Question

quadrilateral cdef has vertices at c(-1, -6), d(-8, 5), e(3, 8), and f(4, -5). is cdef a trapezoid? justify your answer. yes, \\(\overline{cd}\\) is parallel to \\(\overline{ef}\\). yes, \\(\overline{cf}\\) is parallel to \\(\overline{de}\\). no, \\(\overline{cf}\\) is not parallel to \\(\overline{de}\\). no, none of the sides are parallel.

Explanation:

Step1: Recall slope formula

The slope \( m \) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \).

Step2: Calculate slope of \( \overline{CD} \)

For \( C(-1,-6) \) and \( D(-8,5) \),
\( m_{CD}=\frac{5 - (-6)}{-8 - (-1)}=\frac{11}{-7}=-\frac{11}{7} \).

Step3: Calculate slope of \( \overline{EF} \)

For \( E(3,8) \) and \( F(4,-5) \),
\( m_{EF}=\frac{-5 - 8}{4 - 3}=\frac{-13}{1}=-13 \). Wait, no, let's recalculate slopes for other sides. Wait, maybe I made a mistake. Let's check \( \overline{CF} \) and \( \overline{DE} \).

Step4: Calculate slope of \( \overline{CF} \)

For \( C(-1,-6) \) and \( F(4,-5) \),
\( m_{CF}=\frac{-5 - (-6)}{4 - (-1)}=\frac{1}{5} \).

Step5: Calculate slope of \( \overline{DE} \)

For \( D(-8,5) \) and \( E(3,8) \),
\( m_{DE}=\frac{8 - 5}{3 - (-8)}=\frac{3}{11} \). Wait, no, maybe the first pair was wrong. Wait, let's check \( \overline{CD} \) and \( \overline{EF} \) again. Wait, no, maybe the correct pair is \( \overline{CF} \) and \( \overline{DE} \)? Wait, no, let's recalculate \( \overline{CD} \): \( C(-1,-6) \), \( D(-8,5) \): \( y_2 - y_1 = 5 - (-6)=11 \), \( x_2 - x_1=-8 - (-1)=-7 \), so \( m_{CD}=-\frac{11}{7} \). \( \overline{EF} \): \( E(3,8) \), \( F(4,-5) \): \( y_2 - y_1=-5 - 8=-13 \), \( x_2 - x_1=4 - 3=1 \), so \( m_{EF}=-13 \). Not equal. Now \( \overline{CF} \): \( C(-1,-6) \), \( F(4,-5) \): \( m=\frac{-5 - (-6)}{4 - (-1)}=\frac{1}{5} \). \( \overline{DE} \): \( D(-8,5) \), \( E(3,8) \): \( m=\frac{8 - 5}{3 - (-8)}=\frac{3}{11} \). Not equal. Wait, maybe \( \overline{CD} \) and \( \overline{EF} \) was wrong. Wait, no, maybe I mixed up. Wait, let's check \( \overline{CE} \)? No, sides are \( CD, DE, EF, FC \). Wait, trapezoid has at least one pair of parallel sides (slopes equal). Let's recalculate \( \overline{CF} \) and \( \overline{DE} \) again. Wait, \( \overline{CF} \): \( C(-1,-6) \) to \( F(4,-5) \): \( \Delta y=-5 - (-6)=1 \), \( \Delta x=4 - (-1)=5 \), so slope \( 1/5 \). \( \overline{DE} \): \( D(-8,5) \) to \( E(3,8) \): \( \Delta y=8 - 5=3 \), \( \Delta x=3 - (-8)=11 \), slope \( 3/11 \). Not equal. Wait, \( \overline{CD} \): \( C(-1,-6) \) to \( D(-8,5) \): \( \Delta y=5 - (-6)=11 \), \( \Delta x=-8 - (-1)=-7 \), slope \( -11/7 \). \( \overline{EF} \): \( E(3,8) \) to \( F(4,-5) \): \( \Delta y=-5 - 8=-13 \), \( \Delta x=4 - 3=1 \), slope \( -13 \). Not equal. Wait, maybe \( \overline{DE} \) and \( \overline{CF} \) was miscalculated. Wait, no, maybe the correct answer is the second option? Wait, no, let's check again. Wait, maybe I made a mistake in coordinates. \( C(-1,-6) \), \( D(-8,5) \), \( E(3,8) \), \( F(4,-5) \). Let's calculate slope of \( \overline{CF} \): \( ( -5 - (-6) ) / (4 - (-1)) = (1)/5 = 1/5 \). Slope of \( \overline{DE} \): \( (8 - 5)/(3 - (-8)) = 3/11 \). Not equal. Wait, slope of \( \overline{CD} \): \( (5 - (-6))/(-8 - (-1)) = 11/(-7) = -11/7 \). Slope of \( \overline{EF} \): \( (-5 - 8)/(4 - 3) = -13/1 = -13 \). Not equal. Wait, maybe the problem is that the second option says \( \overline{CF} \) is parallel to \( \overline{DE} \), but their slopes are \( 1/5 \) and \( 3/11 \), which are not equal. Wait, no, maybe I messed up the sides. Wait, trapezoid is a quadrilateral with at least one pair of parallel sides. Let's check \( \overline{CD} \) and \( \overline{EF} \): slopes \( -11/7 \) and \( -13 \): not equal. \( \overline{CF} \) and \( \overline{DE} \): \( 1/5 \) and \( 3/11 \): not equal. \( \overline{CD} \) and \( \overline{DE} \): no. Wait, maybe the answer is the second option? Wait, no, maybe I made a mistake. Wait, let's recalculate slope of \( \overline{DE} \): \( D(-8,5) \), \( E(3,8) \): \( (8 - 5) = 3 \), \( (3 - (-8)) = 11 \), so 3/11. Slope of \( \overline{CF} \): \( C(-1,-6) \), \( F(4,-5) \): \( (-5 - (-6)) = 1 \), \( (4 - (-1)) = 5 \), so 1/5. Not equal. Wait, maybe the first option is wrong. Wait, the options are:

1. Yes, \( \overline{CD} \) is parallel to \( \overline{EF} \).

2. Yes, \( \overline{CF} \) is parallel to \( \overline{DE} \).

3. No, \( \overline{CF} \) is not parallel to \( \overline{DE} \).

4. No, none of the sides are parallel.

Wait, maybe I made a mistake in slope calculation. Let's check \( \overline{CF} \) and \( \overline{DE} \) again. Wait, \( \overline{CF} \): from \( C(-1,-6) \) to \( F(4,-5) \): change in y is \( -5 - (-6) = 1 \), change in x is \( 4 - (-1) = 5 \), so slope \( 1/5 = 0.2 \). \( \overline{DE} \): from \( D(-8,5) \) to \( E(3,8) \): change in y is \( 8 - 5 = 3 \), change in x is \( 3 - (-8) = 11 \), so slope \( 3/11 \approx 0.27 \). Not equal. \( \overline{CD} \): from \( C(-1,-6) \) to \( D(-8,5) \): change in y \( 5 - (-6) = 11 \), change in x \( -8 - (-1) = -7 \), slope \( -11/7 \approx -1.57 \). \( \overline{EF} \): from \( E(3,8) \) to \( F(4,-5) \): change in y \( -5 - 8 = -13 \), change in x \( 4 - 3 = 1 \), slope \( -13 \). Not equal. So none of the sides are parallel? But that's the fourth option. Wait, no, maybe I made a mistake. Wait, let's check slope of \( \overline{DE} \) again. \( D(-8,5) \), \( E(3,8) \): \( (8 - 5) = 3 \), \( (3 - (-8)) = 11 \), so 3/11. Slope of \( \overline{CF} \): 1/5. Not equal. Slope of \( \overline{CD} \): -11/7, slope of \( \overline{EF} \): -13. Not equal. So the correct answer is the fourth option? Wait, no, the second option says \( \overline{CF} \) is parallel to \( \overline{DE} \), but their slopes are not equal. Wait, maybe the problem has a typo, or I miscalculated. Wait, let's check \( \overline{CE} \)? No, sides are CD, DE, EF, FC. Wait, maybe the answer is the second option? No, because slopes are not equal. Wait, maybe I messed up the direction of the points. Let's calculate slope of \( \overline{DE} \) as \( (5 - 8)/(-8 - 3) = (-3)/(-11) = 3/11 \), same as before. Slope of \( \overline{CF} \) as \( (-6 - (-5))/(-1 - 4) = (-1)/(-5) = 1/5 \), same as before. So they are not equal. So the correct answer is the second option? No, that can't be. Wait, maybe the question is about \( \overline{CF} \) and \( \overline{DE} \) being parallel, but their slopes are 1/5 and 3/11, which are not equal. So the correct answer is the third option? Wait, no, the third option says "No, \( \overline{CF} \) is not parallel to \( \overline{DE} \)". Wait, but if none of the sides are parallel, then the fourth option is correct. Wait, I'm confused. Let's recalculate all slopes:

- \( \overline{CD} \): \( C(-1,-6) \) to \( D(-8,5) \): \( m = (5 - (-6))/(-8 - (-1)) = 11/(-7) = -11/7 \)
- \( \overline{DE} \): \( D(-8,5) \) to \( E(3,8) \): \( m = (8 - 5)/(3 - (-8)) = 3/11 \)
- \( \overline{EF} \): \( E(3,8) \) to \( F(4,-5) \): \( m = (-5 - 8)/(4 - 3) = -13/1 = -13 \)
- \( \overline{FC} \): \( F(4,-5) \) to \( C(-1,-6) \): \( m = (-6 - (-5))/(-1 - 4) = (-1)/(-5) = 1/5 \) (same as \( \overline{CF} \))

Now, check for parallel sides (equal slopes):

- \( \overline{CD} \) and \( \overline{EF} \): \( -11/7 \approx -1.57 \) vs \( -13 \): not equal.
- \( \overline{CD} \) and \( \overline{DE} \): \( -11/7 \) vs \( 3/11 \): not equal.
- \( \overline{CD} \) and \( \overline{FC} \): \( -11/7 \) vs \( 1/5 \): not equal.
- \( \overline{DE} \) and \( \overline{EF} \): \( 3/11 \) vs \( -13 \): not equal.
- \( \overline{DE} \) and \( \overline{FC} \): \( 3/11 \approx 0.27 \) vs \( 1/5 = 0.2 \): not equal.
- \( \overline{EF} \) and \( \overline{FC} \): \( -13 \) vs \( 1/5 \): not equal.

So none of the sides are parallel, so the correct answer is the fourth option: "No, none of the sides are parallel." Wait, but the options are:

1. Yes, \( \overline{CD} \) is parallel to \( \overline{EF} \).

2. Yes, \( \overline{CF} \) is parallel to \( \overline{DE} \).

3. No, \( \overline{CF} \) is not parallel to \( \overline{DE} \).

4. No, none of the sides are parallel.

So the correct answer is the fourth option. Wait, but maybe I made a mistake. Wait, let's check the slope of \( \overline{DE} \) again. \( D(-8,5) \), \( E(3,8) \): \( (8 - 5) = 3 \), \( (3 - (-8)) = 11 \), so 3/11. Slope of \( \overline{CF} \): \( ( -5 - (-6) ) / (4 - (-1)) = 1/5 \). Not equal. So the third option says "No, \( \overline{CF} \) is not parallel to \( \overline{DE} \)", which is true, but does that mean the trapezoid is not a trapezoid? Wait, a trapezoid needs at least one pair of parallel sides. If \( \overline{CF} \) is not parallel to \( \overline{DE} \), but maybe another pair is parallel. But we saw that \( \overline{CD} \) and \( \overline{EF} \) are not parallel, \( \overline{DE} \) and \( \overline{CF} \) are not parallel, so none of the sides are parallel, so the fourth option is correct.

Answer:

D. No, none of the sides are parallel.