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?

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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