Coordinate Geometry Practice Questions — ACT — Page 12

Question 221Question →

In the standard $(x, y)$ coordinate plane, a line segment has endpoints $A(-2, 3)$ and $B(4, y)$ . If the slope of the line passing through $A$ and $B$ is $-\frac{2}{3}$ , what is the value of $y$ ?

-7

-6

-1

\frac{5}{3}

7

Show answer & explanation

Answer: -1

Answer

The value of

y

is

-1

.

The correct option is the one showing

-1

. By applying the slope formula

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

to the points

A(-2, 3)

and

B(4, y)

with slope

-\frac{2}{3}

, we set up the equation

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

. Simplifying the denominator yields

\frac{y - 3}{6} = -\frac{2}{3}

. Multiplying both sides by

6

gives

y - 3 = -4

, and solving for

y

yields

y = -1

.

Step-by-Step Solution

1

Write down the slope formula and substitute the given values.

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

The slope

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is defined as

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

.

2

Simplify the denominator on the left side of the equation.

\frac{y - 3}{6} = -\frac{2}{3}

Subtracting a negative number is equivalent to addition:

4 - (-2) = 4 + 2 = 6

.

3

Multiply both sides by 6 to isolate the numerator.

y - 3 = -4

Multiplying

-\frac{2}{3}

by

6

yields

-4

.

4

Solve for

y

by adding 3 to both sides of the equation.

y = -1

Isolating

y

gives

y = -4 + 3 = -1

.

Key Concept

Using the slope formula to find a missing coordinate

Estimated Time:1m 0s

Question 222Question →

In the standard $(x, y)$ coordinate plane, a line passes through the points $(-\frac{1}{2}, \frac{2}{3})$ and $(\frac{3}{4}, -\frac{1}{2})$ . What is the slope of this line?

\frac{14}{15}

-\frac{15}{14}

-\frac{9}{10}

-\frac{14}{15}

-\frac{14}{3}

Show answer & explanation

Answer: $-\frac{14}{15}$

Answer

The slope of the line is

-\frac{14}{15}

.

The slope of the line is found by using the slope formula

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

. Substituting the given points

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

and

(\frac{3}{4}, -\frac{1}{2})

into the formula, we find the change in

y

is

-\frac{7}{6}

and the change in

x

is

\frac{5}{4}

. Dividing the change in

y

by the change in

x

yields

-\frac{14}{15}

.

Step-by-Step Solution

1

Identify the coordinates of the two points as

(x_1, y_1) = (-\frac{1}{2}, \frac{2}{3})

and

(x_2, y_2) = (\frac{3}{4}, -\frac{1}{2})

.

x_1 = -\frac{1}{2}

,

y_1 = \frac{2}{3}

,

x_2 = \frac{3}{4}

, and

y_2 = -\frac{1}{2}

To use the slope formula, we must first map the given points to their respective coordinate variables.

2

Apply the slope formula

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

to calculate the change in

y

(rise) and the change in

x

(run).

Rise =

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

; Run = \frac{3}{4} - (-\frac{1}{2}) = \frac{5}{4}

The slope is the ratio of the vertical change to the horizontal change.

3

Divide the change in

y

by the change in

x

and simplify the resulting fraction.

m = \frac{-\frac{7}{6}}{\frac{5}{4}} = -\frac{7}{6} \times \frac{4}{5} = -\frac{28}{30} = -\frac{14}{15}

Dividing fractions is performed by multiplying the numerator fraction by the reciprocal of the denominator fraction.

Key Concept

Calculating the slope of a line between two points containing positive and negative fractional coordinates.

Question 223Question →

A line with a slope of $2.5$ passes through the point $(3, 4)$ in the coordinate plane. What is the y-coordinate of the point on this line whose x-coordinate is $-1$ ?

Show answer & explanation

Answer: -6

Answer

-6

By applying the slope formula

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

with the slope equal to

2.5

, the known point at

(3, 4)

, and the target point at

(-1, y)

, we obtain the linear equation

2.5 = \frac{y - 4}{-1 - 3}

. Simplifying the denominator gives

2.5 = \frac{y - 4}{-4}

. Multiplying both sides by

-4

yields

-10 = y - 4

. Finally, adding

4

to both sides gives

y = -6

.

Step-by-Step Solution

1

Set up the slope formula relation

2.5 = \frac{y - 4}{-1 - 3}

The slope

m

of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is defined as the change in y-values divided by the change in x-values.

2

Solve for the unknown y-coordinate

y = -6

Simplifying the denominator to

-4

, multiplying both sides by

-4

gives

-10 = y - 4

, and adding

4

to both sides isolates

y

.

Key Concept

Finding a missing coordinate on a line given its slope and another coordinate point.

Question 224Question →

A kite has vertices $K(2, 8)$ , $I(6, 3)$ , $T(2, -4)$ , and $E(-2, 3)$ in the standard $(x, y)$ coordinate plane. What is the area of kite $KITE$ , in square units?

Show answer & explanation

Answer: 48

Answer

The area of kite

KITE

is 48 square units.

The area of kite

KITE

is 48 because the horizontal diagonal

IE

has a length of 8 units, the vertical diagonal

KT

has a length of 12 units, and the area of a kite is calculated as half the product of its diagonal lengths:

\frac{1}{2} \times 8 \times 12 = 48

.

Step-by-Step Solution

1

Calculate the lengths of the vertical diagonal

KT

and the horizontal diagonal

IE

.

The length of

KT

is 12 and the length of

IE

is 8.

The vertices

K(2, 8)

and

T(2, -4)

share the same

x

-coordinate, so the diagonal is vertical with length

8 - (-4) = 12

. The vertices

I(6, 3)

and

E(-2, 3)

share the same

y

-coordinate, so the diagonal is horizontal with length

6 - (-2) = 8

.

2

Apply the area formula for a kite:

\text{Area} = \frac{1}{2} d_1 d_2

.

Area = 48

Since the diagonals of a kite are perpendicular, the area is half the product of the lengths of the diagonals:

\frac{1}{2} \times 12 \times 8 = 48

.

Key Concept

Finding the area of a geometric figure on the coordinate plane by using diagonal lengths.

Question 225Question →

In the standard $(x, y)$ coordinate plane, a triangle has vertices at $D(1, -2)$ , $E(4, 7)$ , and $F(6, k)$ . If the line containing the altitude from vertex $F$ to side $DE$ has a $y$ -intercept of $8$ , what is the value of $k$ ?

-10

\frac{2}{3}

6

10

26

Show answer & explanation

Answer: 6

Answer

The value of

k

is

6

.

The slope of side

DE

is calculated as

\frac{7 - (-2)}{4 - 1} = 3

. An altitude is perpendicular to the side it intersects, so the slope of the altitude is the negative reciprocal of

3

, which is

-\frac{1}{3}

. The line containing this altitude has a

y

-intercept of

8

, giving the equation

y = -\frac{1}{3}x + 8

. Since the vertex

F(6, k)

lies on this line, substituting

x = 6

and

y = k

into the equation yields

k = -\frac{1}{3}(6) + 8 = -2 + 8 = 6

. This matches the correct value of

6

.

Step-by-Step Solution

1

Calculate the slope of side

DE

using the coordinates of

D(1, -2)

and

E(4, 7)

.

The slope of

DE

is

3

.

The slope formula is

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

. Substituting the points gives

m_{DE} = \frac{7 - (-2)}{4 - 1} = \frac{9}{3} = 3

.

2

Determine the slope of the altitude line, which is perpendicular to side

DE

.

The slope of the altitude is

-\frac{1}{3}

.

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

3

is

-\frac{1}{3}

.

3

Write the equation of the line containing the altitude using its slope and the given

y

-intercept of

8

.

The equation of the altitude line is

y = -\frac{1}{3}x + 8

.

The slope-intercept form of a linear equation is

y = mx + b

, where

m

is the slope and

b

is the

y

-intercept.

4

Substitute the coordinates of point

F(6, k)

into the equation of the altitude to solve for

k

.

k = 6

.

Since vertex

F

lies on the line containing the altitude, its coordinates must satisfy the equation:

k = -\frac{1}{3}(6) + 8 = -2 + 8 = 6

.

Key Concept

Finding the equation of a line perpendicular to a given line segment and using it to find a missing coordinate of a point on that line.

Question 226Question →

In the standard $(x, y)$ coordinate plane, if a line passes through the point $(1, 3)$ and has an undefined slope, then the point $(1, -5)$ lies on this line.

True

False

Show answer & explanation

Answer: True

Answer

The statement is true because a line with an undefined slope is a vertical line, meaning all points on the line share the same x-coordinate. Since the line passes through

(1, 3)

, the x-coordinate of all points on the line is

1

, which includes the point

(1, -5)

.

A line with an undefined slope is a vertical line. Since it passes through

(1, 3)

, the x-coordinate of every point on this line must be

1

. The point

(1, -5)

has an x-coordinate of

1

, so it lies on this vertical line, making the statement true.

Step-by-Step Solution

1

Identify the geometric properties of a line with an undefined slope.

A line with an undefined slope is a vertical line.

The slope formula is

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

. The slope is undefined when the denominator, representing the change in

x

, is equal to zero. This occurs only for vertical lines.

2

Determine the equation of the vertical line that passes through the point

(1, 3)

.

The equation of the line is

x = 1

.

Since the line is vertical, all points on the line share the same x-coordinate as the given point

(1, 3)

.

3

Verify if the point

(1, -5)

satisfies the equation of the line.

The x-coordinate of the point

(1, -5)

is

1

, which satisfies the equation

x = 1

.

A point lies on a line if its coordinates satisfy the equation of the line.

Key Concept

Slope of a Line

Question 227Question →

A line, $L_1$ , contains the points $(2, 5)$ and $(6, -3)$ in a coordinate plane. Another line, $L_2$ , is perpendicular to $L_1$ and is defined by the equation $ax + 2y = 7$ . What is the value of $a$ ?

-4

-1

1

2

4

Show answer & explanation

Answer: -1

Answer

The value of

a

is

-1

.

The slope of line

L_1

is

-2

. A line perpendicular to it must have a slope that is the negative reciprocal, which is

\frac{1}{2}

. Rewriting

ax + 2y = 7

in slope-intercept form gives

y = -\frac{a}{2}x + \frac{7}{2}

, where the slope is

-\frac{a}{2}

. Setting this slope equal to

\frac{1}{2}

gives

-\frac{a}{2} = \frac{1}{2}

, which simplifies to

a = -1

.

Step-by-Step Solution

1

Find the slope of line

L_1

using the points

(2, 5)

and

(6, -3)

.

m_1 = \frac{-3 - 5}{6 - 2} = \frac{-8}{4} = -2

.

The slope formula is the change in y divided by the change in x.

2

Determine the perpendicular slope for

L_2

.

m_2 = -\frac{1}{m_1} = \frac{1}{2}

.

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

3

Rewrite the equation of

L_2

,

ax + 2y = 7

, in slope-intercept form to find its slope expression.

2y = -ax + 7 \Rightarrow y = -\frac{a}{2}x + \frac{7}{2}

. The slope is

-\frac{a}{2}

.

Slope-intercept form

y = mx + b

allows direct identification of the slope coefficient.

4

Equate the slope of

L_2

to the perpendicular slope and solve for

a

.

-\frac{a}{2} = \frac{1}{2} \Rightarrow a = -1

.

Solving the equation gives the value of

a

required for the lines to be perpendicular.

Key Concept

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

Question 228Question →

In the standard $(x, y)$ coordinate plane, a line segment has endpoints at $P(2, 1)$ and $Q(8, 5)$ . Line $L$ is parallel to segment $PQ$ and passes through the point $(3, -2)$ . If the point $(9, y)$ also lies on line $L$ , what is the value of $y$ ?

Show answer & explanation

Answer: 2

Answer

The value of

y

is

2

.

The slope of segment

PQ

is

\frac{5 - 1}{8 - 2} = \frac{2}{3}

. Since line

L

is parallel to segment

PQ

, its slope is also

\frac{2}{3}

. The slope of line

L

through

(3, -2)

and

(9, y)

is given by

\frac{y - (-2)}{9 - 3} = \frac{y + 2}{6}

. Equating the two slopes yields

\frac{y + 2}{6} = \frac{2}{3}

, which simplifies to

y + 2 = 4

, so

y = 2

.

Step-by-Step Solution

1

Calculate the slope of segment

PQ

.

Slope of

PQ = \frac{2}{3}

Since parallel lines have equal slopes, finding the slope of the reference segment

PQ

is the first step in determining the slope of line

L

.

2

Set up the slope equation for line

L

using its parallel relationship to segment

PQ

.

\frac{y - (-2)}{9 - 3} = \frac{2}{3}

Because line

L

is parallel to segment

PQ

, its slope must also be

\frac{2}{3}

. The slope of line

L

is calculated using the points

(3, -2)

and

(9, y)

.

3

Solve the equation for

y

.

y = 2

Simplifying the numerator gives

y + 2

and the denominator gives

6

. Multiplying both sides of

\frac{y + 2}{6} = \frac{2}{3}

by

6

yields

y + 2 = 4

, which solves to

y = 2

.

Key Concept

Parallel lines have equal slopes in the coordinate plane.

Question 229Question →

In the standard $(x, y)$ coordinate plane, a right triangle has vertices $A(-2, -3)$ , $B(6, -3)$ , and $C(6, y)$ , where $y > 0$ . If the hypotenuse of the triangle has a length of 17 units, what is the value of $y$ ?

6

12

15

18

14

Show answer & explanation

Answer: 12

Answer

The correct value of

y

is

12

.

The horizontal leg

AB

has a length of

6 - (-2) = 8

units. By the Pythagorean theorem, the vertical leg

BC

has a length of

\sqrt{17^2 - 8^2} = 15

units. Since the vertex

B

is at

(6, -3)

and

C

is at

(6, y)

with

y > 0

, the vertical distance is

y - (-3) = 15

, which yields

y = 12

.

Step-by-Step Solution

1

Determine the length of the horizontal leg

AB

.

The length of

AB

is

8

units.

Since vertices

A(-2, -3)

and

B(6, -3)

share the same

y

-coordinate, the segment is horizontal. The length is the positive difference between their

x

-coordinates:

6 - (-2) = 8

.

2

Use the Pythagorean theorem to calculate the length of the vertical leg

BC

.

The length of

BC

is

15

units.

Since

AB

is horizontal and

BC

is vertical (vertices

B

and

C

share the same

x

-coordinate of

6

), the angle at

B

is a right angle. The hypotenuse is

AC = 17

. By the Pythagorean theorem,

AB^2 + BC^2 = AC^2 \implies 8^2 + BC^2 = 17^2 \implies 64 + BC^2 = 289 \implies BC^2 = 225 \implies BC = 15

.

3

Set up an equation using the coordinates to find

y

.

y = 12

The length of the vertical segment

BC

is the difference in

y

-coordinates:

|y - (-3)| = 15

, which simplifies to

y + 3 = 15

since

y > 0

. Solving for

y

yields

y = 12

.

Key Concept

Using coordinate differences and the Pythagorean theorem to determine unknown vertices of geometric figures in the coordinate plane.

Alternative Method

Alternatively, you can apply the distance formula directly between the vertices

A(-2, -3)

and

C(6, y)

with a distance of 17 units:

\sqrt{(6 - (-2))^2 + (y - (-3))^2} = 17 \implies 8^2 + (y + 3)^2 = 289 \implies (y + 3)^2 = 225

, which yields

y + 3 = 15 \implies y = 12

since

y > 0

.

Estimated Time:1m 30s

Question 230Question →

In the standard $(x, y)$ coordinate plane, three vertices of a rectangle are $A(-4, 1)$ , $B(2, 9)$ , and $C(6, 6)$ . What is the area of the rectangle, in square units?

Show answer & explanation

Answer: 50

Answer

50

The length of side

AB

is calculated as

\sqrt{(2 - (-4))^2 + (9 - 1)^2} = \sqrt{36 + 64} = 10

. The length of the adjacent side

BC

is calculated as

\sqrt{(6 - 2)^2 + (6 - 9)^2} = \sqrt{16 + 9} = 5

. The area of the rectangle is the product of these two perpendicular side lengths, which is

10 \times 5 = 50

.

Step-by-Step Solution

1

Calculate the length of side

AB

using the distance formula.

AB = 10

To find one of the side lengths of the rectangle.

2

Calculate the length of side

BC

using the distance formula.

BC = 5

To find the adjacent side length of the rectangle.

3

Multiply the two adjacent side lengths to find the area of the rectangle.

50

The area of a rectangle is equal to the product of its length and width.

Key Concept

Calculating the area of a geometric figure on the coordinate plane by determining its side lengths using the distance formula.

Question 231Question →

A commercial drone starts at an altitude of $150$ meters and descends at a constant rate. After $12$ seconds of descent, its altitude is $114$ meters. If the altitude, $a$ , in meters, is modeled as a linear function of time, $t$ , in seconds, what is the slope of the line representing this relationship in the standard $(t, a)$ coordinate plane?

-36

-\frac{1}{3}

-3

\frac{1}{3}

3

Show answer & explanation

Answer: $-3$

Answer

The slope of the line representing this relationship is

-3

.

The slope of the line represents the rate of change of the altitude with respect to time. By taking the initial point

(0, 150)

and the point after

12

seconds

(12, 114)

, the change in altitude is

114 - 150 = -36

meters. Dividing this by the change in time of

12 - 0 = 12

seconds yields the correct slope of

-3

.

Step-by-Step Solution

1

Identify the coordinate points

(t, a)

from the given information.

The initial state corresponds to the point

(0, 150)

, and the state after

12

seconds corresponds to

(12, 114)

.

Setting up coordinate pairs allows the direct application of the slope formula.

2

Apply the slope formula

m = \frac{a_2 - a_1}{t_2 - t_1}

to find the rate of change.

m = \frac{114 - 150}{12 - 0}

The slope is the change in the vertical variable (altitude) divided by the change in the horizontal variable (time).

3

Calculate the difference and simplify the fraction.

m = \frac{-36}{12} = -3

Subtracting

150

from

114

gives the net change in altitude, which when divided by the duration of

12

seconds yields the rate of change per second.

Key Concept

The slope of a line represents its constant rate of change, defined as the ratio of the change in the dependent variable to the change in the independent variable.

Question 232Question →

In the standard $(x, y)$ coordinate plane, a parallelogram has vertices at $P(-2, -1)$ , $Q(4, -1)$ , $R(6, 4)$ , and $S(0, 4)$ . What is the area of parallelogram $PQRS$ , in square units?

15

18

24

30

40

Show answer & explanation

Answer: 30

Answer

30

The correct answer is 30. The area of a parallelogram is determined by multiplying its base by its perpendicular height. The base segment

PQ

is horizontal and extends from

x = -2

to

x = 4

, giving a length of

4 - (-2) = 6

units. The height is the vertical distance between the line containing the base

PQ

(

y = -1

) and the line containing the opposite side

SR

(

y = 4

). This distance is

4 - (-1) = 5

units. Multiplying the base of 6 units by the height of 5 units yields an area of 30 square units.

Step-by-Step Solution

1

Identify the base of the parallelogram by calculating the length of the horizontal side

PQ

.

PQ = 4 - (-2) = 6

units

The segment

PQ

lies on the horizontal line

y = -1

, so its length is the difference between the x-coordinates of its endpoints.

2

Identify the height of the parallelogram by calculating the vertical distance between the parallel horizontal sides

PQ

(on

y = -1

) and

SR

(on

y = 4

).

height = 4 - (-1) = 5

units

The height of a parallelogram is the perpendicular distance between its parallel bases.

3

Calculate the area of the parallelogram using the formula

Area = \text{base} \times \text{height}

.

Area = 6 \times 5 = 30

square units

Multiplying the base length by the vertical height gives the total area of the parallelogram.

Key Concept

Finding the area of a parallelogram on the coordinate plane using base and height calculations from coordinates.

Question 233Question →

In the standard $(x, y)$ coordinate plane, the vertices of a right triangle are $P(1, 2)$ , $Q(5, 5)$ , and $R(k, 9)$ . If the right angle of the triangle is at vertex $Q$ , what is the value of the constant $k$ ?

Show answer & explanation

Answer: 2

Answer

The value of

k

is

2

.

Because the right angle of the triangle is at vertex

Q

, segment

PQ

must be perpendicular to segment

QR

. The slope of

PQ

is

\frac{5 - 2}{5 - 1} = \frac{3}{4}

. The slope of a perpendicular line is the negative reciprocal, so the slope of

QR

must be

-\frac{4}{3}

. Expressing the slope of

QR

using the coordinates of

Q(5, 5)

and

R(k, 9)

gives

\frac{9 - 5}{k - 5} = \frac{4}{k - 5}

. Setting this equal to

-\frac{4}{3}

and solving for

k

yields

k - 5 = -3

, which means

k = 2

.

Step-by-Step Solution

1

Use the slope formula

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

to calculate the slope of the line segment

PQ

with endpoints

P(1, 2)

and

Q(5, 5)

.

m_{PQ} = \frac{5 - 2}{5 - 1} = \frac{3}{4}

This establishes the direction of the first leg of the right triangle.

2

Find the slope of segment

QR

. Because the right angle is at vertex

Q

, the segment

PQ

is perpendicular to segment

QR

.

m_{QR} = -\frac{4}{3}

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

m_1 \cdot m_2 = -1

).

3

Write the slope of segment

QR

in terms of

k

using coordinates

Q(5, 5)

and

R(k, 9)

.

m_{QR} = \frac{9 - 5}{k - 5} = \frac{4}{k - 5}

This sets up an equation to find the unknown coordinate value.

4

Equate the two expressions for the slope of

QR

and solve for

k

.

\frac{4}{k - 5} = -\frac{4}{3} \Rightarrow k - 5 = -3 \Rightarrow k = 2

Solving the rational equation yields the correct coordinate parameter.

Key Concept

Perpendicular lines in a coordinate plane have slopes that are negative reciprocals of each other.

Estimated Time:1m 30s

Question 234Question →

In the standard $(x, y)$ coordinate plane, a line passes through the points $(a, 2)$ and $(10, a - 1)$ . If the slope of the line is $-\frac{1}{3}$ , what is the value of $a$ ?

Show answer & explanation

Answer: -0.5

Answer

The value of

a

is

-0.5

(or

-\frac{1}{2}

)

Applying the slope formula

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

to the points

(a, 2)

and

(10, a - 1)

with slope

-\frac{1}{3}

yields the equation

-\frac{1}{3} = \frac{a - 3}{10 - a}

. Solving this linear equation correctly yields

a = -0.5

.

Step-by-Step Solution

1

Apply the slope formula using the given coordinates.

-\frac{1}{3} = \frac{(a - 1) - 2}{10 - a}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is defined as

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

.

2

Simplify the numerator.

-\frac{1}{3} = \frac{a - 3}{10 - a}

Subtracting

2

from

a - 1

simplifies the numerator to

a - 3

.

3

Cross-multiply to solve the rational equation.

-1(10 - a) = 3(a - 3)

Multiplying both sides by the denominators eliminates the fractions.

4

Solve the linear equation for

a

.

a = -0.5

Distributing on both sides gives

-10 + a = 3a - 9

. Rearranging terms yields

2a = -1

, which simplifies to

a = -0.5

.

Key Concept

Slope of a Line

Question 235Question →

Three vertices of a parallelogram are $A(1, 3)$ , $B(-2, -1)$ , and $C(4, -1)$ in the standard $(x, y)$ coordinate plane. If the fourth vertex, $D$ , is located in the fourth quadrant, what is the $y$ -coordinate of $D$ ?

Show answer & explanation

Answer: -5

Answer

The

y

-coordinate of the fourth vertex of the parallelogram in the fourth quadrant is

-5

.

By applying the midpoint formula to the diagonals of the three possible parallelograms formed by the vertices

A(1, 3)

,

B(-2, -1)

, and

C(4, -1)

, we find the candidate points for the fourth vertex

D

to be

(7, 3)

,

(-5, 3)

, and

(1, -5)

. The point

(1, -5)

is the only candidate that lies in the fourth quadrant, where

x > 0

and

y < 0

. Therefore, the

y

-coordinate of the fourth vertex is

-5

.

Step-by-Step Solution

1

Set up equations based on the property that the diagonals of a parallelogram bisect each other (have the same midpoint).

Three possible configurations of diagonals lead to three potential sets of coordinates for the fourth vertex

D(x, y)

:

(7, 3)

,

(-5, 3)

, and

(1, -5)

.

Three points in a coordinate plane can form three distinct parallelograms depending on which pairs are connected as diagonals.

2

Identify the signs of the coordinates for each potential vertex to determine which quadrant it lies in.

The point

(7, 3)

is in Quadrant I (

x > 0, y > 0

). The point

(-5, 3)

is in Quadrant II (

x < 0, y > 0

). The point

(1, -5)

is in Quadrant IV (

x > 0, y < 0

).

The fourth quadrant is defined by positive

x

-values and negative

y

-values.

3

Select the correct coordinate value corresponding to the question's requirement.

The

y

-coordinate of the vertex in the fourth quadrant,

(1, -5)

, is

-5

.

The question asks specifically for the

y

-coordinate of the fourth vertex

D

.

Key Concept

Finding a missing vertex of a parallelogram on the coordinate plane using midpoint properties

Question 236Question →

In the standard $(x, y)$ coordinate plane, line $L_1$ passes through the points $(1, 3)$ and $(4, 8)$ . Line $L_2$ is perpendicular to line $L_1$ . If line $L_2$ passes through the points $(5, k)$ and $(10, 1)$ , what is the value of $k$ ?

Show answer & explanation

Answer: 4

Answer

The value of

k

is

4

.

The correct answer is

4

. First, determine the slope of line

L_1

using the points

(1, 3)

and

(4, 8)

, which is

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

. Because line

L_2

is perpendicular to

L_1

, its slope must be the negative reciprocal of

\frac{5}{3}

, which is

-\frac{3}{5}

. Next, set up the slope equation for

L_2

with the points

(5, k)

and

(10, 1)

, yielding

\frac{1 - k}{10 - 5} = -\frac{3}{5}

. Simplifying the equation gives

\frac{1 - k}{5} = -\frac{3}{5}

, which reduces to

1 - k = -3

. Solving for

k

gives

k = 4

.

Step-by-Step Solution

1

Calculate the slope of line

L_1

using the coordinates of the two given points,

(1, 3)

and

(4, 8)

.

The slope of line

L_1

is

\frac{5}{3}

.

Using the slope formula

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

, we find

m_1 = \frac{8 - 3}{4 - 1} = \frac{5}{3}

.

2

Determine the slope of line

L_2

based on the perpendicular relationship between

L_1

and

L_2

.

The slope of line

L_2

is

-\frac{3}{5}

.

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

\frac{5}{3}

is

-\frac{3}{5}

.

3

Use the coordinates

(5, k)

and

(10, 1)

on line

L_2

to write an expression for its slope, set it equal to

-\frac{3}{5}

, and solve for

k

.

k = 4

The slope expression is

\frac{1 - k}{10 - 5} = \frac{1 - k}{5}

. Setting this equal to the perpendicular slope gives

\frac{1 - k}{5} = -\frac{3}{5}

. Multiplying both sides by

5

results in

1 - k = -3

. Adding

k

to both sides and adding

3

to both sides yields

k = 4

.

Key Concept

The slopes of perpendicular lines in a coordinate plane are negative reciprocals of each other, meaning their product is

-1

(

m_1 \cdot m_2 = -1

).

Estimated Time:1m 15s

Question 237Question →

In the standard $(x, y)$ coordinate plane, what is the $y$ -intercept of the perpendicular bisector of the line segment with endpoints $(0, 1)$ and $(2, 5)$ ?

\frac{4}{3}

\frac{5}{2}

\frac{7}{2}

5

-\frac{5}{2}

Show answer & explanation

Answer: $\frac{7}{2}$

Answer

\frac{7}{2}

To find the perpendicular bisector, we first locate the midpoint of the segment with endpoints

(0, 1)

and

(2, 5)

, which is

\left(\frac{0 + 2}{2}, \frac{1 + 5}{2}\right) = (1, 3)

. The slope of this segment is

m = \frac{5 - 1}{2 - 0} = 2

. The slope of the perpendicular bisector is the negative reciprocal,

-\frac{1}{2}

. Using the point-slope form with the midpoint

(1, 3)

and slope

-\frac{1}{2}

, the equation of the perpendicular bisector is

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

. Setting

x = 0

to find the

y

-intercept gives

y = 3 + \frac{1}{2} = \frac{7}{2}

.

Step-by-Step Solution

1

Find the midpoint of the line segment with endpoints

(0, 1)

and

(2, 5)

.

The midpoint is

M = \left(\frac{0 + 2}{2}, \frac{1 + 5}{2}\right) = (1, 3)

.

A perpendicular bisector must pass through the midpoint of the segment it bisects.

2

Calculate the slope of the original line segment.

The slope is

m = \frac{5 - 1}{2 - 0} = \frac{4}{2} = 2

.

The slope of the segment is needed to find the slope of the line perpendicular to it.

3

Determine the slope of the perpendicular bisector.

The perpendicular slope is

m_{\perp} = -\frac{1}{m} = -\frac{1}{2}

.

The slope of a perpendicular line is the negative reciprocal of the original line's slope.

4

Write the equation of the perpendicular bisector and find the

y

-intercept.

Using the point-slope form with point

(1, 3)

and slope

-\frac{1}{2}

gives the equation

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

. Setting

x = 0

to find the

y

-intercept yields

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

, which simplifies to

y = 3 + \frac{1}{2} = \frac{7}{2}

.

The

y

-intercept is the value of the function when

x = 0

.

Key Concept

Perpendicular Bisectors in the Coordinate Plane

Estimated Time:1m 30s

Question 238Question →

On a coordinate map of a state park, a straight hiking trail begins at a campsite located at $(2, -3)$ and ends at a lookout point. A hiker walks along the trail at a constant pace, and after $3$ hours, reaches a trail marker located at $(11, 9)$ . What is the slope of the line on the coordinate map that represents this straight trail?

\frac{4}{3}

\frac{3}{4}

\frac{2}{3}

-\frac{4}{3}

5

Show answer & explanation

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

Answer

\frac{4}{3}

To find the slope of the line representing the trail, we identify the two points through which the line passes:

(2, -3)

and

(11, 9)

. Applying the slope formula

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

, we get

m = \frac{9 - (-3)}{11 - 2} = \frac{12}{9}

. Simplifying the fraction by dividing the numerator and denominator by

3

yields the correct slope of

\frac{4}{3}

.

Step-by-Step Solution

1

Identify the coordinates of the two points on the line representing the trail.

The two points are the campsite at

(2, -3)

and the trail marker at

(11, 9)

.

To find the slope of a straight line, we need the coordinates of any two points that lie on the line.

2

Set up the slope formula.

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

The slope of a line is defined as the change in the

y

-coordinates (rise) divided by the change in the

x

-coordinates (run).

3

Substitute the coordinate values into the formula and simplify.

m = \frac{9 - (-3)}{11 - 2} = \frac{9 + 3}{9} = \frac{12}{9}

Substituting the coordinates correctly handles the subtraction of the negative coordinate.

4

Reduce the fraction to simplest form.

m = \frac{4}{3}

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

3

, yields the final simplified slope value.

Key Concept

Calculating the slope of a line given two points on a coordinate plane

Question 239Question →

An isosceles trapezoid $ABCD$ has vertices $A(-4, -1)$ , $B(6, -1)$ , and $C(3, 3)$ in the standard $(x, y)$ coordinate plane. If the base $AB$ is parallel to the x-axis, which of the following represents the coordinates of vertex $D$ ?

(-7, 3)

(-5, 3)

(-3, 3)

(-1, 3)

(1, 3)

Show answer & explanation

Answer: $(-1, 3)$

Answer

(-1, 3)

The correct answer is the coordinate pair representing

(-1, 3)

. Since the bases are parallel to the x-axis, they are horizontal, meaning

D

must share the same y-coordinate as

C

, which is 3. The vertical line of symmetry passes through the midpoint of the segment

AB

, which is at

x = 1

. The vertex

C(3, 3)

is 2 units to the right of this line of symmetry, so the vertex

D

must be 2 units to the left of the line of symmetry, giving an x-coordinate of

-1

.

Step-by-Step Solution

1

Determine the orientation of the trapezoid and the y-coordinate of the missing vertex.

The base

AB

lies on the horizontal line

y = -1

. Because the bases of a trapezoid are parallel, the second base

CD

must also be horizontal and lie on the line

y = 3

. Therefore, the y-coordinate of vertex

D

is

3

.

Since the trapezoid's bases are parallel to the x-axis, they are horizontal lines, meaning vertices on the same base share the same y-coordinate.

2

Find the equation of the line of symmetry of the isosceles trapezoid.

The midpoint of the base

AB

has an x-coordinate of

\frac{-4 + 6}{2} = 1

. The vertical line of symmetry is

x = 1

.

An isosceles trapezoid is symmetric with respect to the perpendicular bisector of its bases. For horizontal bases, this is a vertical line passing through the midpoint of the base.

3

Use the line of symmetry to find the x-coordinate of vertex

D

.

The x-coordinate of

C(3, 3)

is 3, which is

3 - 1 = 2

units to the right of the line of symmetry

x = 1

. Thus, the x-coordinate of vertex

D

must be 2 units to the left of the line of symmetry:

1 - 2 = -1

. Combining this with the y-coordinate gives

D(-1, 3)

.

Symmetry requires that corresponding vertices on the opposite sides of the line of symmetry are equidistant from it.

Key Concept

Using symmetry and coordinate geometry properties of an isosceles trapezoid to determine the coordinates of a missing vertex.

Estimated Time:1m 30s

Question 240Question →

In the standard $(x, y)$ coordinate plane, if a line $L$ is perpendicular to the line passing through the points $(2, 5)$ and $(2, -3)$ , then the slope of line $L$ is $0$ .

True

False

Show answer & explanation

Answer: True

Answer

The statement is true because the line passing through the points

(2, 5)

and

(2, -3)

is vertical, and any line perpendicular to a vertical line is horizontal, which has a slope of

0

.

The line passing through

(2, 5)

and

(2, -3)

has identical x-coordinates, which defines a vertical line. Any line perpendicular to a vertical line is horizontal, and horizontal lines have a slope of

0

.

Step-by-Step Solution

1

Identify the orientation of the line passing through the points

(2, 5)

and

(2, -3)

.

Since both coordinates share the same x-value (

x = 2

), the line is vertical.

In the standard coordinate plane, a line connecting points with identical x-coordinates is vertical and has an undefined slope.

2

Determine the orientation of line

L

, which is perpendicular to this vertical line.

Line

L

must be a horizontal line.

By geometric definition, a line perpendicular to a vertical line in the coordinate plane is horizontal.

3

Determine the slope of the horizontal line

L

.

The slope of line

L

is

0

.

A horizontal line has no vertical change (rise =

0

) as the horizontal position changes, resulting in a slope of

0

.

Key Concept

The relationship between vertical and horizontal lines in the coordinate plane, specifically that perpendicular lines to vertical lines are horizontal and have a slope of

0

.

Alternative Method

Write the equation of the line passing through

(2, 5)

and

(2, -3)

, which is

x = 2

. Any line perpendicular to

x = 2

must be in the form

y = c

where

c

is a constant. The slope of any line in the form

y = c

is

0

.

Estimated Time:1m 0s

Coordinate Geometry

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution