Question

1. proportional or not? a car travels according to the table below: gallons (x) miles (y) 1 21 2 36 3 56 4 76 what is the ratio for each of these points? is this relationship proportional or not proportional? 9. plotting points plot and label the point (3, 7) using the plot to the right. which quadrant is it in? 10. plotting points plot and label the point (-5, -1) using the plot to the right. which quadrant is it in? 11. the vertices for a triangle are: a(-4,6), b(6,2), and c(-4,-2). the triangle will be translated 5 units up and 4 units left. a. what are the coordinates of a, b, and c? b. how does the length of bc compare to the length of line bc? c. how does the area of triangle abc compare to the area of triangle abc?

Explanation:

Step1: Calculate ratios for proportionality

For $x = 1,y = 21$, ratio is $\frac{y}{x}=\frac{21}{1}=21$. For $x = 2,y = 36$, ratio is $\frac{y}{x}=\frac{36}{2}=18$. For $x = 3,y = 56$, ratio is $\frac{y}{x}=\frac{56}{3}\approx18.67$. For $x = 4,y = 76$, ratio is $\frac{y}{x}=\frac{76}{4}=19$. Since the ratios are not equal, the relationship is not proportional.

Step2: Plot point (3,7)

The $x -$coordinate is 3 and $y -$coordinate is 7. Start at the origin (0,0), move 3 units to the right along the $x -$axis and 7 units up along the $y -$axis. Since both $x>0$ and $y>0$, the point (3,7) is in the first quadrant.

Step3: Plot point (-5,-1)

The $x -$coordinate is - 5 and $y -$coordinate is - 1. Start at the origin (0,0), move 5 units to the left along the $x -$axis and 1 unit down along the $y -$axis. Since both $x<0$ and $y<0$, the point (-5,-1) is in the third quadrant.

Step4: Translate triangle vertices

For a translation 5 units up and 4 units left, the rule is $(x,y)\to(x - 4,y + 5)$.
For point A(-4,6): $A'=(-4-4,6 + 5)=(-8,11)$.
For point B(6,2): $B'=(6-4,2 + 5)=(2,7)$.
For point C(-4,-2): $C'=(-4-4,-2 + 5)=(-8,3)$.

Step5: Compare lengths of BC and B'C'

Translation is a rigid - motion. Rigid motions preserve distance. So the length of $B'C'$ is equal to the length of BC.

Step6: Compare areas of ABC and A'B'C'

Translation is a rigid - motion. Rigid motions preserve area. So the area of triangle $A'B'C'$ is equal to the area of triangle ABC.

Answer:

• The relationship in the table is not proportional.
• The point (3,7) is in the first quadrant.
• The point (-5,-1) is in the third quadrant.
• $A'(-8,11),B'(2,7),C'(-8,3)$
• The length of $B'C'$ is equal to the length of BC.
• The area of triangle $A'B'C'$ is equal to the area of triangle ABC.