Coordinate Geometry Practice Questions — ACT — Page 13

Question 241Question →

In the standard $(x, y)$ coordinate plane, a line passes through the points $(3, -1)$ and $(-1, y)$ . If the slope of the line is $-2$ , what is the value of $y$ ?

1

-9

-7

7

9

Show answer & explanation

Answer: $7$

Answer

The correct value of

y

is

7

.

To find the value of

y

, apply the slope formula

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

. Substituting the coordinates

(3, -1)

and

(-1, y)

alongside the slope of

-2

yields the equation

-2 = \frac{y - (-1)}{-1 - 3}

. Simplifying the denominator gives

-2 = \frac{y + 1}{-4}

. Multiplying both sides of the equation by

-4

results in

8 = y + 1

. Finally, subtracting

1

from both sides gives

y = 7

.

Step-by-Step Solution

1

Set up the slope formula using the given points

(x_1, y_1) = (3, -1)

and

(x_2, y_2) = (-1, y)

.

The formula is

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

, which becomes

-2 = \frac{y - (-1)}{-1 - 3}

.

This establishes the algebraic relationship between the coordinates and the given slope.

2

Simplify the numerator and denominator of the fraction.

-2 = \frac{y + 1}{-4}

Subtracting a negative number is equivalent to addition, and

-1 - 3

simplifies to

-4

.

3

Solve for

y

by isolating the variable.

Multiply both sides by

-4

to get

8 = y + 1

, then subtract

1

from both sides to get

y = 7

.

This isolates the variable

y

to find its numerical value.

Key Concept

Calculating the slope of a line using the coordinates of two points on the line.

Estimated Time:1m 0s

Question 242Question →

In the standard $(x, y)$ coordinate plane, a parallelogram $ABCD$ has vertices $A(-6, -3)$ , $B(1, -3)$ , $C(6, 2)$ , and $D(-1, 2)$ . What is the length of the diagonal $AC$ ?

Show answer & explanation

Answer: 13

Answer

The correct answer is 13.

The length of the diagonal

AC

is found by calculating the distance between the vertices

A(-6, -3)

and

C(6, 2)

. Using the distance formula:

AC = \sqrt{(6 - (-6))^2 + (2 - (-3))^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

.

Step-by-Step Solution

1

Identify the coordinates of the endpoints of diagonal

AC

.

A(-6, -3)

and

C(6, 2)

The diagonal of a polygon connects non-adjacent vertices, so the diagonal

AC

connects vertex

A

to vertex

C

.

2

Use the distance formula to find the length of the segment

AC

.

AC = \sqrt{(6 - (-6))^2 + (2 - (-3))^2}

The distance formula calculates the straight-line distance between two points on the coordinate plane.

3

Simplify the expression to find the final integer length.

13

Calculating the squares and summing them gives

\sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

.

Key Concept

Calculating the length of a geometric segment on the coordinate plane using the distance formula.

Question 243Question →

A line graphed on the standard $(x, y)$ coordinate plane has a slope of $0.5$ and passes through the points $(p, 3)$ and $(7, 2p - 1)$ . What is the value of $p$ ?

Show answer & explanation

Answer: 3

Answer

The value of

p

is

3

.

Applying the slope formula

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

to the points

(p, 3)

and

(7, 2p - 1)

with

m = 0.5

yields the linear equation

0.5 = \frac{2p - 4}{7 - p}

. Cross-multiplying and solving for

p

gives

p = 3

.

Step-by-Step Solution

1

Set up the slope formula using the given points and slope.

0.5 = \frac{(2p - 1) - 3}{7 - p}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

.

2

Simplify the numerator of the fraction.

0.5 = \frac{2p - 4}{7 - p}

Combine the constant terms

-1

and

-3

in the numerator.

3

Multiply both sides by the denominator to clear the fraction.

0.5(7 - p) = 2p - 4

Eliminating the denominator allows solving the linear equation.

4

Distribute the

0.5

on the left side of the equation.

3.5 - 0.5p = 2p - 4

Multiply each term inside the parentheses by

0.5

.

5

Isolate the variable

p

by rearranging terms.

2.5p = 7.5

Add

0.5p

to both sides and add

4

to both sides.

6

Divide by the coefficient of

p

to solve for

p

.

p = 3

Dividing

7.5

by

2.5

yields the final value of the parameter.

Key Concept

Calculating a parameter using the slope formula

Alternative Method

Instead of using the decimal

0.5

, write it as the fraction

\frac{1}{2}

. The equation becomes

\frac{1}{2} = \frac{2p - 4}{7 - p}

. Cross-multiplying gives

1(7 - p) = 2(2p - 4) \Rightarrow 7 - p = 4p - 8 \Rightarrow 15 = 5p \Rightarrow p = 3

.

Estimated Time:1m 30s

Question 244Question →

A circle with center $(5, -6)$ is tangent to the $x$ -axis in the standard $(x, y)$ coordinate plane. Which of the following is the equation of this circle?

x^2 + y^2 + 10x - 12y + 25 = 0

x^2 + y^2 - 10x + 12y + 36 = 0

x^2 + y^2 - 10x + 12y + 25 = 0

x^2 + y^2 - 25 = 0

x^2 + y^2 - 10x + 12y + 55 = 0

Show answer & explanation

Answer: $x^2 + y^2 - 10x + 12y + 25 = 0$

Answer

x^2 + y^2 - 10x + 12y + 25 = 0

The correct equation

x^2 + y^2 - 10x + 12y + 25 = 0

is obtained by identifying that a circle with center

(5, -6)

tangent to the

x

-axis has a radius of

|-6| = 6

. Substituting these into the standard form

(x - h)^2 + (y - k)^2 = r^2

yields

(x - 5)^2 + (y + 6)^2 = 36

. Expanding the terms gives

x^2 - 10x + 25 + y^2 + 12y + 36 = 36

, which simplifies to

x^2 + y^2 - 10x + 12y + 25 = 0

.

Step-by-Step Solution

1

Determine the radius of the circle from the given center and tangency condition.

The center is

(5, -6)

, and the circle is tangent to the

x

-axis (the line

y = 0

). The distance from the center to the

x

-axis is the absolute value of the

y

-coordinate of the center:

|-6| = 6

. Therefore, the radius

r

of the circle is

6

.

A circle tangent to the

x

-axis has a radius equal to the vertical distance from its center to the

x

-axis.

2

Write the equation of the circle in standard form.

(x - 5)^2 + (y - (-6))^2 = 6^2

, which simplifies to

(x - 5)^2 + (y + 6)^2 = 36

.

The standard equation of a circle is

(x - h)^2 + (y - k)^2 = r^2

, where

(h, k)

is the center and

r

is the radius.

3

Expand the standard equation to find the general quadratic form.

Expanding the squared binomials:

(x^2 - 10x + 25) + (y^2 + 12y + 36) = 36

. Grouping terms and simplifying gives

x^2 + y^2 - 10x + 12y + 25 = 0

.

To match the options, we convert the equation from standard form to general form by expanding the terms and setting the equation equal to zero.

Key Concept

The standard equation of a circle with center

(h, k)

and radius

r

is

(x - h)^2 + (y - k)^2 = r^2

. When a circle is tangent to the

x

-axis, its radius is equal to the distance from the center to the

x

-axis, which is

|k|

.

Question 245Question →

A water tank is being drained of its contents at a constant rate. The depth of the water in the tank is $8.2$ feet after $2$ hours of draining, and it is $5.8$ feet after $6$ hours of draining. What is the slope of the line that represents the depth of the water, in feet, as a function of the draining time, in hours?

-\frac{5}{3}

-\frac{3}{5}

-\frac{3}{10}

\frac{3}{5}

\frac{5}{3}

Show answer & explanation

Answer: $-\frac{3}{5}$

Answer

The slope of the line that represents the depth of the water as a function of the draining time is

-\frac{3}{5}

.

The correct answer is

-\frac{3}{5}

. To find the slope, we determine the points

(t, d)

from the given information:

(2, 8.2)

and

(6, 5.8)

. Using the slope formula

m = \frac{d_2 - d_1}{t_2 - t_1}

, we compute

\frac{5.8 - 8.2}{6 - 2} = \frac{-2.4}{4} = -0.6

. Converting

-0.6

to a fraction gives

-\frac{6}{10}

, which simplifies to

-\frac{3}{5}

.

Step-by-Step Solution

1

Identify the coordinates from the problem statement.

The two coordinates

(t, d)

where

t

is time in hours and

d

is depth in feet are

(2, 8.2)

and

(6, 5.8)

.

Since the question asks for the depth as a function of time, time is the independent variable (

x

) and depth is the dependent variable (

y

).

2

Apply the slope formula

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

.

m = \frac{5.8 - 8.2}{6 - 2}

The slope represents the constant rate of change of the water level over time.

3

Simplify the fraction and convert to a simplified ratio.

m = \frac{-2.4}{4} = -0.6 = -\frac{3}{5}

Dividing the change in depth by the change in time yields the slope of the linear relationship.

Key Concept

Slope of a Line

Estimated Time:1m 0s

Question 246Question →

In the standard $(x, y)$ coordinate plane, a triangle has vertices at $P(-3, 1)$ , $Q(5, 1)$ , and $R(1, 7)$ . A line segment connects the midpoint of side $PR$ to the midpoint of side $QR$ . What is the length of this line segment?

4

5

6

8

10

Show answer & explanation

Answer: 4

Answer

4

The midpoints of sides

PR

and

QR

are

(-1, 4)

and

(3, 4)

, respectively. Since both midpoints share a y-coordinate of 4, the segment connecting them is horizontal. The length is the difference between their x-coordinates:

3 - (-1) = 4

. Alternatively, by the Triangle Midsegment Theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. The third side,

PQ

, is horizontal and has a length of

5 - (-3) = 8

. Half of this length is

8 / 2 = 4

.

Step-by-Step Solution

1

Find the coordinates of the midpoint of side

PR

.

Midpoint

M_1 = \left(\frac{-3 + 1}{2}, \frac{1 + 7}{2}\right) = (-1, 4)

The midpoint formula is

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

.

2

Find the coordinates of the midpoint of side

QR

.

Midpoint

M_2 = \left(\frac{5 + 1}{2}, \frac{1 + 7}{2}\right) = (3, 4)

Applying the midpoint formula to vertices

Q(5, 1)

and

R(1, 7)

.

3

Calculate the distance between the two midpoints

M_1(-1, 4)

and

M_2(3, 4)

.

Distance

= 3 - (-1) = 4

Since the y-coordinates are the same, the distance is simply the absolute difference of the x-coordinates.

Key Concept

Midpoint formula and distance calculations on the coordinate plane, or the application of the Triangle Midsegment Theorem.

Alternative Method

By the Triangle Midsegment Theorem, the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length. The third side is

PQ

, which has a horizontal length of

5 - (-3) = 8

. Therefore, the midsegment length is

8 / 2 = 4

.

Estimated Time:1m 30s

Question 247Question →

In the standard $(x, y)$ coordinate plane, a circle with center $(5, -1)$ is tangent to the $x$ -axis at point $P$ . A line passes through point $P$ and the point $(-3, 6)$ . What is the slope of this line?

-\frac{4}{3}

-\frac{7}{8}

-\frac{3}{4}

\frac{3}{4}

-\frac{7}{3}

Show answer & explanation

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

Answer

-\frac{3}{4}

The circle with center

(5, -1)

is tangent to the

x

-axis, meaning the radius is

1

and the point of tangency

P

lies on the

x

-axis directly above the center at

(5, 0)

. The line passes through

P(5, 0)

and

(-3, 6)

. Applying the slope formula

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

gives

m = \frac{6 - 0}{-3 - 5} = \frac{6}{-8} = -\frac{3}{4}

.

Step-by-Step Solution

1

Determine the coordinates of the point of tangency

P

.

P = (5, 0)

Since the circle has center

(5, -1)

and is tangent to the

x

-axis, the point of tangency must lie on the

x

-axis (where

y = 0

). The point on the

x

-axis directly above the center

(5, -1)

is

(5, 0)

.

2

Set up the slope formula for the line passing through point

P(5, 0)

and point

(-3, 6)

.

m = \frac{6 - 0}{-3 - 5}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

.

3

Simplify the expression to find the slope.

m = -\frac{3}{4}

Calculate the difference in the numerator and denominator:

\frac{6}{-8}

, then reduce the fraction to simplest form:

-\frac{3}{4}

.

Key Concept

Calculating the slope of a line using the coordinates of two points, where one point is determined by a geometric tangency property.

Estimated Time:1m 0s

Question 248Question →

In the standard $(x, y)$ coordinate plane, a line with a slope of $\frac{3}{2}$ passes through the points $(k, -2)$ and $(2k, 4)$ . Is the statement ' $k = 4$ ' true or false?

True

False

Show answer & explanation

Answer: True

Answer

The statement is true because solving the slope equation with the given coordinates and slope value yields

k = 4

.

The correct option is true because substituting

k = 4

into the coordinates yields the points

(4, -2)

and

(8, 4)

. Using the slope formula, the slope is calculated as

\frac{4 - (-2)}{8 - 4} = \frac{6}{4} = \frac{3}{2}

, which matches the given slope value.

Step-by-Step Solution

1

State the formula for the slope

m

of a line passing through two points

(x_1, y_1)

and

(x_2, y_2)

.

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

This formula defines the constant rate of change (slope) between any two points on a non-vertical line.

2

Substitute the coordinates

(k, -2)

and

(2k, 4)

and the slope value

\frac{3}{2}

into the formula.

\\frac{3}{2} = \frac{4 - (-2)}{2k - k}

This creates an algebraic equation in terms of

k

that represents the given geometric conditions.

3

Simplify the numerator and denominator of the fraction and solve the resulting equation for

k

.

\frac{3}{2} = \frac{6}{k} \Rightarrow 3k = 12 \Rightarrow k = 4

Simplification reduces the algebraic expression, allowing the variable

k

to be isolated and solved.

Key Concept

Calculating the slope of a line from two coordinate points containing an algebraic parameter.

Estimated Time:1m 30s

Question 249Question →

What is the area, in square units, of the circle defined by the equation $x^2 + y^2 + 8x - 10y + 5 = 0$ in the standard $(x, y)$ coordinate plane?

36\pi

12\pi

6\pi

46\pi

5\pi

Show answer & explanation

Answer: $36\pi$

Answer

The area of the circle is

36\pi

square units.

To find the area of the circle, we first rewrite the given equation

x^2 + y^2 + 8x - 10y + 5 = 0

in standard form

(x-h)^2 + (y-k)^2 = r^2

. Rearranging the terms yields

(x^2 + 8x) + (y^2 - 10y) = -5

. Completing the square for

x

requires adding

(\frac{8}{2})^2 = 16

to both sides, and completing the square for

y

requires adding

(\frac{-10}{2})^2 = 25

to both sides. This gives

(x^2 + 8x + 16) + (y^2 - 10y + 25) = -5 + 16 + 25

, which simplifies to

(x + 4)^2 + (y - 5)^2 = 36

. Thus, the squared radius is

r^2 = 36

. The area of the circle is

\pi r^2 = 36\pi

square units.

Step-by-Step Solution

1

Group the

x

-terms and

y

-terms and move the constant to the right side of the equation.

(x^2 + 8x) + (y^2 - 10y) = -5

This prepares the equation for completing the square.

2

Complete the square for both the

x

and

y

terms by adding

(\frac{8}{2})^2 = 16

and

(\frac{-10}{2})^2 = 25

to both sides.

(x^2 + 8x + 16) + (y^2 - 10y + 25) = -5 + 16 + 25

This allows the quadratic expressions to be written as perfect squares.

3

Simplify the equation into the standard form of a circle

(x-h)^2 + (y-k)^2 = r^2

.

(x + 4)^2 + (y - 5)^2 = 36

This standard form directly reveals that the radius squared (

r^2

) is

36

.

4

Calculate the area of the circle using the formula

A = \pi r^2

.

A = \pi (36) = 36\pi

The area of a circle is calculated by multiplying

\pi

by the square of the radius.

Key Concept

Converting the general equation of a circle to standard form by completing the square to determine its radius and calculate its area.

Estimated Time:1m 30s

Question 250Question →

A scientist records the temperature of a gas compound inside a chamber as it is heated at a constant rate. At a time of $t = 3$ minutes, the temperature of the compound is $-8^\circ\text{C}$ . At $t = 15$ minutes, the temperature is $28^\circ\text{C}$ . What is the rate of change of the temperature with respect to time, in degrees Celsius per minute, of the compound?

\frac{1}{3}

-3

3

\frac{5}{3}

\frac{3}{5}

Show answer & explanation

Answer: $3$

Answer

The rate of change of the temperature of the compound is

3

degrees Celsius per minute.

The rate of change is equivalent to the slope of the line passing through the points

(3, -8)

and

(15, 28)

. Using the slope formula, we compute the ratio of the change in temperature to the change in time:

\frac{28 - (-8)}{15 - 3} = \frac{36}{12} = 3

degrees Celsius per minute.

Step-by-Step Solution

1

Identify the coordinate points from the given information.

The two points are

(3, -8)

and

(15, 28)

, where the first coordinate represents time

t

in minutes and the second coordinate represents temperature

T

in degrees Celsius.

To find the constant rate of change, we need to determine the change in temperature per unit of time, which corresponds to the slope between these two points.

2

Apply the slope formula

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

to calculate the rate of change.

Substituting the coordinates into the formula gives

\frac{28 - (-8)}{15 - 3}

.

The slope formula measures the vertical change (rise) over the horizontal change (run).

3

Simplify the expression to find the final value.

\frac{28 + 8}{12} = \frac{36}{12} = 3

.

Simplifying the numerator and denominator gives the constant rate of change.

Key Concept

The slope of a line represents the constant rate of change of the dependent variable with respect to the independent variable.

Question 251Question →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $x^2 + y^2 - 8x - 6y = 0$ . A point on this circle has an $x$ -coordinate of $1$ and a positive $y$ -coordinate. What is the $y$ -coordinate of this point?

Show answer & explanation

Answer: 7

Answer

The positive y-coordinate of the point on the circle is 7.

Substituting

x = 1

into the circle's equation

x^2 + y^2 - 8x - 6y = 0

gives

1 + y^2 - 8 - 6y = 0

, which simplifies to the quadratic equation

y^2 - 6y - 7 = 0

. Factoring this equation yields

(y - 7)(y + 1) = 0

. The solutions are

y = 7

and

y = -1

. Because the y-coordinate must be positive, the correct value is 7.

Step-by-Step Solution

1

Substitute the x-coordinate

x = 1

into the circle's equation.

1^2 + y^2 - 8(1) - 6y = 0

This sets up the equation to solve for the corresponding y-coordinates.

2

Simplify the equation.

y^2 - 6y - 7 = 0

Combining the constant terms creates a standard quadratic equation in terms of y.

3

Solve the quadratic equation by factoring.

(y - 7)(y + 1) = 0

, yielding

y = 7

or

y = -1

Factoring allows us to find the two possible y-values that satisfy the equation.

4

Choose the positive y-coordinate.

y = 7

The question specifies that the y-coordinate must be positive.

Key Concept

Substituting a known coordinate into a circle's equation to find the other coordinate using quadratic equations.

Question 252Question →

A highway is constructed up a mountain pass at a constant incline. At a distance of $3.6$ miles from the starting toll gate, the elevation of the highway is $1,420$ feet above sea level. At a distance of $8.4$ miles from the toll gate, the elevation is $3,100$ feet above sea level. What is the slope of the highway's elevation profile, in feet per mile?

Show answer & explanation

Answer: 350

Answer

The slope of the highway's elevation profile is 350 feet per mile.

The slope represents the constant rate of change of elevation per mile. By defining the coordinates as

(x_1, y_1) = (3.6, 1420)

and

(x_2, y_2) = (8.4, 3100)

, we calculate the slope using the slope formula. Substituting the values gives

m = \frac{3100 - 1420}{8.4 - 3.6} = \frac{1680}{4.8} = 350

.

Step-by-Step Solution

1

Represent the given data as coordinate points where the x-coordinate represents the distance in miles and the y-coordinate represents the elevation in feet.

The coordinate points are

(3.6, 1420)

and

(8.4, 3100)

.

Slope is the change in the dependent variable (elevation) per unit change in the independent variable (distance).

2

Apply the slope formula

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

to find the rate of change.

m = \frac{3100 - 1420}{8.4 - 3.6} = \frac{1680}{4.8}

The slope represents the constant rate of change of elevation relative to the distance traveled.

3

Perform the division to find the numerical slope.

350

Dividing 1680 by 4.8 yields the exact value of 350.

Key Concept

The slope of a line represents a constant rate of change between two variables, calculated as the change in the vertical coordinate divided by the change in the horizontal coordinate.

Question 253Question →

A delivery truck driver records the odometer reading and the remaining fuel in the gas tank at two points during a long-distance trip. At an odometer reading of $140$ miles, the gas tank contains $18$ gallons of fuel. At an odometer reading of $290$ miles, the tank contains $12$ gallons of fuel. Assuming the fuel is consumed at a constant rate, what is the slope of the line that models the amount of fuel remaining (in gallons) as a function of the odometer reading (in miles)?

-\frac{1}{25}

-25

\frac{1}{25}

25

-\frac{1}{15}

Show answer & explanation

Answer: $-\frac{1}{25}$

Answer

The slope of the line modeling remaining fuel relative to odometer distance is

-\frac{1}{25}

gallons per mile.

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

. Assigning

x

to miles and

y

to gallons gives points

(140, 18)

and

(290, 12)

. Substituting these coordinates yields

\frac{12 - 18}{290 - 140} = \frac{-6}{150} = -\frac{1}{25}

. This negative value represents a rate of decrease of

1

gallon for every

25

miles driven.

Step-by-Step Solution

1

Identify the given coordinate points

(x, y)

where

x

represents the odometer reading in miles and

y

represents the remaining fuel in gallons.

The two points are

(140, 18)

and

(290, 12)

.

The problem specifies fuel remaining as a function of odometer reading, making distance the independent variable (

x

) and fuel the dependent variable (

y

).

2

Apply the slope formula

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

.

m = \frac{12 - 18}{290 - 140} = \frac{-6}{150}

Slope measures the vertical change (change in fuel) divided by the horizontal change (change in distance).

3

Simplify the resulting fraction

\frac{-6}{150}

.

m = -\frac{1}{25}

Dividing both the numerator and denominator by their greatest common divisor,

6

, yields

-\frac{1}{25}

.

Key Concept

Slope as Rate of Change

Estimated Time:1m 15s

Question 254Question →

In the standard $(x, y)$ coordinate plane, the endpoints of a diameter of a circle are $(-2, 5)$ and $(4, -3)$ . Which of the following is the equation of this circle?

(x - 1)^2 + (y - 1)^2 = 25

(x + 1)^2 + (y + 1)^2 = 25

(x - 1)^2 + (y - 1)^2 = 100

(x - 1)^2 + (y - 1)^2 = 5

(x + 1)^2 + (y + 1)^2 = 100

Show answer & explanation

Answer: $(x - 1)^2 + (y - 1)^2 = 25$

Answer

(x - 1)^2 + (y - 1)^2 = 25

The center

(h, k)

of the circle is the midpoint of the diameter endpoints

(-2, 5)

and

(4, -3)

, which is

(1, 1)

. The radius

r

is the distance from the center

(1, 1)

to one endpoint

(4, -3)

, giving

r = \sqrt{(4-1)^2 + (-3-1)^2} = 5

. Thus,

r^2 = 25

. Substituting

h=1

,

k=1

, and

r^2=25

into standard form

(x - h)^2 + (y - k)^2 = r^2

yields

(x - 1)^2 + (y - 1)^2 = 25

.

Step-by-Step Solution

1

Find the center of the circle using the midpoint formula

Center

(h, k) = \left(\frac{-2 + 4}{2}, \frac{5 + (-3)}{2}\right) = (1, 1)

The center of a circle is the midpoint of any of its diameters.

2

Calculate the radius of the circle using the distance formula

r = \sqrt{(4 - 1)^2 + (-3 - 1)^2} = \sqrt{3^2 + (-4)^2} = \sqrt{25} = 5

The radius is the distance from the center to any point on the circle, including the endpoints of the diameter.

3

Write the standard equation of the circle

(x - h)^2 + (y - k)^2 = r^2

(x - 1)^2 + (y - 1)^2 = 5^2 \implies (x - 1)^2 + (y - 1)^2 = 25

Substitute

h = 1

,

k = 1

, and

r = 5

into the standard form of a circle's equation.

Key Concept

Equation of a circle given diameter endpoints

Question 255Question →

In the standard $(x, y)$ coordinate plane, a line passing through the points $(-3, 5)$ and $(1, -3)$ has a slope of $-2$ . Is this statement true or false?

True

False

Show answer & explanation

Answer: True

Answer

True

The statement is true because substituting the coordinates

(-3, 5)

and

(1, -3)

into the slope formula

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

yields

m = \frac{-3 - 5}{1 - (-3)} = \frac{-8}{4} = -2

, which exactly matches the stated slope value.

Step-by-Step Solution

1

Identify the coordinates of the two points on the line.

(x_1, y_1) = (-3, 5)

and

(x_2, y_2) = (1, -3)

The coordinates are needed to calculate the vertical change (rise) and horizontal change (run).

2

Apply the slope formula

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

.

m = \frac{-3 - 5}{1 - (-3)} = \frac{-8}{4} = -2

Subtracting

y_1

from

y_2

yields the change in

y

, and subtracting

x_1

from

x_2

yields the change in

x

. Subtracting a negative number becomes addition:

1 - (-3) = 1 + 3 = 4

.

3

Compare the calculated slope to the value stated in the problem.

The calculated slope is

-2

, which matches the given statement.

Since the calculated value agrees with the statement, the statement is true.

Key Concept

Slope of a Line

Question 256Question →

Line $k$ in the coordinate plane is represented by the equation $4x - 6y = 15$ . If Line $p$ is perpendicular to Line $k$ , what is the slope of Line $p$ ?

-\frac{3}{2}

-\frac{2}{3}

\frac{2}{3}

\frac{3}{2}

-\frac{15}{4}

Show answer & explanation

Answer: $-\frac{3}{2}$

Answer

The slope of Line

p

is

-\frac{3}{2}

.

To find the slope of a line perpendicular to

4x - 6y = 15

, first convert the equation to slope-intercept form (

y = mx + b

). Solving for

y

yields

y = \frac{2}{3}x - \frac{5}{2}

, indicating that the original line has a slope of

\frac{2}{3}

. Perpendicular lines have slopes that are negative reciprocals. The negative reciprocal of

\frac{2}{3}

is

-\frac{3}{2}

, which is the correct answer.

Step-by-Step Solution

1

Convert the equation of Line

k

into slope-intercept form (

y = mx + b

).

Subtract

4x

from both sides to get

-6y = -4x + 15

. Divide all terms by

-6

to get

y = \frac{4}{6}x - \frac{15}{6}

, which simplifies to

y = \frac{2}{3}x - \frac{5}{2}

.

Converting to slope-intercept form directly reveals the slope

m

as the coefficient of

x

.

2

Identify the slope of Line

k

.

The slope of Line

k

is

m_k = \frac{2}{3}

.

The coefficient of

x

in

y = \frac{2}{3}x - \frac{5}{2}

represents the slope.

3

Calculate the negative reciprocal of the slope to find the slope of perpendicular Line

p

.

The slope of Line

p

is

m_p = -\frac{1}{m_k} = -\frac{1}{2/3} = -\frac{3}{2}

.

Perpendicular lines in the coordinate plane have slopes whose product is

-1

(negative reciprocals).

Key Concept

The slope of a line given by

Ax + By = C

is

m = -\frac{A}{B}

. Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocals (

m_1 \cdot m_2 = -1

).

Estimated Time:1m 15s

Question 257Question →

A commercial printing press operates at a constant rate throughout the day. At 10:15 a.m., a total of 180 posters have been printed since the morning shift began. By 11:45 a.m. on the same morning, a total of 540 posters have been printed. What is the printing rate of the press, in posters per hour?

Show answer & explanation

Answer: 240

Answer

The printing rate of the press is 240 posters per hour.

The slope of a linear relationship represents the constant rate of change. The total number of posters increases by

540 - 180 = 360

over an elapsed time of

1.5

hours. Dividing the change in posters by the change in time yields

\frac{360}{1.5} = 240

posters per hour.

Step-by-Step Solution

1

Determine the change in the dependent variable (number of posters printed)

540 - 180 = 360

posters

Slope represents the change in

y

divided by the change in

x

, where

y

is the total posters printed.

2

Determine the change in the independent variable (time in hours)

From 10:15 a.m. to 11:45 a.m. is 1.5 hours

Time must be expressed in hours to match the requested unit of posters per hour.

3

Compute the slope (rate of change)

\frac{360}{1.5} = 240

posters per hour

Dividing total posters printed by total hours gives the constant rate of change.

Key Concept

Interpretation of Slope as a Constant Rate of Change

Question 258Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 - 6x + 8y - 11 = 0$ . What is the distance from the center of this circle to the origin $(0, 0)$ ?

Show answer & explanation

Answer: 5

Answer

The distance from the center of the circle to the origin is 5.

By completing the square on

x^2 - 6x

and

y^2 + 8y

, we rewrite

x^2 + y^2 - 6x + 8y - 11 = 0

as

(x - 3)^2 + (y + 4)^2 = 36

. The center of the circle is

(3, -4)

. The distance from

(3, -4)

to

(0, 0)

is

\sqrt{3^2 + (-4)^2} = \sqrt{25} = 5

.

Step-by-Step Solution

1

Rewrite the general equation of the circle in standard form by completing the square.

(x - 3)^2 + (y + 4)^2 = 36

Grouping

x

and

y

terms and adding

(b/2)^2

to both sides isolates the center coordinates

(h, k)

.

2

Identify the center of the circle from standard form.

Center is

(3, -4)

The standard equation of a circle is

(x - h)^2 + (y - k)^2 = r^2

, where

(h, k)

is the center.

3

Compute the distance between the center

(3, -4)

and the origin

(0, 0)

.

d = \sqrt{3^2 + (-4)^2} = \sqrt{25} = 5

Applying the distance formula

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

yields the distance to the origin.

Key Concept

Completing the square to find the center of a circle and applying the distance formula

Estimated Time:1m 15s

Question 259Question →

In the standard $(x, y)$ coordinate plane, a circle has the equation $x^2 + y^2 + 10x - 4y + 13 = 0$ . A line with a slope of $-\frac{1}{2}$ passes through the center of this circle. What is the $y$ -intercept of this line?

-\frac{9}{2}

-\frac{7}{2}

-\frac{1}{2}

\frac{1}{2}

\frac{9}{2}

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

The

y

-intercept of the line is

-\frac{1}{2}

.

Completing the square for the circle equation yields

(x + 5)^2 + (y - 2)^2 = 16

, which places the center at

(-5, 2)

. Substituting the center

(-5, 2)

and slope

m = -\frac{1}{2}

into the point-slope form

y - 2 = -\frac{1}{2}(x + 5)

simplifies to

y = -\frac{1}{2}x - \frac{1}{2}

, making the

y

-intercept

-\frac{1}{2}

.

Step-by-Step Solution

1

Group terms and complete the square for both

x

and

y

in the circle equation.

(x^2 + 10x + 25) + (y^2 - 4y + 4) = -13 + 25 + 4

, which simplifies to

(x + 5)^2 + (y - 2)^2 = 16

.

Converting the general equation of a circle into standard form

(x - h)^2 + (y - k)^2 = r^2

reveals its center

(h, k)

.

2

Identify the center of the circle from the standard form equation.

The center is

(h, k) = (-5, 2)

.

Comparing

(x + 5)^2 + (y - 2)^2 = 16

to

(x - h)^2 + (y - k)^2 = r^2

gives

h = -5

and

k = 2

.

3

Use the point-slope formula

y - y_1 = m(x - x_1)

with point

(-5, 2)

and slope

m = -\frac{1}{2}

to find the equation of the line.

y - 2 = -\frac{1}{2}(x - (-5)) \implies y - 2 = -\frac{1}{2}x - \frac{5}{2} \implies y = -\frac{1}{2}x - \frac{1}{2}

.

Finding the slope-intercept form

y = mx + b

allows direct identification of the

y

-intercept

b

.

Key Concept

Converting circle equations to standard form to find center coordinates and finding linear equations from slope and point

Estimated Time:1m 15s

Question 260Question →

A circle graphed on the coordinate plane is represented by the equation $x^2 + y^2 - 12x + 16y + 19 = 0$ . What is the length of the radius of the circle?

Show answer & explanation

Answer: 9

Answer

The radius of the circle is 9.

Converting the given general equation

x^2 + y^2 - 12x + 16y + 19 = 0

into standard form

(x - h)^2 + (y - k)^2 = r^2

by completing the square yields

(x - 6)^2 + (y + 8)^2 = 81

. Taking the square root of 81 gives a radius of 9.

Step-by-Step Solution

1

Rearrange the equation to group the x-terms and y-terms together and move the constant term to the right side.

(x^2 - 12x) + (y^2 + 16y) = -19

This sets up the expression for completing the square.

2

Complete the square for both the

x

and

y

variable groups by adding

(12/2)^2 = 36

and

(16/2)^2 = 64

to both sides of the equation.

(x - 6)^2 + (y + 8)^2 = -19 + 36 + 64 = 81

Adding these values balances the equation and converts the quadratic expressions into perfect square binomials.

3

Compare to the standard equation of a circle

(x - h)^2 + (y - k)^2 = r^2

to determine the radius

r

.

r^2 = 81 \implies r = \sqrt{81} = 9

The constant term on the right side of the standard form equation equals the square of the radius.

Key Concept

Standard form equation of a circle and completing the square

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