Explore whether the diagonals have equal length and if they are perpendicular.
Rectangle, see solution.
Practice makes perfect
It is given that one diagonal of a parallelogram has endpoints at P(- 2,5) and R(1,- 1), while the other diagonal has endpoints at Q(1,5) and S(- 2,- 1). Let's determine and then compare their lengths using the Distance Formula.
d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)
We will substitute the given coordinates of the points P, R, Q, and S into the formula and find PR and QS.
Coordinates
P( - 2, 5) and R( 1, - 1)
Q( 1, 5) and S( - 2, - 1)
Substitute values
PR=sqrt(( 1-( - 2))^2+( - 1- 5)^2)
QS=sqrt(( - 2- 1)^2+( - 1- 5)^2)
Subtract terms
PR=sqrt((3)^2+(- 6)^2)
QS=sqrt((- 3)^2+(- 6)^2)
Calculate power
PR=sqrt(9+36)
QS=sqrt(9+36)
Add terms
PR=sqrt(45)
QS=sqrt(45)
As we can see, the length of the diagonals are both equal to sqrt(45). This implies that the parallelogram is a rectangle, which always has two diagonals of equal length. Next, we need to check whether the diagonals are perpendicular. We will use the Slope Formula.
m=y_2-y_1/x_2-x_1
Let's substitute the coordinates of the points and calculate m_(PR) and m_(QS).
Coordinates
P( - 2, 5) and R( 1, - 1)
Q( 1, 5) and S( - 2, - 1)
Substitute values
m_(PR)=- 1- 5/1-( - 2)
m_(QS)=- 1- 5/( - 2)- 1
Subtract terms
m_(PR)=- 6/3
m_(QS)=- 6/- 3
Calculate quotient
m_(PR)=- 2
m_(QS)=2
The diagonals are perpendicular if the product of their slopes is - 1, meaning that the slopes are negative reciprocals. Let's see if this is the case for PR and QS.
m_(PR)* m_(QS)=- 2* 2=- 4 *
We conclude that the diagonals are not perpendicular. Thus, the parallelogram is not a square and is a rectangle. Let's use the given coordinates to graph PQRS on a coordinate plane.
