Coordinate Geometry Practice Questions — ACT — Page 7

Question 121Question →

In the standard $(x, y)$ coordinate plane, a triangle is first reflected across the line $y = -x$ and then translated 4 units to the right and 3 units down. If the vertex $A$ of the triangle is mapped to the final image point $A''(1, -5)$ under this composite transformation, what were the coordinates of the original vertex $A$ ?

(8, -5)

(-2, -3)

(2, 3)

(1, 2)

(9, -4)

Show answer & explanation

Answer: $(2, 3)$

Answer

The original coordinates of vertex

A

are

(2, 3)

.

To find the original coordinates of the vertex, the transformations must be undone in reverse order. First, reverse the translation of 4 units right and 3 units down by moving the final image point

A''(1, -5)

4 units left and 3 units up, yielding the intermediate point

A'(-3, -2)

. Second, reverse the reflection across

y = -x

by applying the rule

(x, y) \rightarrow (-y, -x)

to

A'

, resulting in the original coordinates

(2, 3)

.

Step-by-Step Solution

1

Identify the inverse of each transformation and their correct reverse order. The composite transformation consists of (1) reflection across the line

y = -x

, followed by (2) translation of 4 units to the right and 3 units down. To backtrack from the final image, we must apply the inverse transformations in reverse order: first (1) reverse the translation, then (2) reverse the reflection.

The reverse process is: first translate 4 units left and 3 units up, then reflect across the line

y = -x

.

Reversing the composite transformation requires reversing both the individual operations and the order of their application.

2

Apply the inverse translation to the image point

A''(1, -5)

. Reversing a translation of 4 units right and 3 units down means translating 4 units left and 3 units up, represented by the rule

(x, y) \rightarrow (x - 4, y + 3)

.

The intermediate point is

A' = (1 - 4, -5 + 3) = (-3, -2)

.

This step determines the coordinates of the vertex after the reflection but before the translation.

3

Apply the inverse reflection to

A'(-3, -2)

. Since a reflection is its own inverse, we reflect

A'(-3, -2)

across the line

y = -x

using the rule

(x, y) \rightarrow (-y, -x)

.

The original point is

A = (-(-2), -(-3)) = (2, 3)

.

This step yields the final pre-image coordinates of vertex

A

.

Key Concept

Backtracking a composite transformation in the coordinate plane by applying inverse transformations in reverse order.

Question 122Question →

In the standard $(x, y)$ coordinate plane, a line $L_1$ is perpendicular to the line that contains the points $(3, 5)$ and $(-1, 8)$ . If $L_1$ is also parallel to the line defined by the equation $ax + 6y = 15$ , what is the value of the constant $a$ ?

Show answer & explanation

Answer: -8

Answer

The value of the constant

a

is

-8

.

The slope of the line containing

(3, 5)

and

(-1, 8)

is

-\frac{3}{4}

. The slope of a line perpendicular to it is the negative reciprocal, which is

\frac{4}{3}

. Because line

L_1

is parallel to the line

ax + 6y = 15

, they must have equal slopes. The slope of

ax + 6y = 15

is

-\frac{a}{6}

. Setting the two slopes equal gives

-\frac{a}{6} = \frac{4}{3}

, which yields

a = -8

.

Step-by-Step Solution

1

Calculate the slope of the line containing the points

(3, 5)

and

(-1, 8)

.

The slope is

-\frac{3}{4}

.

Using the slope formula

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

for the points

(3, 5)

and

(-1, 8)

, we get

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

.

2

Determine the slope of line

L_1

using the perpendicular relationship.

The slope of

L_1

is

\frac{4}{3}

.

Since line

L_1

is perpendicular to the line with slope

-\frac{3}{4}

, its slope must be the negative reciprocal, which is

-\frac{1}{-3/4} = \frac{4}{3}

.

3

Express the slope of the line

ax + 6y = 15

in terms of

a

.

The slope is

-\frac{a}{6}

.

Rewriting the equation

ax + 6y = 15

in slope-intercept form (

y = mx + b

) gives

6y = -ax + 15

, which simplifies to

y = -\frac{a}{6}x + \frac{5}{2}

. The slope is the coefficient of

x

, which is

-\frac{a}{6}

.

4

Set the slope of

L_1

equal to the slope of the parallel line to solve for

a

.

a = -8

Because line

L_1

is parallel to the line

ax + 6y = 15

, their slopes are equal:

-\frac{a}{6} = \frac{4}{3}

. Multiplying both sides by

-6

gives

a = -8

.

Key Concept

Parallel lines have equal slopes, and perpendicular lines have slopes that are negative reciprocals of each other.

Estimated Time:1m 30s

Question 123Question →

A linear equation in the standard $(x, y)$ coordinate plane is given by $y = -4$ . Is the statement that the slope of this line is undefined true or false?

True

False

Show answer & explanation

Answer: False

Answer

False

The correct answer is false because the line

y = -4

is horizontal, and all horizontal lines have a slope of

0

. A slope is only undefined for vertical lines, which are written in the form

x = c

.

Step-by-Step Solution

1

Identify the orientation of the line given by the equation

y = -4

.

The equation is of the form

y = c

, where

c

is a constant. This represents a horizontal line where every point on the line has a

y

-coordinate of

-4

.

Since the

y

-value never changes regardless of the

x

-value, the line is parallel to the

x

-axis.

2

Calculate the slope using two points on the line.

Let the two points be

(0, -4)

and

(1, -4)

. The slope

m

is given by

m = \frac{-4 - (-4)}{1 - 0} = \frac{0}{1} = 0

.

The slope formula is

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

.

3

Compare the calculated slope to the statement in the question.

The calculated slope is

0

, which is a defined number. The statement claims the slope is undefined, which is only true for vertical lines. Therefore, the statement is false.

An undefined slope occurs when dividing by zero, which is not the case for a horizontal line.

Key Concept

The slope of any horizontal line is

0

, whereas the slope of any vertical line is undefined.

Question 124Question →

A straight path on a coordinate map is modeled by the equation $2x - y = 8$ . A second path, which is perpendicular to the first path, passes through the point $(3, 5)$ . What is the $y$ -intercept of the second path?

-1

\frac{7}{2}

\frac{13}{2}

\frac{33}{2}

\frac{8}{3}

Show answer & explanation

Answer: $\frac{13}{2}$

Answer

The y-intercept of the second path is 13/2

The y-intercept is 13/2. First, the slope of the given line is found to be 2. The perpendicular line must have a slope of -1/2. Substituting this slope and the point (3, 5) into the slope-intercept form gives the y-intercept of 13/2.

Step-by-Step Solution

1

Find the slope of the first path by rewriting the equation in slope-intercept form (

y = mx + b

).

The equation

2x - y = 8

becomes

y = 2x - 8

, indicating the slope

m_1 = 2

.

Knowing the slope of the first path allows us to determine the slope of any perpendicular path.

2

Calculate the slope of the perpendicular path by taking the negative reciprocal of the first path's slope.

The perpendicular slope

m_2 = -\frac{1}{2}

.

Perpendicular lines have slopes that are negative reciprocals of one another.

3

Use the point-slope formula with the point

(3, 5)

and the perpendicular slope

-\frac{1}{2}

to write the equation of the second path.

The equation is

y - 5 = -\frac{1}{2}(x - 3)

, which simplifies to

y = -\frac{1}{2}x + \frac{3}{2} + 5

.

This establishes the linear relationship of the second path containing the point (3, 5).

4

Find the y-intercept by setting

x = 0

and simplifying.

Setting

x = 0

gives

y = \frac{3}{2} + 5 = \frac{13}{2}

.

The y-intercept of a line is the y-coordinate where the line crosses the y-axis (when x = 0).

Key Concept

Perpendicular lines in a coordinate plane have slopes that are negative reciprocals of each other, and their equations can be solved using point-slope form.

Estimated Time:1m 30s

Question 125Question →

In the standard $(x, y)$ coordinate plane, a line $L$ passes through the point $(2, 3)$ and has a positive $y$ -intercept $(0, b)$ . The area of the triangular region in the first quadrant bounded by line $L$ , the $x$ -axis, and the $y$ -axis is $12$ square units. What is the value of $b$ ?

3

4

6

8

12

Show answer & explanation

Answer: $6$

Answer

The correct value of

b

is

6

.

To find the value of

b

, we can use the intercept form of a linear equation:

\frac{x}{a} + \frac{y}{b} = 1

, where

a

is the

x

-intercept and

b

is the

y

-intercept. The area of the right triangle formed by the axes and the line is given by

\frac{1}{2}ab = 12

, which means

ab = 24

, or

a = \frac{24}{b}

. Since the line passes through the point

(2, 3)

, we substitute these coordinates into the intercept equation to get

\frac{2}{a} + \frac{3}{b} = 1

. Substituting

a = \frac{24}{b}

into this equation gives

\frac{2b}{24} + \frac{3}{b} = 1

, which simplifies to

\frac{b}{12} + \frac{3}{b} = 1

. Multiplying the entire equation by

12b

to clear the denominators results in

b^2 + 36 = 12b

. Rearranging this quadratic equation gives

b^2 - 12b + 36 = 0

, which factors as

(b - 6)^2 = 0

. Solving for

b

yields

b = 6

.

Step-by-Step Solution

1

Express the area of the right triangle in the first quadrant in terms of the

x

-intercept

(a, 0)

and

y

-intercept

(0, b)

.

The area is

\frac{1}{2}ab = 12

, which simplifies to

ab = 24

, or

a = \frac{24}{b}

.

The triangular region is a right triangle with base

a

and height

b

along the coordinate axes.

2

Set up the equation of the line using the intercept form

\frac{x}{a} + \frac{y}{b} = 1

and substitute the given point

(2, 3)

.

Substituting

x = 2

and

y = 3

gives the equation

\frac{2}{a} + \frac{3}{b} = 1

.

Since the line passes through the point

(2, 3)

, this point must satisfy the equation of the line.

3

Substitute

a = \frac{24}{b}

into the intercept equation and solve the resulting quadratic equation for

b

.

Substituting

a

gives

\frac{2}{\frac{24}{b}} + \frac{3}{b} = 1 \implies \frac{b}{12} + \frac{3}{b} = 1

. Multiplying by

12b

yields

b^2 + 36 = 12b \implies b^2 - 12b + 36 = 0 \implies (b - 6)^2 = 0 \implies b = 6

.

Solving the quadratic equation yields the value of the

y

-intercept

b

.

Key Concept

Linear equations and graphing using intercept form and triangle area relations

Alternative Method

An alternative approach is to use the slope formula. The line passes through

(a, 0)

,

(2, 3)

, and

(0, b)

. The slope between

(0, b)

and

(2, 3)

is

\frac{3 - b}{2}

, and the slope between

(0, b)

and

(a, 0)

is

-\frac{b}{a}

. Equating these gives

\frac{3 - b}{2} = -\frac{b}{a} \implies 3a - ab = -2b

. Since the area is

12

, we know

ab = 24

. Substituting

ab = 24

gives

3a - 24 = -2b \implies 3a + 2b = 24

. Since

a = \frac{24}{b}

, we get

3\left(\frac{24}{b}\right) + 2b = 24 \implies \frac{72}{b} + 2b = 24 \implies b^2 - 12b + 36 = 0 \implies b = 6

.

Estimated Time:1m 30s

Question 126Question →

In the standard $(x,y)$ coordinate plane, a parallelogram has vertices $A(1, 2)$ , $B(6, 2)$ , and $C(8, 6)$ . If the fourth vertex, $D$ , has a $y$ -coordinate of $6$ and lies to the left of $C$ , what is the $x$ -coordinate of $D$ ?

-7

3

5

9

13

Show answer & explanation

Answer: $3$

Answer

The correct answer is

3

, which is the

x

-coordinate of vertex

D

.

The correct answer is the coordinate value

3

. Since vertices

A(1, 2)

and

B(6, 2)

share the same

y

-coordinate, the side

AB

is horizontal with a length of

6 - 1 = 5

units. The opposite side

CD

must also be horizontal and have the same length of

5

units. Since vertex

C

is at

(8, 6)

and vertex

D

lies to its left, the

x

-coordinate of

D

is found by subtracting

5

from the

x

-coordinate of

C

, which gives

8 - 5 = 3

.

Step-by-Step Solution

1

Identify the relationship between the vertices of the parallelogram.

Sides

AB

and

CD

are parallel and equal in length.

By definition, opposite sides of a parallelogram are equal in length and parallel.

2

Calculate the horizontal length of the bottom side

AB

.

Length of

AB = 6 - 1 = 5

units.

Since both

A(1,2)

and

B(6,2)

share a

y

-coordinate of

2

, the segment is horizontal, and its length is the difference in their

x

-coordinates.

3

Apply the horizontal translation to find the

x

-coordinate of

D

.

x = 8 - 5 = 3

.

Since

D

lies to the left of

C(8,6)

on the horizontal line

y=6

, we subtract the side length of

5

units from the

x

-coordinate of

C

to get the

x

-coordinate of

D

.

Key Concept

Using the properties of parallelograms and coordinates on a 2D plane to find a missing vertex.

Estimated Time:45s

Question 127Question →

In the standard $(x, y)$ coordinate plane, a point is first dilated by a scale factor of $2$ with the center of dilation at $(1, 1)$ . The resulting intermediate point is then reflected across the line $y = -x$ to yield a final image at $(-5, -9)$ . What were the coordinates of the original point before these two transformations were applied?

(17, 9)

(4, 2)

(5, 3)

(-4, -2)

(-2, -4)

Show answer & explanation

Answer: $(5, 3)$

Answer

The original point before the transformations was

(5, 3)

.

To find the coordinates of the original point, we reverse the transformations in reverse order. First, we reverse the reflection across

y = -x

. Since reflecting

(x', y')

across

y = -x

results in

(-y', -x') = (-5, -9)

, we have

y' = 5

and

x' = 9

, which makes the intermediate point

(9, 5)

. Next, we reverse the dilation centered at

(1, 1)

with a scale factor of

2

. Under this dilation, the original point

(x, y)

maps to

(1 + 2(x - 1), 1 + 2(y - 1)) = (9, 5)

. Solving the equation

1 + 2(x - 1) = 9

gives

x = 5

, and solving

1 + 2(y - 1) = 5

gives

y = 3

. Thus, the original point is

(5, 3)

.

Step-by-Step Solution

1

Determine the intermediate point by reversing the reflection across the line

y = -x

.

The final image is at

(-5, -9)

. The rule for reflection across

y = -x

is

(x', y') \rightarrow (-y', -x')

. To reverse this, we apply the same rule to the final image:

x' = -(-9) = 9

and

y' = -(-5) = 5

. Thus, the intermediate point is

(9, 5)

.

We must work backward from the final result, reversing the second transformation first.

2

Set up the equation to reverse the dilation centered at

(1, 1)

with a scale factor of

2

.

The dilation formula for a point

(x, y)

centered at

(h, k)

with scale factor

c

is

(x', y') = (h + c(x - h), k + c(y - k))

. Substituting

h=1

,

k=1

,

c=2

,

x'=9

, and

y'=5

yields the equations:

9 = 1 + 2(x - 1)

and

5 = 1 + 2(y - 1)

.

The intermediate point is the result of dilating the original point, so we solve for the original coordinates.

3

Solve the coordinate equations to find the original values of

x

and

y

.

For

x

:

8 = 2(x - 1) \Rightarrow 4 = x - 1 \Rightarrow x = 5

. For

y

:

4 = 2(y - 1) \Rightarrow 2 = y - 1 \Rightarrow y = 3

. The original point is

(5, 3)

.

Solving these algebraic equations gives the horizontal and vertical coordinates of the pre-image.

Key Concept

Transformations in the Coordinate Plane

Question 128Question →

In the standard $(x, y)$ coordinate plane, two circles are defined by the equations $x^2 + y^2 - 4x + 6y = 3$ and $x^2 + y^2 + 8x - 2y = 8$ . Is the statement that the line passing through the centers of these two circles has a slope of $-\frac{2}{3}$ true or false?

True

False

Show answer & explanation

Answer: True

Answer

true

The correct answer is true because completing the square for the given circle equations determines the centers to be

(2, -3)

and

(-4, 1)

. Calculating the slope between these two coordinates using the slope formula results in

-\frac{2}{3}

, which matches the statement.

Step-by-Step Solution

1

Complete the square for the first circle's equation to find its center.

The equation

x^2 + y^2 - 4x + 6y = 3

is rewritten as

(x - 2)^2 + (y + 3)^2 = 16

, identifying the center of the first circle as

(2, -3)

.

Converting the circle's equation to the standard form

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

reveals the center coordinates

(h, k)

.

2

Complete the square for the second circle's equation to find its center.

The equation

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

is rewritten as

(x + 4)^2 + (y - 1)^2 = 25

, identifying the center of the second circle as

(-4, 1)

.

Converting the second circle's equation to standard form provides the coordinates of the second point needed to determine the slope.

3

Apply the slope formula

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

to the two center points.

Using the coordinates

(2, -3)

and

(-4, 1)

, we calculate

m = \frac{1 - (-3)}{-4 - 2} = \frac{4}{-6} = -\frac{2}{3}

.

This calculation determines the slope of the line passing through both centers to evaluate the statement.

Key Concept

Slope of a Line

Question 129Question →

A circle in the standard $(x,y)$ coordinate plane is defined by the equation $(x - 8)^2 + (y + 5)^2 = 121$ . What is the diameter of this circle?

Show answer & explanation

Answer: 22

Answer

The diameter of the circle is 22.

By comparing the given equation

(x - 8)^2 + (y + 5)^2 = 121

to the standard circle equation

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

, we find that

r^2 = 121

. Taking the square root of

121

gives a radius of

r = 11

. Since the diameter of a circle is twice the radius, the diameter is

2 \times 11 = 22

.

Step-by-Step Solution

1

Identify the relationship between the circle's equation and its radius squared.

r^2 = 121

The standard equation of a circle is

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

, meaning the constant on the right side of the equation represents the square of the radius.

2

Determine the radius of the circle.

r = 11

Taking the square root of

121

yields the radius of the circle, which must be a positive length.

3

Calculate the diameter of the circle.

d = 22

The diameter of a circle is defined as twice the length of its radius (

d = 2r

).

Key Concept

The standard equation of a circle is

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

, where

r

is the radius and the diameter is

2r

.

Question 130Question →

In the standard $(x, y)$ coordinate plane, a line passes through the points $(2, 5)$ and $(6, k)$ . If the $y$ -intercept of this line is $-1$ , what is the value of $k$ ?

Show answer & explanation

Answer: 17

Answer

The value of

k

is 17.

The correct answer is 17. The y-intercept of

-1

gives the point

(0, -1)

. The slope between

(0, -1)

and

(2, 5)

is

\frac{5 - (-1)}{2 - 0} = 3

. The equation of the line is

y = 3x - 1

. Substituting the point

(6, k)

into the equation gives

k = 3(6) - 1 = 17

.

Step-by-Step Solution

1

Identify the coordinates of the y-intercept.

The point is

(0, -1)

.

The y-intercept is the point where the line crosses the y-axis, which occurs when the x-coordinate is 0.

2

Calculate the slope (

m

) of the line using the points

(0, -1)

and

(2, 5)

.

m = 3

Using the slope formula

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

, we find

m = \frac{5 - (-1)}{2 - 0} = \frac{6}{2} = 3

.

3

Write the equation of the line.

y = 3x - 1

Using the slope-intercept form

y = mx + b

, where the slope

m = 3

and the y-intercept

b = -1

.

4

Substitute the point

(6, k)

into the line equation to solve for

k

.

k = 17

Since the point

(6, k)

lies on the line, substituting

x = 6

and

y = k

yields

k = 3(6) - 1 = 18 - 1 = 17

.

Key Concept

Linear Equations and Graphing

Alternative Method

Since the slope of a straight line is constant, the slope between

(0, -1)

and

(2, 5)

must equal the slope between

(2, 5)

and

(6, k)

. Setting the slopes equal:

\frac{5 - (-1)}{2 - 0} = \frac{k - 5}{6 - 2} \implies 3 = \frac{k - 5}{4} \implies 12 = k - 5 \implies k = 17

.

Estimated Time:1m 0s

Question 131Question →

In the standard $(x, y)$ coordinate plane, line $L_1$ is perpendicular to the line represented by the equation $3x - 2y = 6$ . Line $L_2$ is parallel to $L_1$ and passes through the midpoint of the line segment with endpoints $(-1, 5)$ and $(3, -1)$ . If the equation of $L_2$ is written in the form $Ax + By = 8$ , where $A$ and $B$ are integers, what is the value of $A + B$ ?

Show answer & explanation

Answer: 5

Answer

The value of

A + B

is 5.

The given line

3x - 2y = 6

has a slope of

\frac{3}{2}

. A perpendicular line

L_1

has a slope of

-\frac{2}{3}

. A line

L_2

parallel to

L_1

also has a slope of

-\frac{2}{3}

. The midpoint of the segment with endpoints

(-1, 5)

and

(3, -1)

is calculated as

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

. Using the point-slope form with the slope

-\frac{2}{3}

and point

(1, 2)

yields

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

, which simplifies to

2x + 3y = 8

. Thus,

A = 2

,

B = 3

, and

A + B = 5

.

Step-by-Step Solution

1

Find the slope of the given line

3x - 2y = 6

.

Slope is

\frac{3}{2}

Convert the equation to slope-intercept form

y = mx + b

to find the slope.

2

Determine the slope of line

L_1

.

Slope is

-\frac{2}{3}

Perpendicular lines have slopes that are negative reciprocals of each other.

3

Determine the slope of line

L_2

.

Slope is

-\frac{2}{3}

Parallel lines have the same slope.

4

Calculate the midpoint of the segment with endpoints

(-1, 5)

and

(3, -1)

.

Midpoint is

(1, 2)

Use the midpoint formula:

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

.

5

Write the equation of line

L_2

and convert it to standard form.

Equation is

2x + 3y = 8

Use the point-slope form with slope

-\frac{2}{3}

and midpoint

(1, 2)

, then rearrange to match the target constant 8 on the right-hand side.

6

Identify

A

and

B

, and compute their sum.

A = 2

,

B = 3

, and

A + B = 5

Match the coefficients of

2x + 3y = 8

to

Ax + By = 8

and add them.

Key Concept

Determining equations of parallel and perpendicular lines on the coordinate plane using slopes and midpoints.

Question 132Question →

In the standard $(x, y)$ coordinate plane, a rectangle has vertices at $(1, 2)$ , $(7, 2)$ , $(7, 10)$ , and $(1, 10)$ . What is the length of a diagonal of this rectangle?

Show answer & explanation

Answer: 10

Answer

The length of a diagonal of the rectangle is 10.

The width of the rectangle is

7 - 1 = 6

, and the height is

10 - 2 = 8

. The diagonal forms the hypotenuse of a right triangle with legs of 6 and 8. By the Pythagorean theorem, the length of the diagonal is

\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10

.

Step-by-Step Solution

1

Calculate the horizontal width of the rectangle.

Width =

7 - 1 = 6

The horizontal distance is found by subtracting the x-coordinates of the horizontal vertices.

2

Calculate the vertical height of the rectangle.

Height =

10 - 2 = 8

The vertical distance is found by subtracting the y-coordinates of the vertical vertices.

3

Apply the Pythagorean theorem to find the diagonal length.

Diagonal length =

\sqrt{6^2 + 8^2} = 10

The diagonal of a rectangle forms the hypotenuse of a right triangle with legs equal to the width and the height.

Key Concept

Using coordinate differences to find dimensions of a figure and using the Pythagorean theorem to find its diagonal length.

Question 133Question →

A circle has a center located at the point $(-2, 5)$ and a radius of $6$ units. What is the equation of this circle in the $(x, y)$ coordinate plane?

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

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

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

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

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

Show answer & explanation

Answer: $(x + 2)^2 + (y - 5)^2 = 36$

Answer

The equation of the circle is

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

.

The correct equation is found by substituting the center

(-2, 5)

and radius

6

into the standard equation

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

. This yields

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

, which simplifies to

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

.

Step-by-Step Solution

1

Identify the standard form equation of a circle.

The standard equation is

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

, where

(h, k)

represents the center coordinates and

r

represents the radius.

This general equation is the mathematical template for representing any circle in a coordinate plane.

2

Substitute the given values for the center

(-2, 5)

and the radius

6

into the equation.

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

Plugging in

h = -2

,

k = 5

, and

r = 6

customizes the general formula to this specific circle.

3

Simplify the signs of the terms and calculate the squared radius.

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

Subtracting

-2

simplifies to adding

2

, and squaring the radius of

6

yields

36

.

Key Concept

Standard equation of a circle given its center and radius

Question 134Question →

A line in the standard $(x, y)$ coordinate plane passes through the points $(3, -2)$ and $(7, 6)$ . What is the slope of this line?

Show answer & explanation

Answer: 2

Answer

The slope of the line is

2

.

The correct answer is

2

. 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}

. Substituting

(3, -2)

and

(7, 6)

gives

m = \frac{6 - (-2)}{7 - 3} = \frac{8}{4} = 2

.

Step-by-Step Solution

1

Identify the coordinates of the two points on the line.

(x_1, y_1) = (3, -2)

and

(x_2, y_2) = (7, 6)

The coordinates are needed to apply the slope formula.

2

Apply the slope formula

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

.

m = \frac{6 - (-2)}{7 - 3}

The slope is the ratio of the vertical change (rise) to the horizontal change (run).

3

Simplify the numerator and the denominator, then divide.

m = \frac{8}{4} = 2

Subtracting a negative value is equivalent to addition, which yields the final slope of

2

.

Key Concept

Slope formula

Question 135Question →

In the standard $(x, y)$ coordinate plane, line $l_1$ passes through the points $(1, 2)$ and $(5, r)$ , and line $l_2$ passes through the points $(-1, r)$ and $(3, 8)$ . If the slope of line $l_2$ is $-2$ times the slope of line $l_1$ , what is the value of $r$ ?

6

4

-4

-\frac{4}{3}

14

Show answer & explanation

Answer: $-4$

Answer

The value of

r

is

-4

.

The correct value of

r

is

-4

. The slope of line

l_1

is

m_1 = \frac{r - 2}{4}

and the slope of line

l_2

is

m_2 = \frac{8 - r}{4}

. Given that

m_2 = -2m_1

, substituting these expressions yields

\frac{8 - r}{4} = -2\left(\frac{r - 2}{4}\right)

. Multiplying both sides by

4

gives

8 - r = -2r + 4

. Adding

2r

to both sides and subtracting

8

from both sides results in

r = -4

.

Step-by-Step Solution

1

Find the slope

m_1

of line

l_1

using the coordinates

(1, 2)

and

(5, r)

.

m_1 = \frac{r - 2}{5 - 1} = \frac{r - 2}{4}

The slope formula is

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

.

2

Find the slope

m_2

of line

l_2

using the coordinates

(-1, r)

and

(3, 8)

.

m_2 = \frac{8 - r}{3 - (-1)} = \frac{8 - r}{4}

Applying the slope formula to the points on the second line.

3

Set up the equation using the given relationship

m_2 = -2m_1

.

\frac{8 - r}{4} = -2\left(\frac{r - 2}{4}\right)

We are given that the slope of line

l_2

is

-2

times the slope of line

l_1

.

4

Solve the equation for

r

.

8 - r = -2(r - 2) \Rightarrow 8 - r = -2r + 4 \Rightarrow r = -4

Multiply both sides by

4

and simplify to solve for

r

.

Key Concept

Slope of a Line

Alternative Method

Instead of solving the equation algebraically, you can test the given choices for

r

by calculating the slopes

m_1

and

m_2

for each option and checking if

m_2 = -2m_1

.

Estimated Time:2m 0s

Question 136Question →

In the standard $(x, y)$ coordinate plane, a triangle undergoes a sequence of transformations. First, it is reflected across the line $y = -x$ . Then, the resulting figure is translated $3$ units to the right and $2$ units down. If the vertex $A''$ of the final image is located at $(-1, 5)$ , what were the coordinates of the original vertex $A$ ?

(-7, 4)

(-3, -2)

(7, -4)

(4, -7)

(-4, 7)

Show answer & explanation

Answer: $(-7, 4)$

Answer

The original coordinates of vertex

A

are

(-7, 4)

.

The coordinates of the original vertex are found by reversing the sequence of transformations. Starting from the final image at

(-1, 5)

, we first reverse the translation by moving

3

units to the left and

2

units up, which gives the intermediate point

(-4, 7)

. Then, we apply the reflection across the line

y = -x

to this intermediate point. The reflection rule swaps the coordinates and negates them, mapping

(-4, 7)

to

(-7, 4)

.

Step-by-Step Solution

1

Identify the forward transformation rules for a point

(x, y)

.

The reflection across

y = -x

maps

(x, y) \rightarrow (-y, -x)

. The translation

3

units right and

2

units down maps

(x', y') \rightarrow (x' + 3, y' - 2)

.

Understanding the forward transformations allows us to set up the equations or work backward systematically.

2

Work backward from the final image

A''(-1, 5)

by reversing the translation.

Reversing a translation of

3

units right and

2

units down means translating

3

units left and

2

units up. Applying this to

A''(-1, 5)

yields

(-1 - 3, 5 + 2) = (-4, 7)

.

This gives the intermediate coordinates of the point

A'

before the translation occurred.

3

Reverse the reflection across the line

y = -x

.

A reflection across

y = -x

is its own inverse. Applying the mapping

(x', y') \rightarrow (-y', -x')

to the intermediate point

(-4, 7)

yields

(-7, 4)

.

This step recovers the original pre-image coordinates of vertex

A

.

Key Concept

Reversing composite transformations in the coordinate plane

Estimated Time:1m 30s

Question 137Question →

In a Cartesian coordinate system, the point $A(2, -4)$ is translated $4$ units to the left and $3$ units up to form point $B$ . If point $B$ is then reflected across the line $y = -x$ to form point $C$ , what are the coordinates of point $C$ ?

(1, 2)

(-1, -2)

(7, -6)

(7, 2)

(-2, 1)

Show answer & explanation

Answer: $(1, 2)$

Answer

(1, 2)

First, the point

(2, -4)

is translated

4

units left and

3

units up by subtracting

4

from the

x

-coordinate and adding

3

to the

y

-coordinate, which yields

(-2, -1)

. Next, this point is reflected across the line

y = -x

by swapping the coordinates and negating both, which results in the point

(1, 2)

.

Step-by-Step Solution

1

Apply the translation to point

A(2, -4)

.

Point

B

has coordinates

(-2, -1)

.

Translating a point

4

units to the left subtracts

4

from the

x

-coordinate, and translating

3

units up adds

3

to the

y

-coordinate:

(2 - 4, -4 + 3) = (-2, -1)

.

2

Apply the reflection across the line

y = -x

to point

B(-2, -1)

.

Point

C

has coordinates

(1, 2)

.

Reflecting a point

(x, y)

across the line

y = -x

swaps the coordinates and negates both, mapping

(x, y) \rightarrow (-y, -x)

. Thus,

(-2, -1) \rightarrow (-(-1), -(-2)) = (1, 2)

.

Key Concept

Composite transformations involving translations and reflections in the coordinate plane.

Question 138Question →

A scientist is monitoring the temperature of a chemical solution. At time $t = \frac{1}{3}$ hours, the temperature is $2^\circ\text{C}$ . At time $t = \frac{1}{2}$ hours, the temperature is $5^\circ\text{C}$ . Assuming the temperature increases at a constant rate, what was the initial temperature of the solution, in degrees Celsius?

-4

\frac{2}{9}

3

8

30

Show answer & explanation

Answer: -4

Answer

-4

The linear relationship is modeled by

T(t) = mt + b

. Calculating the slope

m

gives

\frac{5 - 2}{1/2 - 1/3} = \frac{3}{1/6} = 18

. Substituting the point

(1/3, 2)

into the equation yields

2 = 18(1/3) + b

, which simplifies to

2 = 6 + b

, giving

b = -4

. Thus, the initial temperature at

t = 0

is

-4^\circ\text{C}

.

Step-by-Step Solution

1

Find the constant rate of change (slope,

m

) using the points

(\frac{1}{3}, 2)

and

(\frac{1}{2}, 5)

.

m = \frac{5 - 2}{\frac{1}{2} - \frac{1}{3}} = \frac{3}{\frac{1}{6}} = 18

The rate of change represents the slope of the linear relationship between time and temperature.

2

Use the slope-intercept form

T(t) = mt + b

and the point

(\frac{1}{3}, 2)

to solve for the vertical intercept (

b

), which represents the initial temperature.

2 = 18(\frac{1}{3}) + b \Rightarrow 2 = 6 + b \Rightarrow b = -4

The initial temperature occurs at time

t = 0

, which corresponds to the

y

-intercept (

b

) of the linear equation.

Key Concept

Determining a linear equation from two points to find the initial value (vertical intercept)

Question 139Question →

What is the slope of the line represented by the equation $5x + 3y = -12$ in the standard $(x, y)$ coordinate plane?

\frac{5}{3}

-\frac{3}{5}

-\frac{5}{3}

\frac{3}{5}

-5

Show answer & explanation

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

Answer

-\frac{5}{3}

The correct answer is found by converting the standard form equation

5x + 3y = -12

into slope-intercept form (

y = mx + b

). Subtracting

5x

from both sides gives

3y = -5x - 12

. Dividing both sides by

3

yields

y = -\frac{5}{3}x - 4

. Thus, the slope of the line is the coefficient of

x

, which is

-\frac{5}{3}

.

Step-by-Step Solution

1

Start with the given equation in standard form.

5x + 3y = -12

Identify the equation to be transformed.

2

Subtract

5x

from both sides of the equation to isolate the term containing

y

.

3y = -5x - 12

Move the

x

term to the right side of the equation.

3

Divide every term in the equation by

3

to solve for

y

in slope-intercept form (

y = mx + b

).

y = -\frac{5}{3}x - 4

Isolate

y

so that the coefficient of

x

represents the slope.

4

Identify the coefficient of

x

, which is the slope

m

.

m = -\frac{5}{3}

In the form

y = mx + b

, the slope is

m

.

Key Concept

To find the slope of a line from its linear equation, rewrite the equation in slope-intercept form,

y = mx + b

, where

m

represents the slope.

Question 140Question →

In the standard $(x, y)$ coordinate plane, a triangle has vertex $A(3, -4)$ . If the triangle is reflected across the line $y = x$ and then translated $5$ units to the left and $2$ units up, what are the coordinates of the image of vertex $A$ ?

(1, 1)

(-2, -2)

(-2, 6)

(-9, 5)

(-1, -1)

Show answer & explanation

Answer: $(-9, 5)$

Answer

(-9, 5)

To find the coordinates of the image, we apply the transformations in the specified order. First, reflecting the point

(3, -4)

across the line

y = x

swaps the

x

- and

y

-coordinates, resulting in

(-4, 3)

. Second, translating the point

5

units to the left subtracts

5

from the

x

-coordinate, and translating

2

units up adds

2

to the

y

-coordinate:

(-4 - 5, 3 + 2) = (-9, 5)

.

Step-by-Step Solution

1

Reflect the point across the line

y = x

(-4, 3)

The coordinate rule for a reflection across the line

y = x

is

(x, y) \rightarrow (y, x)

. Applying this to the vertex

A(3, -4)

swaps the coordinates to yield

(-4, 3)

.

2

Translate the point

5

units to the left and

2

units up

(-9, 5)

Translating a point

5

units left subtracts

5

from the

x

-coordinate, and translating

2

units up adds

2

to the

y

-coordinate:

(-4 - 5, 3 + 2) = (-9, 5)

.

Key Concept

Applying composite transformations in the coordinate plane including reflections and translations.

Estimated Time:1m 0s

Coordinate Geometry

Answer

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