All practice questions

Question 2441Question →

The circle defined by the equation $x^2 + y^2 - 6x + 8y - 11 = 0$ is plotted in the standard $(x, y)$ coordinate plane. A second circle has a center that is the reflection of the first circle's center across the line $y = x$ . If the radius of the second circle is twice the radius of the first circle, which of the following is the equation of the second circle?

(x - 4)^2 + (y + 3)^2 = 144

(x + 4)^2 + (y - 3)^2 = 12

(x - 3)^2 + (y + 4)^2 = 144

(x + 4)^2 + (y - 3)^2 = 144

(x + 4)^2 + (y - 3)^2 = 72

Show answer & explanation

Answer: $(x + 4)^2 + (y - 3)^2 = 144$

Answer

(x + 4)^2 + (y - 3)^2 = 144

Completing the square on

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

gives

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

. The center is

(3, -4)

and the radius is

6

. Reflecting

(3, -4)

across the line

y = x

swaps the coordinates, yielding the new center

(-4, 3)

. Doubling the radius yields

12

. The equation of the second circle is therefore

(x - (-4))^2 + (y - 3)^2 = 12^2

, which simplifies to

(x + 4)^2 + (y - 3)^2 = 144

.

Step-by-Step Solution

1

Group the

x

and

y

terms of the given equation

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

and add 11 to both sides.

(x^2 - 6x) + (y^2 + 8y) = 11

To prepare the terms for completing the square.

2

Complete the square for both the

x

and

y

groups by adding

(6/2)^2 = 9

and

(8/2)^2 = 16

to both sides.

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

To express the circle's equation in standard form

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

.

3

Identify the center and radius of the first circle from the standard form.

Center is

(3, -4)

and radius is

r_1 = \sqrt{36} = 6

.

These characteristics are needed to find the properties of the second circle.

4

Reflect the center

(3, -4)

across the line

y = x

by swapping the

x

and

y

coordinates, and double the radius.

New center is

(-4, 3)

and new radius is

r_2 = 2 \times 6 = 12

.

To satisfy the geometric transformations specified in the problem statement.

5

Write the standard equation of the second circle using the new center

(-4, 3)

and new radius

12

.

(x + 4)^2 + (y - 3)^2 = 144

Substituting

h = -4

,

k = 3

, and

r = 12

into the standard equation

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

.

Key Concept

Completing the square to find standard circle characteristics, and applying reflections across the line

y = x

.

Estimated Time:2m 0s

Question 2442Question →

In the standard $(x,y)$ coordinate plane, a line segment $OQ$ connects the origin $O(0,0)$ to a point $Q$ in the first quadrant. The segment $OQ$ has a length of $10$ units and makes an angle of $60^\circ$ with the positive $x$ -axis. An isosceles right triangle $\triangle OQR$ is constructed such that the right angle is at $Q$ , the leg $QR$ has a length of $10$ units, and point $R$ lies in the first quadrant. What are the coordinates of point $R$ ?

(5\sqrt{3} - 5, 5\sqrt{3} + 5)

(5\sqrt{3} + 5, 5 - 5\sqrt{3})

(5 + 5\sqrt{3}, 5\sqrt{3} - 5)

(20 - 10\sqrt{3}, 10)

(10 + 10\sqrt{3}, 10\sqrt{3} - 10)

Show answer & explanation

Answer: $(5 + 5\sqrt{3}, 5\sqrt{3} - 5)$

Answer

The coordinates

(5 + 5\sqrt{3}, 5\sqrt{3} - 5)

The correct answer is

(5 + 5\sqrt{3}, 5\sqrt{3} - 5)

because constructing two helper

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangles allows us to determine both the coordinates of

Q

as

(5, 5\sqrt{3})

and the horizontal and vertical shifts to

R

as

+5\sqrt{3}

and

-5

respectively.

Step-by-Step Solution

1

Project point

Q

onto the

x

-axis to form a

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangle.

The horizontal leg is

10 \cos(60^\circ) = 5

and the vertical leg is

10 \sin(60^\circ) = 5\sqrt{3}

.

The hypotenuse

OQ

has a length of

10

and makes a

60^\circ

angle with the positive

x

-axis.

2

Determine the coordinates of point

Q

.

Q = (5, 5\sqrt{3})

.

Point

Q

is in the first quadrant, so both coordinates are positive.

3

Determine the orientation of segment

QR

.

Segment

QR

must make a

30^\circ

angle below the horizontal line passing through

Q

(going down and to the right).

Since

\triangle OQR

is a right isosceles triangle with the right angle at

Q

,

QR

is perpendicular to

OQ

and has length

10

. To keep

R

in the first quadrant,

QR

must rotate clockwise from

OQ

by

90^\circ

.

4

Construct a helper

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangle under

QR

to find the changes in

x

and

y

.

The horizontal change is

+10 \cos(30^\circ) = +5\sqrt{3}

and the vertical change is

-10 \sin(30^\circ) = -5

.

The hypotenuse of this triangle is

QR = 10

, and the angle with the horizontal is

30^\circ

.

5

Calculate the coordinates of

R

by applying the changes to the coordinates of

Q

.

R = (5 + 5\sqrt{3}, 5\sqrt{3} - 5)

.

Add the horizontal change to

x_Q

and the vertical change to

y_Q

.

Key Concept

Solving coordinate geometry problems using

30^\circ\text{-}60^\circ\text{-}90^\circ

special right triangles.

Question 2443Question →

A regular octagon has 8 congruent interior angles. What is the degree measure of one of these interior angles?

Show answer & explanation

Answer: 135

Answer

The degree measure of one interior angle of a regular octagon is 135 degrees.

The interior angles of a regular octagon sum to

1080^\circ

because

(8-2) \times 180^\circ = 1080^\circ

. Since a regular octagon has 8 equal angles, dividing

1080^\circ

by

8

results in

135^\circ

per interior angle.

Step-by-Step Solution

1

Identify the number of sides in a regular octagon.

The number of sides,

n

, is

8

.

An octagon is defined as a polygon with 8 sides and 8 angles.

2

Calculate the sum of the interior angles of the octagon.

The sum of the interior angles is

1080^\circ

.

The sum of the interior angles of any convex polygon with

n

sides is

(n-2) \times 180^\circ

. For

n = 8

, this is

(8 - 2) \times 180^\circ = 6 \times 180^\circ = 1080^\circ

.

3

Find the measure of one interior angle by dividing the sum by the number of angles.

Each interior angle measures

135^\circ

.

A regular polygon has congruent interior angles. Therefore, dividing the total sum of

1080^\circ

by the

8

congruent angles yields the measure of each individual interior angle:

1080^\circ \div 8 = 135^\circ

.

Key Concept

The sum of the interior angles of an

n

-sided convex polygon is

(n-2) \times 180^\circ

. For a regular polygon, each interior angle has a measure of

\frac{(n-2) \times 180^\circ}{n}

.

Question 2444Question →

In the standard $(x, y)$ coordinate plane, where $1$ unit represents $1$ kilometer, a playground is located at $(1, 10)$ and a picnic shelter is located at $(5, -2)$ . A water fountain is to be placed exactly halfway between the playground and the picnic shelter. What is the distance, in kilometers, from the origin $(0, 0)$ to the water fountain?

Show answer & explanation

Answer: 5

Answer

The distance from the origin to the water fountain is

5

kilometers.

The midpoint of the segment representing the water fountain is determined to be

(3, 4)

. The distance from the origin

(0, 0)

to

(3, 4)

is found using the Pythagorean relation, yielding

\sqrt{3^2 + 4^2} = 5

kilometers.

Step-by-Step Solution

1

Find the coordinates of the midpoint between

(1, 10)

and

(5, -2)

.

The midpoint is

(3, 4)

.

The midpoint formula is

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

. Substituting the given endpoints gives

M = \left(\frac{1 + 5}{2}, \frac{10 + (-2)}{2}\right) = (3, 4)

.

2

Calculate the distance from the origin

(0, 0)

to the midpoint

(3, 4)

.

The distance is

5

kilometers.

The distance formula from the origin to any point

(x, y)

is

d = \sqrt{x^2 + y^2}

. Substituting the coordinates

(3, 4)

yields

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

.

Key Concept

Using the midpoint formula to find the center of a line segment, followed by the distance formula from the origin to that point.

Question 2445Question →

A triangle has side lengths of $7$ , $x+2$ , and $2x-1$ , where $x$ is an integer. What is the total number of possible values for $x$ ?

Show answer & explanation

Answer: 7

Answer

The total number of possible integer values for

x

is

7

.

According to the Triangle Inequality Theorem, a triangle is formed if and only if the sum of any two side lengths is strictly greater than the third side length. Solving the three inequalities

7 + (x+2) > 2x-1

,

7 + (2x-1) > x+2

, and

(x+2) + (2x-1) > 7

gives

x < 10

,

x > -4

, and

x > 2

. The overlapping interval is

2 < x < 10

. The integers in this range are

\{3, 4, 5, 6, 7, 8, 9\}

, which gives a total of

7

possible integer values.

Step-by-Step Solution

1

Set up the three inequalities required by the Triangle Inequality Theorem.

1)

7 + (x+2) > 2x-1

2)

7 + (2x-1) > x+2

3)

(x+2) + (2x-1) > 7

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the remaining side.

2

Solve the first inequality:

7 + (x+2) > 2x-1

.

x < 10

Simplifying the left side yields

x + 9 > 2x - 1

. Subtracting

x

and adding

1

to both sides results in

10 > x

, which means

x < 10

.

3

Solve the second inequality:

7 + (2x-1) > x+2

.

x > -4

Simplifying the left side yields

2x + 6 > x + 2

. Subtracting

x

and

6

from both sides results in

x > -4

.

4

Solve the third inequality:

(x+2) + (2x-1) > 7

.

x > 2

Simplifying the left side yields

3x + 1 > 7

. Subtracting

1

and dividing by

3

results in

x > 2

.

5

Determine the combined range for

x

and identify the valid integers.

The combined range is

2 < x < 10

. The valid integers are

3, 4, 5, 6, 7, 8,

and

9

.

To satisfy all three inequalities,

x

must be greater than

2

, greater than

-4

, and less than

10

, which simplifies to

2 < x < 10

.

6

Count the total number of valid integer values for

x

.

There are

7

integer values.

Counting the elements of the set

\{3, 4, 5, 6, 7, 8, 9\}

yields a total of

7

values.

Key Concept

Triangle Inequality Theorem

Estimated Time:1m 30s

Question 2446Question →

In the standard $(x, y)$ coordinate plane, a circular running track is modeled by the equation $(x - 2)^2 + (y - 3)^2 = 25$ . A straight pathway, modeled by the line $y = 2x - 1$ , cuts through the track. What is the distance, in coordinate units, between the two points where the pathway intersects the track?

5

2\sqrt{5}

10

20

50

Show answer & explanation

Answer: $10$

Answer

The distance between the two intersection points is

10

.

The circle

(x - 2)^2 + (y - 3)^2 = 25

has a center of

(2, 3)

and a radius of

r = 5

. Since the line

y = 2x - 1

passes through

(2, 3)

, the line contains a diameter of the circle. The distance between the two intersection points is therefore the diameter of the circle, which is

2r = 10

.

Step-by-Step Solution

1

Identify the center and radius of the circle from its equation

(x - 2)^2 + (y - 3)^2 = 25

.

The center is

(2, 3)

and the radius is

r = \sqrt{25} = 5

.

To understand the geometry of the circle and find its radius.

2

Determine if the line

y = 2x - 1

passes through the center of the circle

(2, 3)

by substituting the coordinates into the equation.

3 = 2(2) - 1 \implies 3 = 3

, which is true.

If the line passes through the center, the distance between the intersection points is simply the diameter of the circle.

3

Calculate the diameter of the circle.

\text{Diameter} = 2r = 2(5) = 10

.

The distance between two opposite points on a circle passing through the center is equal to the diameter.

Key Concept

Systems of Linear and Non-Linear Equations

Question 2447Question →

The quadratic equation $0.4x^2 - 2x + c = 0$ has two real solutions. If the difference between these two solutions is exactly $3$ , what is the value of $c$ ?

Show answer & explanation

Answer: 1.6

Answer

1.6

By using the relation between the roots of a quadratic equation

ax^2 + bx + c = 0

, we find that the difference of the roots is given by

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

. Substituting

a = 0.4

,

b = -2

, and the difference of

3

yields

3 = \frac{\sqrt{4 - 1.6c}}{0.4}

. Multiplying by

0.4

and squaring both sides gives

1.44 = 4 - 1.6c

. Solving this linear equation gives

c = 1.6

. We can verify this result by substituting

c = 1.6

back into the original equation:

0.4x^2 - 2x + 1.6 = 0

simplifies to

x^2 - 5x + 4 = 0

, which factors as

(x-1)(x-4) = 0

. The roots are

1

and

4

, and their difference is

4 - 1 = 3

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation.

a = 0.4

,

b = -2

, and the constant term is

c

.

To apply formulas relating the roots to the coefficients of the equation.

2

Apply the formula for the difference of the roots.

|x_1 - x_2| = \frac{\sqrt{b^2 - 4ac}}{|a|}

The problem states the difference between the two solutions is

3

.

3

Substitute the known values into the formula and solve.

3 = \frac{\sqrt{(-2)^2 - 4(0.4)c}}{0.4} \Rightarrow 1.2 = \sqrt{4 - 1.6c}

To isolate the square root expression containing the unknown variable.

4

Square both sides and solve the linear equation for

c

.

1.44 = 4 - 1.6c \Rightarrow 1.6c = 2.56 \Rightarrow c = 1.6

To eliminate the square root and find the value of

c

.

Key Concept

Quadratic Equations and the Quadratic Formula

Estimated Time:1m 30s

Question 2448Question →

A rectangular swimming pool has a length of $3x - 2$ meters and a width of $2x - 5$ meters. A concrete walkway with a uniform width of $2$ meters surrounds the pool. Which of the following expressions represents the area, in square meters, of the walkway?

20x - 12

-18x + 8

20x + 8

12x^2 - 18x + 8

12x^2 - 12

Show answer & explanation

Answer: $20x - 12$

Answer

The expression representing the area of the walkway is

20x - 12

.

The correct answer is the expression

20x - 12

. The total area including the walkway is

(3x + 2)(2x - 1) = 6x^2 + x - 2

. The pool's area is

(3x - 2)(2x - 5) = 6x^2 - 19x + 10

. Subtracting the pool's area from the total area and distributing the negative sign to all terms gives:

(6x^2 + x - 2) - (6x^2 - 19x + 10) = 6x^2 + x - 2 - 6x^2 + 19x - 10 = 20x - 12

.

Step-by-Step Solution

1

Determine the dimensions of the entire area including the pool and the walkway.

The total length is

(3x - 2) + 2(2) = 3x + 2

meters, and the total width is

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

meters.

Since the walkway surrounds the pool on all sides, we add twice the width of the walkway to both the length and width of the pool.

2

Calculate the total area including the walkway.

Total Area =

(3x + 2)(2x - 1) = 6x^2 - 3x + 4x - 2 = 6x^2 + x - 2

square meters.

The area is found by multiplying the total length and the total width using binomial multiplication.

3

Calculate the area of the pool itself.

Pool Area =

(3x - 2)(2x - 5) = 6x^2 - 15x - 4x + 10 = 6x^2 - 19x + 10

square meters.

The area of the pool is the product of its length and width.

4

Subtract the pool area from the total area to find the area of the walkway.

Walkway Area =

(6x^2 + x - 2) - (6x^2 - 19x + 10) = 6x^2 + x - 2 - 6x^2 + 19x - 10 = 20x - 12

square meters.

Subtracting the inner pool area from the total area yields the remaining area of the walkway surrounding it.

Key Concept

Calculating the difference between two binomial products to find the area of a border.

Question 2449Question →

For a certain four-sided convex polygon, the ratio of its interior angle measures is $1:2:3:4$ . What is the degree measure of the smallest angle?

18^\circ

54^\circ

36^\circ

72^\circ

90^\circ

Show answer & explanation

Answer: $36^\circ$

Answer

36^\circ

The correct answer is

36^\circ

because the sum of the interior angles of a four-sided polygon is

360^\circ

. The ratio

1:2:3:4

means the angles can be represented as

x

,

2x

,

3x

, and

4x

. Their sum is

10x = 360^\circ

, which yields

x = 36^\circ

. The smallest angle is

x

, which is

36^\circ

.

Step-by-Step Solution

1

Determine the sum of the interior angles of a four-sided convex polygon.

The sum of the interior angles of a four-sided polygon is

360^\circ

.

The sum of the interior angles of a polygon with

n

sides is given by the formula

(n-2) \times 180^\circ

. For a four-sided polygon,

(4-2) \times 180^\circ = 360^\circ

.

2

Find the sum of the parts of the given ratio.

1 + 2 + 3 + 4 = 10

parts

To distribute the total angle measure proportionally, the individual parts of the ratio

1:2:3:4

must be summed.

3

Calculate the measure of one part of the ratio.

360^\circ \div 10 = 36^\circ

Dividing the total sum of the interior angles by the total number of parts determines the degree measure of a single part.

4

Find the measure of the smallest angle.

1 \times 36^\circ = 36^\circ

The smallest angle corresponds to the smallest part of the ratio, which is 1.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and this total can be divided proportionally using a given ratio to find individual angle measures.

Estimated Time:1m 0s

Question 2450Question →

In the standard $(x, y)$ coordinate plane, a circle is tangent to the $x$ -axis at the point $(5, 0)$ . If the circle is also tangent to the line $y = \frac{4}{3}x$ and its center lies in the first quadrant, what is the $y$ -coordinate of the center of the circle?

Show answer & explanation

Answer: 2.5

Answer

The

y

-coordinate of the center of the circle is

2.5

.

A circle tangent to the

x

-axis at

(5, 0)

has a center along the vertical line

x = 5

. Since the center is in the first quadrant, its coordinates can be represented as

(5, k)

where

k > 0

, and the radius is

k

. The distance from the center

(5, k)

to the line

4x - 3y = 0

must also equal the radius

k

. Using the point-to-line distance formula, we write

\frac{|4(5) - 3k|}{\sqrt{4^2 + (-3)^2}} = k

, which simplifies to

|20 - 3k| = 5k

. Since the center must lie in the first quadrant (

k > 0

), we solve

20 - 3k = 5k

to find

k = 2.5

. The alternative case

20 - 3k = -5k

gives

k = -10

, which lies in the fourth quadrant and is thus excluded.

Step-by-Step Solution

1

Identify the coordinates of the center and the radius of the circle based on the

x

-axis tangency.

Center:

(5, k)

and Radius:

r = k

(where

k > 0

).

A circle tangent to the

x

-axis at

(5, 0)

has its center directly above or below this point on the line

x = 5

. Since the center is in the first quadrant, its

y

-coordinate

k

must be positive, making the radius

k

.

2

Use the distance from the center to the second tangent line to write an equation for

k

.

\frac{|4(5) - 3k|}{\sqrt{4^2 + (-3)^2}} = k

The distance from the center

(5, k)

to the tangent line

4x - 3y = 0

must equal the radius of the circle.

3

Simplify the equation and solve for

k

.

|20 - 3k| = 5k

, leading to

k = 2.5

or

k = -10

.

Simplifying the denominator yields

5

. Multiplying both sides by

5

gives the absolute value equation, which resolves to

20 - 3k = 5k

or

20 - 3k = -5k

.

4

Select the valid solution using the quadrant constraint.

k = 2.5

The center must lie in the first quadrant, which requires

k > 0

. Thus,

k = -10

is discarded, and the correct value is

2.5

.

Key Concept

The relationship between a circle's center, its radius, and its tangent lines in the coordinate plane.

Question 2451Question →

A rectangular piece of paper $ABCD$ has dimensions $AB = 12$ inches and $BC = 9$ inches. The paper is folded so that vertex $A$ falls directly on vertex $C$ , creating a crease $EF$ where $E$ lies on $AB$ and $F$ lies on $CD$ . What is the length of the crease $EF$ , in inches?

Show answer & explanation

Answer: 11.25

Answer

The length of the crease is 11.25 inches.

By interpreting the fold geometrically, we establish that the crease is the perpendicular bisector of the diagonal. We can solve for the segments along the side using a right triangle formed at the corner, and then construct a second right triangle using an altitude to find the length of the crease hypotenuse.

Step-by-Step Solution

1

Identify the relationship created by the fold.

The crease

EF

is the perpendicular bisector of the diagonal

AC

, meaning

AE = EC = x

.

When a point is folded onto another, the crease line acts as the perpendicular bisector of the segment connecting the two points.

2

Set up an expression for the remaining part of the side

AB

.

Since

AB = 12

and

AE = x

, the length of

EB = 12 - x

.

The point

E

lies on segment

AB

, dividing it into

AE

and

EB

.

3

Use the Pythagorean Theorem in right triangle

EBC

to solve for

x

.

x^2 = (12 - x)^2 + 9^2 \implies x^2 = 144 - 24x + x^2 + 81 \implies 24x = 225 \implies x = 9.375

.

The triangle

EBC

is a right triangle with legs

EB

and

BC

, and hypotenuse

EC

.

4

Form a second right triangle to find the length of the crease

EF

.

Draw

FG \perp AB

with

G

on

AB

. This forms right triangle

EGF

with legs

FG = 9

and

EG = AB - EB - DF = 12 - 2.625 - 2.625 = 6.75

.

Constructing an altitude from

F

to

AB

allows us to create a right triangle that has the crease

EF

as its hypotenuse.

5

Apply the Pythagorean Theorem to right triangle

EGF

to calculate the final length of

EF

.

EF = \sqrt{6.75^2 + 9^2} = \sqrt{45.5625 + 81} = \sqrt{126.5625} = 11.25

inches.

The hypotenuse of right triangle

EGF

represents the length of the crease.

Key Concept

Applying the Pythagorean Theorem to geometric folds and multi-step right triangle relationships

Question 2452Question →

On a map of a town laid out on a standard coordinate grid, a straight path has an $x$ -intercept of $-6$ and a $y$ -intercept of $4$ . A landmark is located on this path at a point where the $x$ -coordinate is $9$ . What is the $y$ -coordinate of the landmark?

Show answer & explanation

Answer: 10

Answer

The

y

-coordinate of the landmark is 10.

The correct answer is 10. The linear path passes through the points

(-6, 0)

and

(0, 4)

. Using the slope formula, the slope is

m = \frac{4 - 0}{0 - (-6)} = \frac{2}{3}

. The slope-intercept equation of the line is

y = \frac{2}{3}x + 4

. Substituting the landmark's

x

-coordinate of

9

into the equation gives

y = \frac{2}{3}(9) + 4 = 10

.

Step-by-Step Solution

1

Identify the coordinates of the points representing the intercepts on the coordinate grid.

The points are

(-6, 0)

and

(0, 4)

.

An

x

-intercept of

-6

corresponds to the coordinate point

(-6, 0)

, and a

y

-intercept of

4

corresponds to the coordinate point

(0, 4)

.

2

Calculate the slope of the linear path.

The slope is

\frac{2}{3}

.

Using the slope formula

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

, substitute the points

(-6, 0)

and

(0, 4)

to get

m = \frac{4 - 0}{0 - (-6)} = \frac{4}{6} = \frac{2}{3}

.

3

Write the linear equation representing the path.

The equation is

y = \frac{2}{3}x + 4

.

With a slope of

m = \frac{2}{3}

and a

y

-intercept of

b = 4

, the equation in slope-intercept form (

y = mx + b

) is

y = \frac{2}{3}x + 4

.

4

Find the

y

-coordinate of the landmark by substituting its

x

-coordinate.

The

y

-coordinate is 10.

Substitute

x = 9

into the equation:

y = \frac{2}{3}(9) + 4 = 6 + 4 = 10

.

Key Concept

Linear Equations and Graphing

Question 2453Question →

The position of a particle moving along a straight line is described by the polynomial $s(t) = (2t^2 - 3t + 4)(t + 3)$ , where $t$ represents the time in seconds and $s(t)$ represents the displacement in meters. When the polynomial is expanded and written in the standard form $at^3 + bt^2 + ct + d$ , what is the value of $b + c$ ?

Show answer & explanation

Answer: -2

Answer

The value of

b + c

is

-2

.

Expanding the polynomial

s(t) = (2t^2 - 3t + 4)(t + 3)

yields

2t^3 + 3t^2 - 5t + 12

. Comparing this to the standard form

at^3 + bt^2 + ct + d

, we find

b = 3

and

c = -5

. Summing these coefficients gives

b + c = 3 + (-5) = -2

.

Step-by-Step Solution

1

Multiply each term of the trinomial

2t^2 - 3t + 4

by each term of the binomial

t + 3

.

2t^3 + 6t^2 - 3t^2 - 9t + 4t + 12

Apply the distributive property to expand the product of the polynomials.

2

Combine the like terms (the

t^2

terms and the

t

terms).

2t^3 + 3t^2 - 5t + 12

Group and add the coefficients of like terms to simplify the polynomial.

3

Identify the coefficients

b

(for the

t^2

term) and

c

(for the

t

term) and find their sum.

b = 3

and

c = -5

, so

b + c = 3 + (-5) = -2

Compare the simplified polynomial to the standard form

at^3 + bt^2 + ct + d

to find the coefficients, then calculate the requested sum.

Key Concept

Operations on Polynomials

Alternative Method

Instead of expanding the whole expression, you can find the individual coefficients directly. The

t^2

term is formed by

(2t^2)(3) + (-3t)(t) = 6t^2 - 3t^2 = 3t^2

, so

b = 3

. The

t

term is formed by

(-3t)(3) + (4)(t) = -9t + 4t = -5t

, so

c = -5

. Thus,

b + c = 3 - 5 = -2

.

Estimated Time:1m 30s

Question 2454Question →

For the quadratic equation $1.5x^2 - 5x + c = 0$ , where $c$ is a real constant, the equation has two non-real complex solutions. Which of the following inequalities represents all possible values of $c$ ?

c > \frac{25}{6}

c < \frac{25}{6}

c > -\frac{25}{6}

c > \frac{50}{3}

c < -\frac{25}{6}

Show answer & explanation

Answer: $c > \frac{25}{6}$

Answer

c > \frac{25}{6}

For a quadratic equation to have two non-real complex solutions, its discriminant must be negative. Substituting

a = 1.5

,

b = -5

, and the constant

c

into the discriminant formula

b^2 - 4ac < 0

gives

25 - 6c < 0

. Solving this inequality results in the requirement that the constant must be strictly greater than twenty-five sixths.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation

1.5x^2 - 5x + c = 0

in the standard form

ax^2 + bx + c = 0

.

a = 1.5

,

b = -5

, and

c = c

.

These coefficients are required to compute the discriminant.

2

Set up the condition for the quadratic equation to have two non-real complex solutions using the discriminant

\Delta = b^2 - 4ac

.

The discriminant must be strictly negative:

(-5)^2 - 4(1.5)(c) < 0

.

A quadratic equation has non-real complex solutions if and only if its discriminant is negative.

3

Simplify the inequality and solve for

c

.

25 - 6c < 0 \implies 25 < 6c \implies c > \frac{25}{6}

.

Isolating

c

yields the range of values that satisfy the condition.

Key Concept

A quadratic equation

ax^2 + bx + c = 0

has two non-real complex solutions if and only if its discriminant,

\Delta = b^2 - 4ac

, is strictly less than zero.

Estimated Time:1m 30s

Question 2455Question →

Points $K(-2, 5)$ and $L(6, -1)$ are the endpoints of segment $KL$ on a coordinate plane. What is the distance from the midpoint of segment $KL$ to the point $P(5, 7)$ ?

\sqrt{10}

\sqrt{34}

8

10

\sqrt{130}

Show answer & explanation

Answer: $\sqrt{34}$

Answer

The correct answer is the square root of 34.

To find the distance from the midpoint of segment

KL

to the point

P(5, 7)

, we first calculate the midpoint

M

of

KL

. Using the midpoint formula, the coordinates are

x = \frac{-2 + 6}{2} = 2

and

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

, giving

M(2, 2)

. Next, we apply the distance formula to find the distance between

M(2, 2)

and

P(5, 7)

, which is

d = \sqrt{(5 - 2)^2 + (7 - 2)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}

. This confirms the correct distance is the square root of 34.

Step-by-Step Solution

1

Calculate the coordinates of the midpoint of segment

KL

using the midpoint formula:

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

.

The midpoint is

M(2, 2)

since

x = \frac{-2 + 6}{2} = 2

and

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

.

Finding the midpoint of the segment is required before finding the distance from it to point

P

.

2

Calculate the distance between the midpoint

M(2, 2)

and the point

P(5, 7)

using the distance formula:

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

.

The distance is

\sqrt{34}

since

d = \sqrt{(5 - 2)^2 + (7 - 2)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34}

.

This determines the final distance required by the question.

Key Concept

Applying both the midpoint and distance formulas sequentially to solve coordinate geometry problems.

Question 2456Question →

In $\triangle PQR$ , the measure of $\angle P$ is $50^\circ$ . The angle bisectors of $\angle PQR$ and $\angle PRQ$ intersect at point $I$ inside the triangle. What is the measure of $\angle QIR$ ?

65^\circ

100^\circ

115^\circ

125^\circ

130^\circ

Show answer & explanation

Answer: $115^\circ$

Answer

115 degrees

To find the measure of the angle at the intersection, we first use the Triangle Angle-Sum Theorem on the larger triangle. The sum of the interior angles in the larger triangle is 180 degrees. Since the angle at one vertex is 50 degrees, the sum of the other two angles must be 130 degrees. The angle bisectors divide these two angles in half, meaning the sum of the two half-angles in the smaller triangle is half of 130 degrees, which is 65 degrees. Applying the Triangle Angle-Sum Theorem to the smaller triangle, the sum of its interior angles is also 180 degrees. Subtracting the sum of the two half-angles (65 degrees) from 180 degrees gives 115 degrees for the angle at the intersection.

Step-by-Step Solution

1

Find the sum of the remaining interior angles of the triangle.

The sum of the angles at Q and R is 130 degrees.

The sum of all interior angles in any triangle is always 180 degrees, and the angle at P is given as 50 degrees.

2

Determine the sum of the bisected angles in the smaller triangle.

The sum of the half-angles is 65 degrees.

Since the lines QI and RI are angle bisectors, the sum of the interior angles of the smaller triangle at vertices Q and R is half the sum of the angles at Q and R of the larger triangle.

3

Calculate the measure of the angle at the intersection point.

The angle at the intersection point is 115 degrees.

The sum of the interior angles in the smaller triangle is also 180 degrees, so the unknown angle is found by subtracting the sum of the two half-angles from 180 degrees.

Key Concept

The Triangle Angle-Sum Theorem states that the sum of the measures of the interior angles of a triangle is always 180 degrees. Angle bisectors divide an angle into two equal parts.

Alternative Method

Alternatively, one can assign specific values to the angles that satisfy the given conditions. Since the sum of the angles in the large triangle must be 180 degrees and the angle at vertex P is 50 degrees, the sum of the other two angles must be 130 degrees. If we assume the triangle is isosceles with the two unknown angles being equal, each of those angles is 65 degrees. Their bisectors would each form an angle of 32.5 degrees with the base. In the smaller triangle, the sum of these two bisected angles is 65 degrees, which leaves 115 degrees for the angle at the intersection point.

Estimated Time:1m 15s

Question 2457Question →

In parallelogram $ABCD$ , the measure of interior angle $A$ is $72^\circ$ . What is the measure, in degrees, of interior angle $B$ ?

Show answer & explanation

Answer: 108

Answer

The measure of interior angle

B

is

108

degrees.

In parallelogram

ABCD

, interior angles

A

and

B

are consecutive angles. A fundamental property of parallelograms is that consecutive angles are supplementary (their measures sum to

180^\circ

). Therefore, the measure of angle

B

is calculated as

180^\circ - 72^\circ = 108^\circ

.

Step-by-Step Solution

1

Use the consecutive angles property of parallelograms.

The sum of consecutive interior angles

A

and

B

is

180^\circ

.

Since opposite sides of a parallelogram are parallel, consecutive interior angles are supplementary.

2

Set up the equation and solve for the unknown angle.

m\angle B = 108^\circ

Subtract

72^\circ

from

180^\circ

.

Key Concept

Properties of parallelograms (consecutive angles are supplementary)

Question 2458Question →

A square in the standard $(x, y)$ coordinate plane has vertices at $A(1, 1)$ , $B(3, 1)$ , $C(3, 3)$ , and $D(1, 3)$ . The square is dilated by a scale factor of $3$ with the center of dilation at point $A(1, 1)$ . The resulting image is then translated $2$ units left and $4$ units up. What are the coordinates of the final image of vertex $C$ ?

(7, 13)

(9, 3)

(1, 19)

(5, 11)

(11, 13)

Show answer & explanation

Answer: $(5, 11)$

Answer

The final coordinates of the image of vertex

C

are

(5, 11)

.

To find the coordinates of the image of vertex

C(3, 3)

after the dilation, we calculate the horizontal and vertical distances from the center of dilation

A(1, 1)

to

C(3, 3)

. Both distances are

2

units. Since the scale factor is

3

, these distances are tripled to

6

units. Adding these to the coordinates of the center

A(1, 1)

gives the dilated point

(1 + 6, 1 + 6) = (7, 7)

. Next, we apply the translation of

2

units left (subtracting

2

from the

x

-coordinate) and

4

units up (adding

4

to the

y

-coordinate) to

(7, 7)

, which yields the final coordinates

(7 - 2, 7 + 4) = (5, 11)

.

Step-by-Step Solution

1

Find the coordinates of the image of vertex

C(3, 3)

after a dilation with scale factor

3

centered at

A(1, 1)

.

The coordinates after dilation are

(7, 7)

.

The horizontal and vertical distances from the center

A(1, 1)

to

C(3, 3)

are both

2

units (

3 - 1 = 2

). Multiplying these distances by the scale factor of

3

yields

6

units. Adding these new distances to the coordinates of the center

A(1, 1)

gives

(1 + 6, 1 + 6) = (7, 7)

.

2

Apply a translation of

2

units left and

4

units up to the point

(7, 7)

.

The final coordinates are

(5, 11)

.

Translating

2

units left subtracts

2

from the

x

-coordinate (

7 - 2 = 5

). Translating

4

units up adds

4

to the

y

-coordinate (

7 + 4 = 11

).

Key Concept

Transformations in the Coordinate Plane

Estimated Time:1m 30s

Question 2459Question →

In the right trapezoid $ABCD$ below, $AB$ is parallel to $CD$ , and the measures of $\angle A$ and $\angle D$ are both $90^\circ$ . The length of $CD$ is $7$ , the length of $BC$ is $8$ , and the measure of $\angle B$ is $60^\circ$ . What is the length of the diagonal $BD$ ?

Show answer & explanation

Answer: 13

Answer

13

By drawing altitude

CE

perpendicular to

AB

, we form rectangle

AECD

and right triangle

\triangle CEB

. Since

\angle B = 60^\circ

,

\triangle CEB

is a

30^\circ

-

60^\circ

-

90^\circ

triangle with hypotenuse

BC = 8

. The side opposite

30^\circ

is

BE = 8/2 = 4

, and the side opposite

60^\circ

is

CE = 4\sqrt{3}

. Since opposite sides of rectangle

AECD

are equal, we find

AD = CE = 4\sqrt{3}

and

AE = CD = 7

. Thus,

AB = AE + BE = 7 + 4 = 11

. Finally, we apply the Pythagorean Theorem to right triangle

\triangle DAB

:

BD^2 = AD^2 + AB^2 = (4\sqrt{3})^2 + 11^2 = 48 + 121 = 169

, yielding

BD = 13

.

Step-by-Step Solution

1

Decompose the trapezoid by drawing an altitude from

C

perpendicular to

AB

, meeting it at

E

.

This forms a rectangle

AECD

and a right triangle

CEB

.

Decomposing the figure allows us to use right triangle trigonometry and parallel line relationships to determine missing side lengths.

2

Calculate the lengths of the legs of right triangle

CEB

.

BE = 4

and

CE = 4\sqrt{3}

.

Since

\angle B = 60^\circ

,

\triangle CEB

is a

30^\circ

-

60^\circ

-

90^\circ

triangle. The shorter leg

BE

is half the hypotenuse

BC

, and the longer leg

CE

is the shorter leg times

\sqrt{3}

.

3

Determine the lengths of

AD

and

AB

.

AD = 4\sqrt{3}

and

AB = 11

.

In the rectangle

AECD

, opposite sides are equal, so

AD = CE = 4\sqrt{3}

and

AE = CD = 7

. Thus, the base

AB = AE + BE = 7 + 4 = 11

.

4

Use the Pythagorean Theorem in right triangle

DAB

to solve for

BD

.

BD = 13

.

In right triangle

DAB

, the legs are

AD = 4\sqrt{3}

and

AB = 11

. The hypotenuse

BD

satisfies

BD^2 = AD^2 + AB^2 = (4\sqrt{3})^2 + 11^2 = 48 + 121 = 169

, so

BD = \sqrt{169} = 13

.

Key Concept

Solving multi-step geometry problems by decomposing shapes into rectangles and special right triangles (

30^\circ

-

60^\circ

-

90^\circ

), then applying the Pythagorean Theorem.

Question 2460Question →

A convex hexagon has four interior angles that measure $100^\circ$ , $115^\circ$ , $125^\circ$ , and $140^\circ$ . The remaining two interior angles are in the ratio $3:5$ . What is the degree measure of the largest interior angle of this hexagon?

90^\circ

140^\circ

144^\circ

150^\circ

160^\circ

Show answer & explanation

Answer: $150^\circ$

Answer

The degree measure of the largest interior angle is

150^\circ

.

The sum of the interior angles of a hexagon is

(6-2) \times 180^\circ = 720^\circ

. Subtracting the four given angles (

100^\circ

,

115^\circ

,

125^\circ

, and

140^\circ

) from

720^\circ

leaves a remaining sum of

240^\circ

. Since the two remaining angles are in the ratio

3:5

, they can be written as

3x

and

5x

, where

3x + 5x = 240^\circ

. Solving for

x

gives

8x = 240^\circ \implies x = 30^\circ

. The two remaining angles are

90^\circ

and

150^\circ

. Comparing all six interior angles of the hexagon, the largest measure is

150^\circ

.

Step-by-Step Solution

1

Calculate the sum of the interior angles of a hexagon.

The sum is

(6-2) \times 180^\circ = 720^\circ

.

The sum of the interior angles of any convex

n

-sided polygon is given by

(n-2) \times 180^\circ

.

2

Find the sum of the four given interior angles.

100^\circ + 115^\circ + 125^\circ + 140^\circ = 480^\circ

.

This determines the total measure of the known angles.

3

Determine the remaining sum of the two unknown angles.

720^\circ - 480^\circ = 240^\circ

.

Subtracting the sum of the known angles from the total sum yields the combined measure of the remaining two angles.

4

Set up an equation using the ratio

3:5

to find the measures of the two remaining angles.

3x + 5x = 240^\circ \implies 8x = 240^\circ \implies x = 30^\circ

. The two angles are

3(30^\circ) = 90^\circ

and

5(30^\circ) = 150^\circ

.

Representing the two angles in terms of a common variable

x

allows us to solve for their individual measures.

5

Identify the largest interior angle among all six angles of the hexagon.

The largest angle is

150^\circ

.

Comparing all six angles (

90^\circ

,

100^\circ

,

115^\circ

,

125^\circ

,

140^\circ

, and

150^\circ

), the maximum value is

150^\circ

.

Key Concept

Calculating the interior angle sum of a polygon and solving for unknown angles using ratios.

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

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Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

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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

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