Coordinate Geometry: Distance, Slope, Parallel & Perpendicular Lines

Comprehensive flashcards covering warm-up problems, the distance formula, average rate of change, equations of parallel and perpendicular lines, and geometric figure classification.

How do you simplify the radical expression $\sqrt{18}$?

$$3\sqrt{2}$$

What is the slope of the line that passes through the points $(-2, 4)$ and $(4, -4)$?

$$-\frac{4}{3}$$

Ms. Gmiter records the total number of students scoring an A on the final exam after $x$ years: Year 1 has $10$, Year 2 has $25$, Year 3 has $62$, Year 4 has $80$, and Year 5 has $150$. What is the average rate of change from year 2 to year 4?

$27.5\text{ students per year}$ (calculated as $\frac{80 - 25}{4 - 2} = \frac{55}{2}$)

How do you simplify the radical expression $\sqrt{45}$?

$$3\sqrt{5}$$

What is the Distance Formula used to calculate the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$?

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

What is the distance between the ordered pairs $(8, -1)$ and $(2, -9)$?

$10$ (calculated as $\sqrt{(2 - 8)^2 + (-9 - (-1))^2} = \sqrt{(-6)^2 + (-8)^2} = \sqrt{100}$)

The total number of cars in a parking garage was recorded as: Feb 1 ($288$), Feb 8 ($274$), Feb 15 ($484$), and Feb 22 ($330$). What was the average rate of change in total cars per day between February 8 and February 22?

$4\,\text{cars per day}$ (calculated as $\frac{330 - 274}{22 - 8} = \frac{56}{14}$)

What is the slope of the line given by the equation $x - 3y = -12$?

$\frac{1}{3}$ (rearranged to slope-intercept form $y = \frac{1}{3}x + 4$)

Based on the provided graph of $f(x)$, what is the best estimate of the average rate of change of $f(x)$ over the interval $-60 \le x \le -20$?

$-2$ (calculated as $\frac{-40 - 40}{-20 - (-60)} = \frac{-80}{40}$)

What geometric properties define parallel lines on a coordinate plane?

Parallel lines are an equal distance from each other at all points, have the same slope, and will never intersect.

What is the slope of a line parallel to the line given by the equation $3y = 4x - 18$?

$\frac{4}{3}$ (dividing by $3$ gives $y = \frac{4}{3}x - 6$)

What is the equation of the line that is parallel to $y = -4x + 10$ and has a y-intercept of $(0, -8)$?

$$y = -4x - 8$$

What is the equation in slope-intercept form of the line parallel to $y = \frac{1}{4}x - 9$ that passes through the point $(8, 1)$?

$y = \frac{1}{4}x - 1$ (using $m = \frac{1}{4}$ and solving $1 = \frac{1}{4}(8) + b$)

What are the orientations and slopes of lines parallel to the x-axis and y-axis?

A line parallel to the x-axis is horizontal and has a slope of $0$. A line parallel to the y-axis is vertical and has an undefined slope.

Natasha says the line $x = 7$ is parallel to the x-axis and has a slope of zero. Do you agree or disagree, and why?

Disagree. The line $x = 7$ is a vertical line parallel to the y-axis, and its slope is undefined.

Line $p$ contains points $(-3, 2)$ and $(6, -1)$, while line $m$ contains points $(-6, 0)$ and $(-3, -1)$. Are lines $p$ and $m$ parallel?

Yes. Both lines have an identical slope of $-\frac{1}{3}$ (line $p$: $\frac{-1 - 2}{6 - (-3)} = -\frac{3}{9}$; line $m$: $\frac{-1 - 0}{-3 - (-6)} = -\frac{1}{3}$).

How are perpendicular lines defined, and how do you find the slope of a line perpendicular to a given line?

Perpendicular lines intersect to form a right angle ($90^\circ$). Their slopes are opposite reciprocals, found by flipping the fraction and changing the sign.

What is the slope of a line perpendicular to the equation $y = 17 - 10x$?

$\frac{1}{10}$ (the original slope is $-10$, so the opposite reciprocal is $\frac{1}{10}$)

What is the equation in slope-intercept form of a line perpendicular to $y = -\frac{1}{3}x - 7$ that passes through the point $(-2, 7)$?

$y = 3x + 13$ (the perpendicular slope is $3$, and solving $7 = 3(-2) + b$ yields $b = 13$)

What are the orientations and slopes of lines perpendicular to the x-axis and y-axis?

A line perpendicular to the x-axis is vertical and has an undefined slope. A line perpendicular to the y-axis is horizontal and has a slope of $0$.

What is the distance between the points $(-2, 5)$ and $(1, -1)$ in simplest radical form?

$3\sqrt{5}$ (calculated as $\sqrt{(1 - (-2))^2 + (-1 - 5)^2} = \sqrt{3^2 + (-6)^2} = \sqrt{45}$)

A gym membership costs $\$25$ a month with a one-time joining fee of $\$60$. If Jalen joins the gym for a year and a half, what will be his total cost?

$\$510$ (calculated as \60 + 18 \times \$25 = \$60 + \$450$)

In coordinate geometry proofs, how are congruent sides, parallel sides, and right angles determined?

Congruent sides are proven using the Distance Formula to show equal side lengths; parallel sides have the same slope; right angles are created by perpendicular lines with opposite reciprocal slopes.

Triangle RST has vertices $R(7, -4)$, $S(12, -12)$, and $T(2, -12)$. What type of triangle is it, and why?

It is an isosceles triangle because two sides are congruent: $RS = \sqrt{5^2 + (-8)^2} = \sqrt{89}$ and $RT = \sqrt{(-5)^2 + (-8)^2} = \sqrt{89}$, while $ST = 10$.

What is the area of the right triangle with vertices $A(2, 1)$, $B(7, 1)$, and $C(7, 4)$?

$7.5$ (base $AB = 5$, height $BC = 3$, and area $A = \frac{1}{2}bh = \frac{1}{2}(5)(3)$)

Three vertices of a rectangle are $(-2, 0)$, $(4, 2)$, and $(5, -1)$. What are the coordinates of the fourth vertex?

$$(-1, -3)$$

What type of figure is quadrilateral RSTU with vertices $R(-3, 4)$, $S(1, 6)$, $T(2, 3)$, and $U(-2, 1)$, and what are its side lengths?

It is a parallelogram. Its side lengths are $RS = TU = 2\sqrt{5}$ (or $\sqrt{20}$) and $ST = UR = \sqrt{10}$.

If the endpoints of a circle's diameter are $J(-5, 3)$ and $K(7, -3)$, what is the length of the diameter?

$6\sqrt{5}$ (or $\sqrt{180} \approx 13.4$, calculated as $\sqrt{(7 - (-5))^2 + (-3 - 3)^2} = \sqrt{12^2 + (-6)^2}$)