Question

Three vertices of parallelogram PRTV are P(-4,-4), R(-10,0), and V(5,-1). Find the coordinates of vertex T.

Answer 1

To find the coordinates of vertex T in parallelogram PRTV, we can use the property of parallelograms that states opposite sides are congruent and parallel. By finding the vector PR and adding it to the coordinates of vertex V, we can determine that T is located at (-1,3).

Step-by-step explanation:

The coordinates of vertex T can be found by using the property of parallelograms which states that opposite sides are congruent and parallel. We can use the coordinates of two of the vertices to find the vector that represents one of the sides, and then add that vector to the coordinates of the third vertex to find the coordinates of T.

Let's find the vector PR first. PR = R - P = (-10,0) - (-4,-4) = (-10+4,0-(-4)) = (-6,4).

Now, to find the coordinates of T, we add the vector PR to the coordinates of V. T = V + PR = (5,-1) + (-6,4) = (5-6,-1+4) = (-1,3). Therefore, the coordinates of vertex T are (-1,3).

Answer 2

Slope of TV = Slope of PR
(y + 1)/(x - 5) = (0 + 4)/(-10 + 4)
(y + 1)/(x - 5) = -4/6
6(y + 1) = -4(x - 5)
6y + 6= -4x + 20
equation #1: 6y + 4x = 14

Slope of RT = Slope of PV
(y - 0)/(x + 10) = (-1 + 4)/(5 + 4)
(y - 0)/(x + 10) = 3/9
3(y - 0) = 9(x + 10)
3y = 9x +90
equation #2: 3y = 9x +90

We now we 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: 6y + 4x = 14
equation #2: 3y - 9x = 90

Multiply equation 2 by -2

equation #1: 6y + 4x = 14
equation #2: -6y + 18x = -180

Add together Eq. 1 & 2

22x = -166
x = -7.5

6y +4(-7.5) = 14
6y = 44
y = 7 1/3

(-7 1/2, 7 1/3)