b This is the same thing as a reflection in the x-axis.
A
a Parallelogram.
Explanation: See solution.
B
b P'(1,-5)
Practice makes perfect
a Let's plot the points on graph paper and connect them.
This looks like a parallelogram which is a quadrilateral with two pairs of parallel sides. To determine if they are parallel, we can substitute the segments into the Slope Formula and simplify.
Segment
Points
y_2-y_1/x_2-x_1
m
MN
M(-3,6), N(2,8)
8- 6/2-( - 3)
2/5
QP
Q(- 4,3), P(1,5)
5- 3/1-( - 4)
2/5
MQ
Q(- 4,3), M(- 3,6)
6- 3/- 3-( - 4)
3
NP
P(1,5), N(2,8)
8- 5/2- 1
3
As we can see, opposite sides are parallel which means this is in fact a parallelogram.
A parallelogram can also be a rhombus if all sides are congruent. Therefore, let's calculate the side lengths using the Distance Formula.
Segment
Points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
d
MN
N(2,8), M(- 3,6)
sqrt(( 2-( - 3))^2+( 8- 6)^2)
sqrt(29)
QP
Q(- 4,3), P(1,5)
sqrt(( - 4- 1)^2+( 3- 5)^2)
sqrt(29)
MQ
M(- 3,6), Q(- 4,3)
sqrt(( - 3-( - 4))^2+( 6- 3)^2)
sqrt(10)
NP
P(1,5), N(2,8)
sqrt(( 1- 2)^2+( 5- 8)^2)
sqrt(10)
Since sqrt(29)≠sqrt(10), we know that MNPQ is in fact a parallelogram.
b The function x→ x, y→ - y tells us two things.
x → x: the $x-$coordinate is unchanged
y→ - y: the $y-$coordinate changes sign.
This is the same thing as a reflection in the x-axis.
