c Yes. Two lines are perpendicular if the product of their slopes is - 1.
D
d A=50 units^2
Practice makes perfect
a Let's begin by drawing the quadrilateral in a diagram using the given points.
b We can find the perimeter by calculating the length of each of the four sides and adding them together. Let's calculate the length of the side EF using the Distance Formula. The endpoints of this side are E(- 3,6) and F(- 7,3).
We now know that EF=5. We can continue by calculating the lengths of the other three sides.
Side
FG
GH
HE
Measure
10
5
10
To find the perimeter, we calculate the sum of the lengths of the four sides
P&=EF+FG+GH+HE
&=5+10+5+10
&=30 units
c To tell if the quadrilateral is a rectangle, we need to find if the angles are right angles. Because the opposite sides have equal lengths, if one angle is 90^(∘), all of them are. Let's see if angle E is right by checking if the sides are perpendicular. For this, we need to know the slopes of the sides EF and EH. Let's find the slope of EF.
Two lines are perpendicular if their slopes are negative reciprocals. This is true if the product of the slopes is - 1.
m_(EF)* m_(EH)=3/4 * - 4/3=- 1
Therefore, these two sides of the quadrilateral are perpendicular and, by the reasoning explained above, so are the remaining sides.
d In Part C, we found that quadrilateral EFGH is a rectangle. Now, let's think about whether or not it's a square. A square must have all four sides equal. We found the following side lengths for our quadrilateral.
EF&=5
FG&=10
GH&=5
HE&=10
Because there are two different side lengths, we know that the shape is not a square. However, since the quadrilateral is still a rectangle, we can use the formula A=l w to calculate the area. This rectangle has the length l= 10 and the width w= 5.
A&=l* w
A&= 10 * 5
A&=50 units^2
