 + y = 1

y=2

So the 4th vertex will be (-1,2)

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