Question 1

Which of the following is the correct formula for the distance between two points $$A(x_1$, y_1$)$$ and $$B(x_2$, y_2$) in $$the coordinate plane?

Accepted Answer

$$\sqrt{(x_2$ - x_1$)^2$$ + ($$y_2$$ - $$y_1)^2$}$$

Question 2

Given the points $$A(2, 3)$$ and $$B(5, 7)$$, what is the distance between them?

Accepted Answer

$$\sqrt{(5-2)^2$ + (7-3)^2$} = \sqrt{9 + 16} = \sqrt{25} = 5$$

Question 3

What is the midpoint of the segment joining the points $$A(-4, 6)$$ and $$B(2, -2)$$?

Accepted Answer

$$(-1, 2)$$

Question 4

If the equation of a line is $$y = 2x + 3$$, what is the slope of the line?

Accepted Answer

$$2$$

Question 5

Which of the following points lies on the line $$y = -x + 4$$?

Accepted Answer

$$(2, 2)$$

Question 6

A city plans to build a straight road between two locations at $$(1, 2)$$ and $$(7, 8). $$What is the equation of the road in slope-intercept form?

Accepted Answer

Question 7

Given the points $$A(0, 0)$$, $$B(4, 0)$$, and $$C(4, 3)$$, what is the perimeter of triangle $$ABC$$?

Accepted Answer

$$12$$

Question 8

Which of the following is the correct formula for the slope $$m of a $$line passing through points $$(x_1$, y_1$)$$ and $$(x_2$, y_2$)$$?

Accepted Answer

$$m = \frac{y_2$ - y_1$}{x_2$ - x_1$}$$

Question 9

If a line passes through $$(2, 5)$$ and has a slope of $$-3$$, what is its equation in point-slope form?

Accepted Answer

$$y - 5 = -3(x - 2)$$

Question 10

A biologist tracks the migration of birds using the coordinates $$(3, 7)$$ and $$(6, 1). $$What is the slope of the path?

Accepted Answer

$$\frac{1 - 7}{6 - 3} = \frac{-6}{3} = -2$$

Question 11

Which of the following equations represents a line parallel to $$y = 4x - 2$$?

Accepted Answer

$$y = 4x + 5$$

Question 12

A medical researcher plots the growth of bacteria at $$(0, 2)$$ and $$(5, 12). $$What is the equation of the line passing through these points in slope-intercept form?

Accepted Answer

$$y = 2x + 2$$

Question 13

Which of the following is the correct formula for the midpoint $$M$$ between $$(x_1$, y_1$)$$ and $$(x_2$, y_2$)$$?

Accepted Answer

$$M = \left( \frac{x_1$ + x_2$}{2}, \frac{y_1$ + y_2$}{2} \right)$$

Question 14

A line passes through $$(1, 2)$$ and $$(4, 8). $$What is its slope?

Accepted Answer

$$\frac{8 - 2}{4 - 1} = \frac{6}{3} = 2$$

Question 15

Which of the following lines is perpendicular to $$y = \frac{1}{2}x + 3$$?

Accepted Answer

$$y = -2x + 1$$

Question 16

A chemist plots the temperature change over time and finds the line $$y = -5x + 20. $$What does the slope represent in this context?

Accepted Answer

The temperature decreases by 5 units per time unit.

Question 17

If the distance between two points is $$10$$ and their coordinates are $$(x, 3)$$ and $$(7, y)$$, which equation must be true?

Accepted Answer

$$\sqrt{(x - 7)^2$ + (3 - y)^2$} = 10$$

Question 18

A historian plots the locations of two ancient cities at $$(2, 5)$$ and $$(8, 11). $$What is the midpoint between the cities?

Accepted Answer

$$(5, 8)$$

Question 19

Which of the following is the correct equation for a line with slope $$m$$ passing through $$(x_1$, y_1$)$$?

Accepted Answer

$$y - y_1$ = m(x - x_1$)$$

Graphs and Equations of Circles in the Rectangular Coordinate System

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