m=y_2-y_1/x_2-x_1
In this formula, (x_1,y_1) and (x_2,y_2) are points on the line or segment. Using this formula, we can evaluate the slope of each side of a quadrilateral.
Let's start with finding the slope of AB. To do this, we will substitute ( -3, 2) and ( 1,3) into the above formula.
We will find the slopes of the rest of sides in the same way. Let's recall that if the lines or segments are parallel, they have the same slope.
Side
Points
Slope Formula
Slope
AB
( -3, 2) & ( 1,3)
3- 2/1-( -3)
m_(AB)=1/4
BC
( 1, 3) & ( 5, )
- 3/5- 1
m_(BC)=3/4
CD
( 5, 0) & ( 5,-4)
-4- 0/5- 5
Undefined
DA
( 5, -4) & ( -3,2)
2-( -4)/-3- 5
m_(DA)=3/4
Since m_(AB)=m_(DA), segments AB and DA are parallel. This means that this quadrilateral is a trapezoid, as it has one pair of opposite parallel sides. To determine if this trapezoid is isosceles, we need to check if its legs are congruent. We can use the Distance Formula.
MN=sqrt((x_N-x_M)^2+(y_N-y_M)^2)
Let's evaluate this formula for AB and CD.
Segment
Points
Distance Formula
Length
AB
( -3, 2) & ( 1,3)
sqrt(( 1-( -3))^2+(3- 2)^2)
AB=sqrt(17)
CD
( 5, 0) & ( 5,-4)
sqrt(( 5- 5)^2+(-4- 0)^2)
CD=4
Since legs of this trapezoid have different measures, trapezoid ABCD is not isosceles.
b According to the Trapezoid Midsegment Theorem, the midsegment is parallel to both bases of a trapezoid. As we recalled in Part A, parallel segments have the same slope. Let's take a look at the given equation.
y=- x+1 ⇒ y= -1x+1
In the given equation, the slope is -1 and it is not the same as the slope of the bases, which is 34 (we found this in the previous part). Therefore, the midsegment is not contained in the line with equation y=- x+1.
c We will begin with recalling the Trapezoid Midsegment Theorem. This theorem tells us that if x is the midsegment of a trapezoid, then this midsegment is parallel to both bases of this figure, and x is the mean of the lengths of the bases of the trapezoid. Let's call the bases a and b.
x=1/2(a+b)
To find the length of the midsegment in ABCD, we need to start with evaluating the lengths of the bases in this trapezoid. To do this, let's use the Distance Formula we recalled in Part A.
Segment
Points
Distance Formula
Length
BC
( 1, 3) & ( 5, )
sqrt(( 5- 1)^2+( - 3)^2)
BC=5
AD
( -3, 2) & ( 5,-4)
sqrt(( 5-( -3))^2+(-4- 2)^2)
AD=10
Now, as we know the lengths of the bases, we can evaluate the length of the midsegment of this trapezoid. Let's substitute 5 for a and 10 for b in the formula.
