To determine if opposite sides are parallel, we can substitute the segment's endpoints in the Slope Formula and simplify. If opposite sides have the same slope, they are parallel.
segment
points
y_2-y_1/x_2-x_1
m
AB
A(1,4), B(6,6)
6- 4/6- 1
2/5
DC
D(- 1,- 1), C(4,1)
1-( - 1)/4-( - 1)
2/5
DA
D(- 1,- 1), A(1,4)
4-( - 1)/1-( - 1)
5/2
CB
C(4,1), B(6,6)
6- 1/6- 4
5/2
As we can see, opposite sides are parallel which means we can say that this is a parallelogram.
To investigate if it is a rhombus, we have to calculate the side's lengths using the Distance Formula.
segment
points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
d
AB
A(1,4), B(6,6)
sqrt(( 1- 6)^2+( 4- 6)^2)
sqrt(29)
DC
D(- 1,- 1), C(4,1)
sqrt(( - 1- 4)^2+( - 1- 1)^2)
sqrt(29)
DA
D(- 1,- 1), A(1,4)
sqrt(( - 1- 1)^2+( - 1- 4)^2)
sqrt(29)
CB
C(4,1), B(6,6)
sqrt(( 4- 6)^2+( 1- 6)^2)
sqrt(29)
Since all sides have the same length, this is in fact a rhombus.
b Let's first draw the diagonals DB and CA.
Both of these lines are straight lines which means we can write them on Slope-Intercept form.
y=mx+b
In this equation, m is the line's slope and b is the y-intercept. The slope, we can find by using the Slope Formula.
Segment
Points
y_2-y_1/x_2-x_1
m
DB
D(- 1,- 1), B(6,6)
6-( - 1)/6-( - 1)
1
CA
C(4,1), A(1,4)
4- 1/1- 4
- 1
With this information, we have half of what we need to write the equations.
DB:& y= x+b
CA:& y= - x+b
Finally, we must find the y-intercept by substituting any of the points on the lines into the equations and solving for b. For example, we can substitute B(6,6) in the equation for DB and C(4,1) in the equation for CA.
Segment
y=mx+b
Substitute point
Solve for b
DB
y= x+b
6= 6+b
b=0
CA
y= - x+b
1= - 4+b
b=5
Now we can complete the equations.
DB:& y= x
CA:& y= - x+5
c If the lines are perpendicular, the product of their slopes equals - 1.
m_1m_2=- 1
From Part B we know the slopes. By substituting these into this equation, we can determine if the lines are perpendicular.
