Question

what is the slope of a line through points (1,4) and (7,8)?
a 6
b 4
c 1
d 4
what characteristic does a line have if its slope is zero?
a the line is decreasing
b the line is increasing
c the line is vertical
d the line is horizontal
what feature do parallel lines share when graphed in a coordinate plane?
a equal lengths
b equal slopes
c intersecting points
d equal distances
for the points a(-1,1), b(1,5) and c(8,1), which formula would you use to prove that triangle abc is isosceles?
a midpoint formula
b slope formula
c area formula
d distance formula
what is the length of a line - segment between points (2, - 2) and (-4,6)?
a 10.6 units
b 12 units
c 6.1 units
d 4 units
how can you determine if two line - segments are parallel using their coordinates?
a by determining their midpoints
b by checking if they intersect
c by showing their slopes are equal
d by measuring their lengths
what characteristic of a slope indicates a decreasing line?
a zero slope
b negative slope

Explanation:

Step1: Recall slope formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $(1,4)$ and $(7,8)$, $x_1 = 1,y_1=4,x_2 = 7,y_2 = 8$.
$m=\frac{8 - 4}{7 - 1}=\frac{4}{6}=\frac{2}{3}$. But this is not in the options. Let's answer the multiple - choice questions one by one.

1. For the question "What is the slope of a line through points $(1,4)$ and $(7,8)$?":

• Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{8 - 4}{7 - 1}=\frac{4}{6}=\frac{2}{3}$ (not in options, might be a mis - type in question setup).

1. For "What characteristic does a line have if its slope is positive?":

• A positive - slope line rises from left to right, which means the line is increasing. So the answer is b. The line is increasing.

1. For "What feature do parallel lines share when graphed in a coordinate plane?":

• Parallel lines have equal slopes. So the answer is b. Equal slopes.

1. For "For the points $A(-1,1),B(1,5)$ and $C(8,1)$, which formula would you use to prove that triangle $ABC$ is isosceles?":

• An isosceles triangle has two equal sides. We use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ to find the lengths of the sides of the triangle. So the answer is d. Distance formula.

1. For "What is the length of a line segment between points $(2,-2)$ and $(-4,6)$?":

• Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, where $x_1 = 2,y_1=-2,x_2=-4,y_2 = 6$.
• $d=\sqrt{(-4 - 2)^2+(6+2)^2}=\sqrt{(-6)^2+8^2}=\sqrt{36 + 64}=\sqrt{100}=10$. But this is not in the options.

1. For "How can you determine if two line segments are parallel using their coordinates?":

• We can determine if two line segments are parallel by showing that their slopes are equal. So the answer is c. By showing their slopes are equal.

1. For "What characteristic of a slope indicates a decreasing line?":

• A negative slope indicates a line that is decreasing (falls from left to right). So the answer is b. Negative slope.

Answer:

1. (No correct option based on calculation)
2. b. The line is increasing
3. b. Equal slopes
4. d. Distance formula
5. (No correct option based on calculation)
6. c. By showing their slopes are equal
7. b. Negative slope