All practice questions

If $a > 1.5$ is a real number that satisfies the equation $4^a - 3(2^{a+1}) + 8 = 0$ , what is the value of $a$ ?

Show answer & explanation

Answer: 2

Answer

The correct answer is 2.

Substituting

u = 2^a

turns the equation into

u^2 - 6u + 8 = 0

. Factoring gives

(u-2)(u-4)=0

, meaning

u=2

or

u=4

. Reversing the substitution gives

2^a = 2 \implies a=1

and

2^a = 4 \implies a=2

. Since

a > 1.5

, the correct value is 2.

Step-by-Step Solution

1

Express the equation in terms of base 2.

(2^a)^2 - 6(2^a) + 8 = 0

Since 4 is

2^2

and

2^{a+1} = 2 \cdot 2^a

, expressing all terms in base 2 allows for algebraic substitution.

2

Substitute

u = 2^a

to form a quadratic equation.

u^2 - 6u + 8 = 0

Substitution simplifies the exponential equation into a standard quadratic form.

3

Factor the quadratic equation.

(u-2)(u-4) = 0

Factoring allows us to find the roots of the quadratic equation.

4

Solve for the variable

a

.

a = 1

or

a = 2

Solving

2^a = 2

yields

a = 1

, and solving

2^a = 4

yields

a = 2

.

5

Apply the given constraint on

a

.

a = 2

The problem states that

a > 1.5

, so

a = 1

is discarded and

a = 2

is the correct value.

Key Concept

Solving exponential equations using quadratic substitution

Alternative Method

Instead of using substitution, test values for

a

. Since

a > 1.5

, testing small integer values starting with

a=2

shows

4^2 - 3(2^3) + 8 = 16 - 24 + 8 = 0

, validating that

a=2

is the solution.

Estimated Time:1m 30s

Question 2222Question →

A community theater group sells tickets for a play. Student tickets cost $12$ each, and adult tickets cost $18$ each. For a Friday night performance, the theater sold a total of $150$ tickets. The revenue from the adult tickets sold was $360$ more than the revenue from the student tickets sold. What is the absolute difference between the number of student tickets and the number of adult tickets sold for this performance?

78

72

54

30

6

Show answer & explanation

Answer: 6

Answer

6

The correct answer is 6. By setting the number of student tickets to

s

, the number of adult tickets is

150 - s

. Setting up the revenue relationship gives

18(150 - s) = 12s + 360

. Solving this equation gives

s = 78

student tickets and

150 - 78 = 72

adult tickets. The absolute difference between the two values is

78 - 72 = 6

.

Step-by-Step Solution

1

Define variables for the unknown quantities.

Let

s

be the number of student tickets sold. Since the total number of tickets sold was

150

, the number of adult tickets sold is

150 - s

.

Establishing single-variable representations allows us to model the entire situation using one equation.

2

Write an equation representing the revenue relationship.

The revenue from student tickets is

12s

and the revenue from adult tickets is

18(150 - s)

. The equation is:

18(150 - s) = 12s + 360

.

The problem states that the adult ticket revenue was

360

more than the student ticket revenue.

3

Solve the linear equation for

s

.

2700 - 18s = 12s + 360 \implies 2340 = 30s \implies s = 78

.

Solving the equation gives the exact number of student tickets sold.

4

Calculate the number of adult tickets sold.

Adult tickets sold =

150 - 78 = 72

.

Subtracting the number of student tickets from the total ticket count yields the number of adult tickets.

5

Find the absolute difference between the two quantities.

|78 - 72| = 6

.

The question asks for the absolute difference between the number of student tickets and the number of adult tickets sold.

Key Concept

Translating and Solving Algebraic Word Problems

Estimated Time:1m 30s

Question 2223Question →

A square $PQRS$ has vertices at $P(2, 2)$ , $Q(6, 2)$ , $R(6, 6)$ , and $S(2, 6)$ in the coordinate plane. The square is first rotated $90^\circ$ counterclockwise about the origin, and then translated such that the final image of vertex $R$ is located at $(1, 5)$ . What are the coordinates of the final image of vertex $P$ after this sequence of transformations?

(-3, 9)

(5, 1)

(-9, 3)

(5, 9)

(-3, 1)

Show answer & explanation

Answer: $(5, 1)$

Answer

(5, 1)

To find the coordinates of the final image of vertex P, we perform the transformations step-by-step. First, a

90^\circ

counterclockwise rotation about the origin maps any point

(x, y)

to

(-y, x)

. Under this rotation,

P(2, 2)

maps to

P'(-2, 2)

and

R(6, 6)

maps to

R'(-6, 6)

. Second, we find the translation vector by comparing the rotated vertex

R'(-6, 6)

to its final position

R''(1, 5)

. The change in the x-coordinate is

1 - (-6) = 7

, and the change in the y-coordinate is

5 - 6 = -1

. This represents a translation of

7

units right and

1

unit down, or the vector

(7, -1)

. Finally, applying this translation to the intermediate point

P'(-2, 2)

gives

P''(-2 + 7, 2 - 1) = (5, 1)

.

Step-by-Step Solution

1

Apply the

90^\circ

counterclockwise rotation about the origin to the vertices

P(2, 2)

and

R(6, 6)

.

Using the rotation rule

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

, vertex

P(2, 2)

maps to

P'(-2, 2)

and vertex

R(6, 6)

maps to

R'(-6, 6)

.

This establishes the intermediate positions of the vertices before the translation is applied.

2

Determine the translation vector

(h, k)

that maps the intermediate point

R'(-6, 6)

to the final point

R''(1, 5)

.

Solving the equations

-6 + h = 1

and

6 + k = 5

gives

h = 7

and

k = -1

. The translation vector is

(7, -1)

.

The translation shifts all points in the plane by the same distance and direction.

3

Apply the translation vector

(7, -1)

to the intermediate vertex

P'(-2, 2)

to find the final image

P''

.

P'' = (-2 + 7, 2 - 1) = (5, 1)

.

Applying the same translation to the rotated position of vertex P yields its final coordinates.

Key Concept

Composite transformations in the coordinate plane

Estimated Time:2m 0s

Question 2224Question →

If $w$ and $z$ are positive real numbers and $k$ is a constant such that the expression $\frac{(w^{3/2} z^{-1})^4}{(w z^k)^2}$ is equivalent to $w^4 z^6$ , what is the value of $k$ ?

Show answer & explanation

Answer: -5

Answer

The value of

k

is

-5

.

Applying the exponent rules, the expression simplifies to

w^4 z^{-4-2k}

. Equating the exponent of

z

to the exponent in the target expression

w^4 z^6

gives

-4-2k = 6

, which solves to

k = -5

.

Step-by-Step Solution

1

Apply the power of a product rule to the expression in the numerator

(w^{3/2} z^{-1})^4 = w^6 z^{-4}

When raising a product to a power, multiply the exponent of each factor by the outer exponent:

(x^a y^b)^c = x^{ac} y^{bc}

.

2

Apply the power of a product rule to the expression in the denominator

(w z^k)^2 = w^2 z^{2k}

Multiply the exponent of each factor in the denominator by

2

.

3

Divide the numerator by the denominator using the quotient rule for exponents

\frac{w^6 z^{-4}}{w^2 z^{2k}} = w^4 z^{-4-2k}

When dividing terms with the same base, subtract the exponent in the denominator from the exponent in the numerator:

\frac{x^a}{x^b} = x^{a-b}

.

4

Equate the exponent of

z

in the simplified expression to the exponent of

z

in the target expression

-4 - 2k = 6

Since the simplified expression is equivalent to

w^4 z^6

, the exponents of the corresponding variable bases must be equal.

5

Solve the linear equation for

k

k = -5

Add

4

to both sides of the equation to get

-2k = 10

, then divide both sides by

-2

.

Key Concept

Properties of Exponents in Algebraic Expressions

Question 2225Question →

Two volunteer teams, Team A and Team B, are packing food boxes for a local shelter. Team A packs at a constant rate of $15$ boxes per hour and begins packing at 8:00 a.m. Team B packs at a constant rate of $20$ boxes per hour and begins packing at 9:30 a.m. If both teams pack continuously at their respective rates, how many hours after Team A begins will both teams have packed the same total number of boxes?

Show answer & explanation

Answer: 6

Answer

Both teams will have packed the same total number of boxes

6

hours after Team A begins.

The correct answer is

6

. Let

x

be the number of hours Team A packs. Team B begins 1.5 hours later, so Team B packs for

x - 1.5

hours. Setting their total boxes packed equal gives

15x = 20(x - 1.5)

. Distributing yields

15x = 20x - 30

. Subtracting

20x

from both sides gives

-5x = -30

, which simplifies to

x = 6

.

Step-by-Step Solution

1

Define the variable for time and identify the time difference.

Let

x

be the number of hours Team A packs. Since Team B starts 1 hour and 30 minutes (which is

1.5

hours) later, Team B's time is represented as

x - 1.5

hours.

Establishing correct algebraic representations for time is necessary to set up the equation.

2

Set up an equation equating the total boxes packed by both teams.

The equation is

15x = 20(x - 1.5)

.

Since both teams pack a constant number of boxes per hour, multiplying their rate by their active time gives the total boxes packed.

3

Solve the linear equation for

x

.

Distribute the 20:

15x = 20x - 30

. Subtract

20x

from both sides:

-5x = -30

. Divide by

-5

:

x = 6

.

Solving the equation gives the number of hours after Team A starts when their packed boxes are equal.

Key Concept

Translating real-world rates and time shifts into linear equations and solving them.

Question 2226Question →

In the standard $(x, y)$ coordinate plane, a line segment has midpoint $M(4, 1)$ and one endpoint $P(x, 5)$ . If the total length of the line segment is $10$ units, and $x < 4$ , what is the value of $x$ ?

Show answer & explanation

Answer: 1

Answer

1

The midpoint

M

divides the line segment into two equal segments, meaning the distance from endpoint

P

to midpoint

M

is half of the total length of the segment,

\frac{10}{2} = 5

units. Applying the distance formula between

P(x, 5)

and

M(4, 1)

gives

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

. Squaring both sides yields

(x - 4)^2 + 16 = 25

, which simplifies to

(x - 4)^2 = 9

. Solving for

x

gives

x - 4 = 3

or

x - 4 = -3

, meaning

x = 7

or

x = 1

. The problem specifies that

x < 4

, so the only valid value is

1

.

Step-by-Step Solution

1

Find the distance from the endpoint to the midpoint.

The distance

PM

is

5

units.

The midpoint divides the line segment into two equal parts, so the distance from any endpoint to the midpoint is half of the total length of the segment:

\frac{10}{2} = 5

.

2

Set up the distance formula equation for the segment

PM

.

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

Using the coordinates of

P(x, 5)

and

M(4, 1)

with the distance formula

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

.

3

Simplify the equation and square both sides.

(x - 4)^2 + 16 = 25

Calculating

(5 - 1)^2 = 16

and squaring both sides of the equation to eliminate the radical.

4

Isolate the squared term and solve for

x

.

x = 7

or

x = 1

Subtracting

16

from both sides gives

(x - 4)^2 = 9

. Taking the square root gives

x - 4 = 3

or

x - 4 = -3

, yielding

x = 7

or

x = 1

.

5

Apply the constraint

x < 4

.

x = 1

Since the question states that

x < 4

, we choose

x = 1

instead of

x = 7

.

Key Concept

The midpoint divides a segment into two segments of equal length, and the distance between two points

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

Question 2227Question →

In the standard $(x, y)$ coordinate plane, a line passes through the point $(3, 1)$ and has a slope of $-\frac{2}{3}$ . If the line intersects the $x$ -axis at the point $(p, 0)$ , what is the value of $p$ ?

Show answer & explanation

Answer: 4.5

Answer

The value of

p

is

4.5

.

The line equation is

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

. By substituting

y = 0

for the

x

-intercept, the equation becomes

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

. Multiplying both sides by

-3

gives

3 = 2(p - 3)

, which simplifies to

2p = 9

, resulting in

p = 4.5

.

Step-by-Step Solution

1

Write the point-slope equation of the line.

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

We use the point-slope form

y - y_1 = m(x - x_1)

with the point

(3, 1)

and slope

-\frac{2}{3}

.

2

Substitute the point

(p, 0)

into the line's equation.

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

The

x

-intercept

(p, 0)

lies on the line, so its coordinates must satisfy the equation of the line.

3

Solve for the variable

p

.

p = 4.5

Multiply by

-3

to get

3 = 2(p - 3)

, add

6

to both sides to get

2p = 9

, and divide by

2

.

Key Concept

Linear equations and graphing, specifically point-slope form and x-intercept calculation.

Question 2228Question →

An ellipse in the standard $(x, y)$ coordinate plane is defined by the equation $\frac{(x-4)^2}{169} + \frac{(y+3)^2}{144} = 1$ . What is the distance between the two foci of this ellipse?

Show answer & explanation

Answer: 10

Answer

The distance between the two foci of the ellipse is 10.

By comparing the given equation to the standard form of an ellipse, we find

a^2 = 169

and

b^2 = 144

. The distance from the center to each focus,

c

, is given by

c = \sqrt{a^2 - b^2} = \sqrt{169 - 144} = \sqrt{25} = 5

. The total distance between the two foci is

2c = 2(5) = 10

.

Step-by-Step Solution

1

Identify

a^2

and

b^2

from the given equation of the ellipse.

a^2 = 169

and

b^2 = 144

The standard equation of a horizontal ellipse is

\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1

, where

a^2

is the larger denominator.

2

Calculate the value of

c

, the distance from the center to a focus.

c = 5

For an ellipse, the focal distance

c

is related to the semi-major axis

a

and semi-minor axis

b

by the equation

c^2 = a^2 - b^2

.

3

Calculate the distance between the two foci, which is

2c

.

10

The distance between the two foci of an ellipse is twice the distance from the center to each focus (

2c

).

Key Concept

Focal distance of an ellipse

Estimated Time:1m 15s

Question 2229Question →

The measures of the interior angles of a triangle are in the ratio $2:3:5$ . What is the measure, in degrees, of the largest exterior angle of this triangle?

90^\circ

108^\circ

126^\circ

144^\circ

150^\circ

Show answer & explanation

Answer: $144^\circ$

Answer

144^\circ

The sum of the interior angles of a triangle is

180^\circ

. Given the ratio

2:3:5

, the angles are

2x

,

3x

, and

5x

. Adding these gives

10x = 180^\circ

, which solves to

x = 18^\circ

. The smallest interior angle is

2(18^\circ) = 36^\circ

. Since an exterior angle is supplementary to its adjacent interior angle, the largest exterior angle is the supplement of the smallest interior angle:

180^\circ - 36^\circ = 144^\circ

.

Step-by-Step Solution

1

Represent the three interior angles of the triangle in terms of a variable

x

based on the given ratio.

The angles can be expressed as

2x

,

3x

, and

5x

.

A ratio of

2:3:5

means the angles are multiples of a common factor

x

.

2

Set up and solve an equation for

x

using the triangle angle sum theorem.

2x + 3x + 5x = 180^\circ \implies 10x = 180^\circ \implies x = 18^\circ

.

The sum of the interior angles of any triangle is always

180^\circ

.

3

Determine the measures of all three interior angles.

The angles are

2(18^\circ) = 36^\circ

,

3(18^\circ) = 54^\circ

, and

5(18^\circ) = 90^\circ

.

This identifies the individual interior angle measures to find the smallest one.

4

Calculate the measure of the largest exterior angle by finding the supplement of the smallest interior angle.

180^\circ - 36^\circ = 144^\circ

.

An interior angle and its adjacent exterior angle form a linear pair and sum to

180^\circ

. To maximize the exterior angle, we must subtract the smallest interior angle.

Key Concept

Interior and exterior angle relationships in triangles and ratio partitioning

Alternative Method

By the Exterior Angle Theorem, an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. To find the largest exterior angle, sum the two largest interior angles. The two largest interior angles in ratio units are

3

parts and

5

parts, summing to

8

parts. Since the total sum of the interior angles is

10

parts (

180^\circ

), each part is

18^\circ

. Thus, the largest exterior angle is

8 \times 18^\circ = 144^\circ

.

Estimated Time:1m 0s

Question 2230Question →

What is the complete solution set, expressed in interval notation, for the inequality $\left| \frac{2-3t}{4} \right| > 2$ ?

(-\infty, -2)

\left(-2, \frac{10}{3}\right)

(-\infty, -8) \cup (8, \infty)

(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)

\left(-\infty, -\frac{10}{3}\right) \cup (2, \infty)

Show answer & explanation

Answer: $(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)$

Answer

The complete solution set is

(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)

The correct answer shows

(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)

because solving the absolute value inequality

|2-3t| > 8

requires splitting it into two inequalities:

2-3t > 8

and

2-3t < -8

. Solving the first yields

t < -2

after reversing the inequality sign when dividing by

-3

. Solving the second yields

t > \frac{10}{3}

after also reversing the inequality sign. The union of these two intervals is

(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)

.

Step-by-Step Solution

1

Multiply both sides of the inequality by

4

to isolate the absolute value term.

|2-3t| > 8

Eliminating the denominator simplifies the absolute value expression.

2

Split the absolute value inequality into two separate linear inequalities representing the positive and negative cases.

2-3t > 8

or

2-3t < -8

An absolute value greater than a positive number

c

is equivalent to the expression being greater than

c

or less than

-c

.

3

Solve the first inequality: subtract

2

from both sides, then divide by

-3

and reverse the inequality sign.

-3t > 6 \implies t < -2

Dividing an inequality by a negative number requires flipping the inequality sign.

4

Solve the second inequality: subtract

2

from both sides, then divide by

-3

and reverse the inequality sign.

-3t < -10 \implies t > \frac{10}{3}

Dividing an inequality by a negative number requires flipping the inequality sign.

5

Combine the two solutions using interval notation.

(-\infty, -2) \cup \left(\frac{10}{3}, \infty\right)

The union of the two intervals represents the complete set of values that satisfy either inequality.

Key Concept

Solving absolute value inequalities of the form

|ax+b| > c

by splitting them into two cases and reversing the inequality sign when dividing by a negative number.

Estimated Time:1m 30s

Question 2231Question →

In the standard $(x, y)$ coordinate plane, a line with a non-zero slope $m$ passes through the point $(8, 2)$ . If the product of the line's $x$ -intercept and its $y$ -intercept is $64$ , what is the value of $m$ ?

Show answer & explanation

Answer: -0.25

Answer

The value of the slope

m

is

-0.25

(or

-\frac{1}{4}

).

By writing the line as

y = mx + b

, the given point

(8, 2)

establishes that

b = 2 - 8m

. Since the

x

-intercept is

-\frac{b}{m}

, the product of the intercepts is

-\frac{b^2}{m}

. Setting this product to

64

gives

-\frac{(2-8m)^2}{m} = 64

, which simplifies to the quadratic equation

16m^2 + 8m + 1 = 0

. Factoring this perfect square trinomial gives

(4m + 1)^2 = 0

, which yields the unique solution

m = -0.25

.

Step-by-Step Solution

1

Express the

y

-intercept in terms of

m

.

b = 2 - 8m

The line equation is

y = mx + b

and it passes through

(8, 2)

, so

2 = 8(m) + b

.

2

Express the

x

-intercept in terms of

m

.

x = -\frac{b}{m}

The

x

-intercept is the value of

x

when

y = 0

in the line equation

y = mx + b

.

3

Write the product of the intercepts and set it equal to

64

.

b^2 = -64m

The product of the intercepts is

b \cdot \left(-\frac{b}{m}\right) = -\frac{b^2}{m} = 64

.

4

Substitute

b = 2 - 8m

into the product equation and simplify to a quadratic equation.

64m^2 + 32m + 4 = 0

Substituting

b

gives

(2 - 8m)^2 = -64m

. Expanding and combining like terms yields the quadratic equation.

5

Solve the quadratic equation for

m

.

m = -0.25

Dividing by

4

gives

16m^2 + 8m + 1 = (4m + 1)^2 = 0

, which has the single real solution

m = -0.25

.

Key Concept

Using linear equation forms, point-slope relationships, and intercept properties to solve coordinate geometry problems.

Question 2232Question →

In the standard $(x, y)$ coordinate plane, what is the distance between the points $(1, \frac{1}{2})$ and $(5, \frac{7}{2})$ ?

7

2\sqrt{5}

5

25

2\sqrt{13}

Show answer & explanation

Answer: $5$

Answer

The distance between the two points is

5

.

The distance

d

between

(x_1, y_1)

and

(x_2, y_2)

is given by the formula

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

. Substituting

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

and

(5, \frac{7}{2})

gives

d = \sqrt{(5-1)^2 + (\frac{7}{2} - \frac{1}{2})^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5

.

Step-by-Step Solution

1

Calculate the horizontal change (difference in

x

-coordinates)

x_2 - x_1 = 5 - 1 = 4

This determines the horizontal leg of the right triangle formed between the two points.

2

Calculate the vertical change (difference in

y

-coordinates)

y_2 - y_1 = \frac{7}{2} - \frac{1}{2} = \frac{6}{2} = 3

This determines the vertical leg of the right triangle formed between the two points.

3

Apply the Pythagorean theorem (distance formula) to find the distance

d

d = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5

The straight-line distance is the hypotenuse of the right triangle with legs of lengths 4 and 3.

Key Concept

The Distance Formula

Estimated Time:45s

Question 2233Question →

Line $P$ is perpendicular to a line with a slope of $3$ . If line $P$ passes through the point $(2, 4)$ , what is the $y$ -intercept of line $P$ ?

-2

-\frac{2}{3}

\frac{3}{2}

\frac{14}{3}

10

Show answer & explanation

Answer: $\frac{14}{3}$

Answer

The

y

-intercept of line

P

is

\frac{14}{3}

.

The slope of a line perpendicular to a line with a slope of

3

is its negative reciprocal,

-\frac{1}{3}

. Using the point-slope form with the point

(2, 4)

gives the equation

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

. Setting

x = 0

to find the

y

-intercept yields

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

, which simplifies to

y = \frac{14}{3}

.

Step-by-Step Solution

1

Determine the slope of line

P

.

The slope of line

P

is

-\frac{1}{3}

.

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

3

is

-\frac{1}{3}

.

2

Write the equation of line

P

in point-slope form using the point

(2, 4)

.

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

The point-slope form is

y - y_1 = m(x - x_1)

, where

m

is the slope and

(x_1, y_1)

is the given point.

3

Solve for the

y

-intercept by setting

x = 0

.

y = \frac{14}{3}

The

y

-intercept occurs where

x = 0

. Substituting

x = 0

gives

y - 4 = -\frac{1}{3}(0 - 2) \Rightarrow y - 4 = \frac{2}{3} \Rightarrow y = \frac{2}{3} + 4 = \frac{14}{3}

.

Key Concept

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

Question 2234Question →

In the standard $(x,y)$ coordinate plane, a right isosceles triangle $PQR$ has its right angle at vertex $Q(2, -1)$ and another vertex at $P(-1, 3)$ . If the third vertex $R(x, y)$ is located in the first quadrant, what is the value of $3x - y$ ?

16

12

-2

8

21

Show answer & explanation

Answer: 16

Answer

16

The correct answer is 16. To find the coordinates of vertex

R(x, y)

, we find that the vector

\vec{QP} = (-3, 4)

has a length of 5 and a slope of

-4/3

. Since

\triangle PQR

is a right isosceles triangle with the right angle at

Q

, the segment

QR

must be perpendicular to

QP

(slope of

3/4

) and equal in length (

QR = 5

). This yields two possible locations for

R

:

(6, 2)

and

(-2, -4)

. Because

R

must lie in the first quadrant, its coordinates are

(6, 2)

. Evaluating

3x - y

gives

3(6) - 2 = 16

.

Step-by-Step Solution

1

Calculate the vector

\vec{QP}

and the length of segment

QP

.

\vec{QP} = P - Q = (-1 - 2, 3 - (-1)) = (-3, 4)

. The distance is

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

.

Since the triangle is right isosceles with the right angle at

Q

, the leg

QR

must be perpendicular to

QP

and have the same length of 5.

2

Find the slope of

QP

and determine the slope of the perpendicular line containing

QR

.

Slope of

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

. The perpendicular slope of

QR

is the negative reciprocal, which is

\frac{3}{4}

.

Perpendicular lines on the coordinate plane have slopes that are negative reciprocals of each other.

3

Determine the possible coordinates of

R

by scaling the unit perpendicular vector.

The vector

\vec{QR}

must be of the form

(4k, 3k)

for some scalar

k

. Since the length is 5,

(4k)^2 + (3k)^2 = 25 \implies 25k^2 = 25 \implies k = \pm 1

. This yields two possible coordinates for

R

:

R = (2+4, -1+3) = (6, 2)

or

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

.

Using the slope components and the distance constraint ensures

QR

is perpendicular and equal in length to

QP

.

4

Apply the quadrant constraint and evaluate the final expression

3x - y

.

Since

R

is in the first quadrant,

R = (6, 2)

with

x=6, y=2

. Thus,

3x - y = 3(6) - 2 = 16

.

The coordinates of a point in the first quadrant must both be positive.

Key Concept

Using slopes, vectors, and distance formulas to determine the vertices of geometric figures on the coordinate plane.

Question 2235Question →

In $\triangle ABC$ , the measures of the three exterior angles (one at each vertex) are in the ratio $4:5:6$ . What is the measure of the smallest interior angle of $\triangle ABC$ ?

36^\circ

48^\circ

84^\circ

96^\circ

176^\circ

Show answer & explanation

Answer: $36^\circ$

Answer

The correct answer is

36^\circ

.

The correct answer is

36^\circ

. The sum of the three exterior angles of a triangle is

360^\circ

. Given the ratio

4:5:6

, we set up the equation

4k + 5k + 6k = 360^\circ

, which simplifies to

15k = 360^\circ

, yielding

k = 24^\circ

. The largest exterior angle is

6(24^\circ) = 144^\circ

. Since the interior and exterior angles at a vertex are supplementary, the smallest interior angle corresponds to the largest exterior angle, which is

180^\circ - 144^\circ = 36^\circ

.

Step-by-Step Solution

1

Recall the sum of the exterior angles of a triangle.

The sum of the exterior angles of any convex polygon (including a triangle) is

360^\circ

.

This is a fundamental geometric property of exterior angles.

2

Set up an algebraic equation to find the value of each part in the ratio.

Let the measures of the exterior angles be

4k

,

5k

, and

6k

. Then:

4k + 5k + 6k = 360^\circ \implies 15k = 360^\circ \implies k = 24^\circ

.

The sum of the ratio parts must equal the total sum of the exterior angles.

3

Identify which exterior angle corresponds to the smallest interior angle.

Since an interior angle and its adjacent exterior angle are supplementary (

180^\circ

sum), the smallest interior angle must correspond to the largest exterior angle, which is

6k = 6(24^\circ) = 144^\circ

.

As the exterior angle increases, the supplementary interior angle decreases.

4

Calculate the smallest interior angle.

180^\circ - 144^\circ = 36^\circ

.

Subtracting the largest exterior angle from

180^\circ

yields the smallest interior angle.

Key Concept

The sum of the exterior angles of a triangle is

360^\circ

, and the interior and exterior angles at each vertex are supplementary.

Estimated Time:1m 30s

Question 2236Question →

A drone's position on a coordinate grid is represented by the point $(3, -5)$ . If the drone undergoes a translation of $4$ units to the left and $2$ units up, what are the coordinates of its new position on the grid?

(7, -7)

(-1, -7)

(7, -3)

(1, -1)

(-1, -3)

Show answer & explanation

Answer: $(-1, -3)$

Answer

(-1, -3)

The correct answer is

(-1, -3)

. A horizontal translation of

4

units left subtracts

4

from the original

x

-coordinate of

3

, resulting in a new

x

-coordinate of

3 - 4 = -1

. A vertical translation of

2

units up adds

2

to the original

y

-coordinate of

-5

, resulting in a new

y

-coordinate of

-5 + 2 = -3

. This places the drone at

(-1, -3)

.

Step-by-Step Solution

1

Calculate the new x-coordinate

x' = 3 - 4 = -1

Translating

4

units to the left means subtracting

4

from the initial

x

-coordinate of

3

.

2

Calculate the new y-coordinate

y' = -5 + 2 = -3

Translating

2

units up means adding

2

to the initial

y

-coordinate of

-5

.

3

Combine the new coordinates to write the final position

(-1, -3)

Combining the calculated

x

-coordinate and

y

-coordinate yields the final position of the drone.

Key Concept

Performing translations in the coordinate plane

Estimated Time:45s

Question 2237Question →

In the standard $(x, y)$ coordinate plane, triangle $XYZ$ has vertices $X(1, 2)$ , $Y(4, 2)$ , and $Z(1, 6)$ . The triangle is reflected across the line $y = x$ , and then the resulting image is dilated by a scale factor of $3$ with the center of dilation at $(2, 1)$ to form triangle $X''Y''Z''$ . What are the coordinates of the vertex $Z''$ ?

(12, 0)

(14, 1)

(16, -1)

(18, 3)

(-22, -5)

Show answer & explanation

Answer: $(14, 1)$

Answer

The correct coordinates of the vertex

Z''

are

(14, 1)

.

The correct coordinate pair is

(14, 1)

. Swapping the coordinates of

Z(1, 6)

across the reflection line

y = x

gives

Z'(6, 1)

. The dilation of

Z'(6, 1)

by a scale factor of 3 relative to the center

(2, 1)

involves scaling the displacement vector

(4, 0)

to

(12, 0)

and adding the center back, resulting in

(14, 1)

.

Step-by-Step Solution

1

Reflect the point

Z(1, 6)

across the line

y = x

.

The intermediate image is

Z'(6, 1)

.

Reflecting a point across the line

y = x

swaps its

x

- and

y

-coordinates.

2

Calculate the displacement of

Z'(6, 1)

relative to the center of dilation

(2, 1)

.

(6 - 2, 1 - 1) = (4, 0)

.

To perform a dilation centered at a point other than the origin, the point's coordinate must first be measured relative to the center.

3

Multiply the displacement by the scale factor of

3

.

3 \times (4, 0) = (12, 0)

.

Dilation scales the distance from the center of dilation by the given factor.

4

Add the scaled displacement back to the center of dilation

(2, 1)

to find the final absolute coordinates.

(2 + 12, 1 + 0) = (14, 1)

.

This translates the relative coordinates back into the standard coordinate plane.

Key Concept

Applying composite transformations in the coordinate plane, specifically combining a line reflection and a dilation about a non-origin center.

Estimated Time:2m 0s

Question 2238Question →

An exterior angle of a triangle measures $135^\circ$ . The two nonadjacent interior angles of the triangle have measures in the ratio $2:3$ . What is the measure, in degrees, of the largest interior angle of the triangle?

Show answer & explanation

Answer: 81

Answer

The measure of the largest interior angle of the triangle is 81 degrees.

According to the Exterior Angle Theorem, the sum of the two nonadjacent interior angles equals the measure of the exterior angle:

2x + 3x = 135

, which simplifies to

5x = 135

and gives

x = 27

. The nonadjacent interior angles are

2(27) = 54^\circ

and

3(27) = 81^\circ

. The adjacent interior angle is

180^\circ - 135^\circ = 45^\circ

. Comparing the three angles (

45^\circ

,

54^\circ

,

81^\circ

), the largest is

81^\circ

.

Step-by-Step Solution

1

Use the Exterior Angle Theorem to relate the two nonadjacent interior angles to the given exterior angle.

Equation:

2x + 3x = 135

The theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of its two nonadjacent interior angles.

2

Solve the linear equation for

x

.

x = 27

Combine like terms to get

5x = 135

, then divide both sides by 5.

3

Calculate the measures of the two nonadjacent interior angles.

First angle:

2(27) = 54^\circ

; Second angle:

3(27) = 81^\circ

Multiply each part of the ratio by the scale factor

x = 27

.

4

Calculate the measure of the remaining interior angle.

Adjacent angle:

180^\circ - 135^\circ = 45^\circ

An interior angle and its adjacent exterior angle lie on a straight line and are supplementary (they sum to

180^\circ

).

5

Compare the three interior angle measures to find the largest.

The largest angle is

81^\circ

.

Comparing

45^\circ

,

54^\circ

, and

81^\circ

shows that

81^\circ

is the maximum value.

Key Concept

Exterior Angle Theorem and interior angle relationships in a triangle

Alternative Method

Find the adjacent interior angle first:

180^\circ - 135^\circ = 45^\circ

. Since the sum of all interior angles in a triangle is

180^\circ

, the sum of the remaining two interior angles must be

180^\circ - 45^\circ = 135^\circ

. Set up the ratio equation

2x + 3x = 135

and solve for

x

to find the other two angles.

Estimated Time:1m 15s

Question 2239Question →

A local community garden allocates a plot of land for different types of vegetables. Tomatoes occupy $\frac{2}{5}$ of the total area, peppers occupy $\frac{1}{3}$ of the total area, and the remaining 80 square feet are planted with herbs. What is the total area, in square feet, of the community garden plot?

100

120

128

200

300

Show answer & explanation

Answer: 300

Answer

300 square feet

Let

x

be the total area of the garden plot. Since the garden is divided into tomatoes (

\frac{2}{5}

of the total area), peppers (

\frac{1}{3}

of the total area), and herbs (

80

square feet), the sum of these parts equals the total area:

\frac{2}{5}x + \frac{1}{3}x + 80 = x

. Finding a common denominator of 15 for the fractions gives

\frac{6}{15}x + \frac{5}{15}x + 80 = x

, which simplifies to

\frac{11}{15}x + 80 = x

. Subtracting

\frac{11}{15}x

from both sides yields

80 = \frac{4}{15}x

. Multiplying both sides by the reciprocal

\frac{15}{4}

gives

x = 80 \times \frac{15}{4} = 300

square feet.

Step-by-Step Solution

1

Define the variable and translate the given fractions and numbers into an equation. Let

x

represent the total area of the garden plot.

\frac{2}{5}x + \frac{1}{3}x + 80 = x

The sum of the areas of the tomato plot, the pepper plot, and the herb plot must equal the total area of the garden.

2

Find a common denominator to combine the fractional coefficients of

x

. The least common multiple of 5 and 3 is 15. Convert

\frac{2}{5}

and

\frac{1}{3}

to fractions with a denominator of 15.

\frac{6}{15}x + \frac{5}{15}x + 80 = x \implies \frac{11}{15}x + 80 = x

Combining like terms simplifies the expression to help isolate the variable.

3

Subtract

\frac{11}{15}x

from both sides of the equation to group all

x

terms on one side.

80 = x - \frac{11}{15}x \implies 80 = \frac{4}{15}x

This isolates the constant term on one side of the equation.

4

Multiply both sides of the equation by the reciprocal of

\frac{4}{15}

, which is

\frac{15}{4}

, to solve for

x

.

x = 80 \times \frac{15}{4} = 20 \times 15 = 300

This yields the value of the total area of the garden plot.

Key Concept

Solving linear equations in one variable with fractional coefficients

Question 2240Question →

A pet care service, Canine Care, charges a flat monthly registration fee of $25 plus$ 15 per dog walk. A competing service, Paws & Claws, charges a flat monthly registration fee of $45 plus$ 10 per dog walk. If a client's total monthly cost with Canine Care is $30 more than their total monthly cost with Paws & Claws for the same number of walks, how many dog walks did the client's dog receive during that month?

Show answer & explanation

Answer: 10

Answer

The client's dog received 10 dog walks during the month.

The correct answer is 10. By translating the verbal descriptions into algebraic expressions, we find that Canine Care's monthly cost is

15w + 25

and Paws & Claws' monthly cost is

10w + 45

. Setting up the equation where Canine Care is

30

more than Paws & Claws gives

15w + 25 = 10w + 45 + 30

. Simplifying the right side results in

15w + 25 = 10w + 75

. Subtracting

10w

from both sides gives

5w + 25 = 75

. Subtracting

25

from both sides gives

5w = 50

, and dividing by

5

yields

w = 10

.

Step-by-Step Solution

1

Define the variable and write the cost expression for Canine Care.

15w + 25

, where

w

is the number of walks.

To represent the total monthly cost of Canine Care algebraically based on the flat fee and per-walk rate.

2

Write the cost expression for Paws & Claws.

10w + 45

, where

w

is the number of walks.

To represent the total monthly cost of Paws & Claws algebraically based on the flat fee and per-walk rate.

3

Set up the equation relating the two costs.

15w + 25 = 10w + 45 + 30

The problem states the cost of Canine Care is $30 more than the cost of Paws & Claws.

4

Simplify the equation and solve for

w

.

5w = 50 \Rightarrow w = 10

Combine like terms and isolate the variable

w

to find the number of dog walks.

Key Concept

Translating and Solving Algebraic Word Problems

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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