All practice questions

Question 321Question →

For the quadratic equation $2x^2 + bx + 16 = 0$ , where $b$ is a positive constant, the sum of the squares of the two complex solutions is equal to $-7$ . What is the value of $b$ ?

Show answer & explanation

Answer: 6

Answer

The value of the positive constant

b

is

6

.

By Vieta's formulas, the sum of the roots of the quadratic equation

2x^2 + bx + 16 = 0

is

-\frac{b}{2}

and the product is

8

. Using the identity

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

, we substitute

-7

for the sum of the squares, yielding

-7 = \left(-\frac{b}{2}\right)^2 - 2(8)

. This simplifies to

-7 = \frac{b^2}{4} - 16

, which leads to

\frac{b^2}{4} = 9

and

b^2 = 36

. Since

b

is a positive constant,

b = 6

.

Step-by-Step Solution

1

Find the sum and product of the roots in terms of

b

using Vieta's formulas.

x_1 + x_2 = -\frac{b}{2}

and

x_1 x_2 = 8

Vieta's formulas state that for a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is

-\frac{b}{a}

and the product of the roots is

\frac{c}{a}

.

2

Relate the sum of the squares of the roots to their sum and product.

x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2x_1 x_2

This algebraic identity allows us to express the sum of the squares of the roots using the sum and product of the roots.

3

Substitute the known values into the identity and solve for

b

.

-7 = \left(-\frac{b}{2}\right)^2 - 2(8) \implies -7 = \frac{b^2}{4} - 16 \implies \frac{b^2}{4} = 9 \implies b^2 = 36 \implies b = 6

Substituting the given sum of squares (

-7

), sum (

-\frac{b}{2}

), and product (

8

) produces a single-variable equation that can be solved for the positive constant

b

.

Key Concept

Using Vieta's formulas and algebraic identities to relate the roots of a quadratic equation to its coefficients.

Question 322Question →

In the standard $(x, y)$ coordinate plane, a line is defined by the equation $2x + 3y = 22$ . A second line passes through the origin $(0, 0)$ and intersects the first line at a point where $x = 5$ . What is the slope of this second line?

Show answer & explanation

Answer: 0.8

Answer

The slope of the second line is

0.8

(or

\frac{4}{5}

).

The intersection point has an

x

-coordinate of

5

. Substituting this into

2x + 3y = 22

gives

2(5) + 3y = 22

, which simplifies to

3y = 12

, or

y = 4

. Thus, the intersection point is

(5, 4)

. The second line passes through

(0, 0)

and

(5, 4)

. Using the slope formula, the slope is

\frac{4 - 0}{5 - 0} = 0.8

.

Step-by-Step Solution

1

Substitute

x = 5

into the equation of the first line to find the

y

-coordinate of the intersection point.

The intersection point is

(5, 4)

.

Since the two lines intersect at

x = 5

, the intersection point must satisfy the equation of the first line.

2

Use the slope formula to find the slope of the line connecting

(0, 0)

and

(5, 4)

.

The slope is

0.8

.

The second line passes through the origin

(0, 0)

and the intersection point

(5, 4)

, so its slope is the ratio of the change in

y

to the change in

x

.

Key Concept

Finding the slope of a line given two points on the coordinate plane, where one point is determined by the intersection of two linear paths.

Alternative Method

Since the second line passes through the origin

(0, 0)

, its equation is of the form

y = mx

, where

m

is the slope. At the intersection point

(5, y)

, we have

y = 5m

. We can substitute this directly into the first line's equation:

2x + 3y = 22 \Rightarrow 2(5) + 3(5m) = 22 \Rightarrow 10 + 15m = 22 \Rightarrow 15m = 12 \Rightarrow m = \frac{12}{15} = 0.8

.

Estimated Time:1m 30s

Question 323Question →

A coordinate grid is laid over a map of a city. A subway station is located at $S(-1, 9)$ and a bus terminal is located at $B(7, -1)$ . A passenger transfer center is built at the midpoint of the line segment connecting the subway station and the bus terminal. What is the distance, in grid units, between the transfer center and a parking garage located at $(7, 7)$ ?

Show answer & explanation

Answer: 5

Answer

The distance between the passenger transfer center and the parking garage is 5 grid units.

First, the midpoint is determined by averaging the

x

-coordinates and

y

-coordinates of the endpoints:

x = \frac{-1 + 7}{2} = 3

and

y = \frac{9 - 1}{2} = 4

, giving the transfer center coordinates of

(3, 4)

. Second, the distance between

(3, 4)

and

(7, 7)

is found using the distance formula:

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

.

Step-by-Step Solution

1

Calculate the coordinates of the midpoint of the line segment connecting the subway station

S(-1, 9)

and the bus terminal

B(7, -1)

.

The midpoint is

M(3, 4)

.

The transfer center is located exactly halfway between the two endpoints, which corresponds to the midpoint.

2

Calculate the distance between the midpoint

M(3, 4)

and the parking garage at

(7, 7)

.

The distance is 5.

Applying the distance formula to

M(3, 4)

and

(7, 7)

yields the straight-line distance.

Key Concept

Distance and Midpoint Formulas

Alternative Method

Instead of using the distance formula, one can draw a right triangle on the grid with vertices at

(3, 4)

,

(7, 7)

, and

(7, 4)

. The horizontal leg has a length of

|7 - 3| = 4

, and the vertical leg has a length of

|7 - 4| = 3

. Using the Pythagorean theorem, the hypotenuse (distance) is

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

.

Estimated Time:1m 30s

Question 324Question →

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , the measure of $\angle A$ is $30^\circ$ , and the length of leg $BC$ is $12$ . An altitude $BD$ is drawn perpendicular to the hypotenuse $AC$ . Let $E$ be the midpoint of the altitude $BD$ . A line passing through $E$ is perpendicular to $BD$ and intersects the leg $AB$ at $G$ and the leg $BC$ at $F$ . What is the length of segment $GF$ ?

Show answer & explanation

Answer: 12

Answer

The length of segment

GF

is

12

.

The correct answer is

12

. By analyzing the geometric properties of the 30-60-90 right triangle

ABC

, the altitude

BD

is found to be

6\sqrt{3}

, making the half-segment

BE = 3\sqrt{3}

. The perpendicular line at

E

creates two smaller 30-60-90 right triangles,

\triangle BEF

and

\triangle BEG

. Solving for the legs along the line gives

EF = 3

and

EG = 9

, which sum to

12

.

Step-by-Step Solution

1

Find the length of altitude

BD

in right triangle

ABC

.

BD = 6\sqrt{3}

In right triangle

ABC

, we have

\angle B = 90^\circ

,

\angle A = 30^\circ

, and

\angle C = 60^\circ

. The altitude

BD

forms a smaller 30-60-90 right triangle

BCD

with hypotenuse

BC = 12

. Since

BD

is opposite the

60^\circ

angle

\angle C

, we have

BD = BC \sin(60^\circ) = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3}

.

2

Calculate the length of segment

BE

.

BE = 3\sqrt{3}

Since

E

is the midpoint of the altitude

BD

, we divide the length of

BD

by 2:

BE = \frac{6\sqrt{3}}{2} = 3\sqrt{3}

.

3

Determine the length of segment

EF

in right triangle

BEF

.

EF = 3

Since the line

GF

is perpendicular to

BD

,

\angle BEF = 90^\circ

. In right triangle

BCD

, we have

\angle DBC = 30^\circ

, which means

\angle EBF = 30^\circ

. This makes

\triangle BEF

a 30-60-90 right triangle where

BE

is adjacent to the

30^\circ

angle and

EF

is opposite to it. Thus,

EF = \frac{BE}{\sqrt{3}} = \frac{3\sqrt{3}}{\sqrt{3}} = 3

.

4

Determine the length of segment

EG

in right triangle

BEG

.

EG = 9

Since

\angle BEG = 90^\circ

and

\angle ABD = 90^\circ - \angle DBC = 60^\circ

, the angle

\angle EBG = 60^\circ

. This makes

\triangle BEG

a 30-60-90 right triangle where

BE

is adjacent to the

60^\circ

angle and

EG

is opposite to it. Thus,

EG = BE \sqrt{3} = 3\sqrt{3} \times \sqrt{3} = 9

.

5

Calculate the total length of segment

GF

.

GF = 12

Since

G

,

E

, and

F

are collinear and

E

lies between

G

and

F

, the length of segment

GF

is the sum of

EG

and

EF

:

GF = 9 + 3 = 12

.

Key Concept

Using properties of 30-60-90 special right triangles to find segment lengths in complex geometric configurations.

Question 325Question →

What is the greatest integer value of $x$ that satisfies the inequality $3 - 2x > 10$ ?

Show answer & explanation

Answer: -4

Answer

The correct answer is

-4

.

Subtracting 3 from both sides of

3 - 2x > 10

gives

-2x > 7

. When dividing both sides by

-2

, the inequality sign must be flipped, yielding

x < -3.5

. The greatest integer less than

-3.5

is

-4

.

Step-by-Step Solution

1

Subtract 3 from both sides of the inequality to isolate the variable term.

-2x > 7

Subtracting 3 from both sides keeps the inequality balanced while moving the constant term to the right side.

2

Divide both sides by

-2

and reverse the inequality sign.

x < -3.5

Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign to maintain a true statement.

3

Identify the greatest integer that satisfies the inequality.

-4

The integers that are strictly less than

-3.5

are

-4, -5, -6, \dots

. The largest (greatest) of these integers is

-4

.

Key Concept

Solving linear inequalities by applying the sign-reversal rule when dividing by a negative number and identifying integer boundary values.

Question 326Question →

In the standard $(x, y)$ coordinate plane, the circle defined by the equation $x^2 + y^2 - 12x - 4y + 31 = 0$ is translated $3$ units to the right and $5$ units down. A line passing through the origin with a non-zero slope $m$ is tangent to this translated circle. What is the value of $m$ ?

Show answer & explanation

Answer: -0.75

Answer

The correct answer is -0.75.

The correct answer is -0.75. By completing the square on the original equation, we find the circle

(x-6)^2 + (y-2)^2 = 9

with center

(6, 2)

and radius

3

. Translating the circle shifts the center to

(9, -3)

. A line passing through the origin with slope

m

has the equation

mx - y = 0

. For this line to be tangent to the circle, its perpendicular distance from the center

(9, -3)

must equal the radius

3

. Using the distance formula, we get the equation

\frac{|9m + 3|}{\sqrt{m^2 + 1}} = 3

. Simplifying gives

|3m + 1| = \sqrt{m^2 + 1}

. Squaring both sides yields

9m^2 + 6m + 1 = m^2 + 1

, which simplifies to

8m^2 + 6m = 0

. The non-zero solution is

m = -0.75

.

Step-by-Step Solution

1

Complete the square for the original circle equation

x^2 + y^2 - 12x - 4y + 31 = 0

.

(x - 6)^2 + (y - 2)^2 = 9

.

Grouping the

x

and

y

terms and completing the square yields the standard form of a circle, showing that the original center is

(6, 2)

and the radius

r = \sqrt{9} = 3

.

2

Apply the translation of

3

units right and

5

units down to the center

(6, 2)

.

New center is

(6 + 3, 2 - 5) = (9, -3)

.

Translating a point in the coordinate plane shifts the coordinates directly:

x

-coordinate increases by

3

and

y

-coordinate decreases by

5

. The radius remains

3

.

3

Set up the distance from the new center

(9, -3)

to the tangent line

y = mx

.

Distance equation:

\frac{|9m - (-3)|}{\sqrt{m^2 + 1}} = 3

.

The line

y = mx

can be rewritten as

mx - y = 0

. The perpendicular distance from the center to a tangent line must equal the radius of the circle, which is

3

.

4

Simplify the distance equation and solve for

m

.

|3m + 1| = \sqrt{m^2 + 1}

.

Factoring out

3

from the numerator and dividing both sides by

3

simplifies the equation before squaring.

5

Square both sides of the equation and solve the resulting quadratic equation.

2m(4m + 3) = 0 \implies m = 0

or

m = -0.75

.

Squaring both sides yields

(3m + 1)^2 = m^2 + 1 \implies 9m^2 + 6m + 1 = m^2 + 1 \implies 8m^2 + 6m = 0

. Since we are looking for a non-zero slope, the only valid solution is

m = -0.75

.

Key Concept

Using completing the square, coordinate translations, and the point-to-line distance formula to solve circle tangency problems

Question 327Question →

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , the measure of $\angle A$ is $60^\circ$ , and the hypotenuse $AC$ has a length of $20$ centimeters. Point $D$ lies on leg $BC$ such that the length of segment $BD$ is $2\sqrt{3}$ centimeters. A line segment $DE$ is drawn perpendicular to $AC$ such that $E$ lies on $AC$ . What is the length, in centimeters, of segment $AE$ ?

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. By solving for the angles and side lengths of the two nested

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

right triangles, we find that the segment

BC

is

10\sqrt{3}

cm, making

DC = 8\sqrt{3}

cm. Using the ratio of sides for the smaller right triangle

DEC

, we find

EC = 12

cm, which leaves

AE = 20 - 12 = 8

cm.

Step-by-Step Solution

1

Determine the third angle of right triangle

ABC

.

\angle C = 30^\circ

The sum of angles in a triangle is

180^\circ

. Since

\angle B = 90^\circ

and

\angle A = 60^\circ

, we have

\angle C = 180^\circ - 90^\circ - 60^\circ = 30^\circ

.

2

Calculate the length of the side

BC

.

BC = 10\sqrt{3}

cm

In the

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

triangle

ABC

, the side

BC

is opposite the

60^\circ

angle, so its length is the hypotenuse

AC

multiplied by

\sin(60^\circ)

or

\frac{\sqrt{3}}{2}

. Thus,

BC = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3}

.

3

Find the length of segment

DC

.

DC = 8\sqrt{3}

cm

Since point

D

lies on segment

BC

, the length of

DC

is the total length of

BC

minus the length of

BD

. Since

BD = 2\sqrt{3}

, we have

DC = 10\sqrt{3} - 2\sqrt{3} = 8\sqrt{3}

.

4

Determine the properties of the right triangle

DEC

.

\triangle DEC

is a

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

triangle with hypotenuse

DC = 8\sqrt{3}

cm.

Since segment

DE

is perpendicular to

AC

,

\angle DEC = 90^\circ

. Triangle

DEC

shares the angle

\angle C = 30^\circ

with triangle

ABC

, which makes it a

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

triangle where

DC

is the hypotenuse.

5

Calculate the length of segment

EC

.

EC = 12

cm

In the

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

triangle

DEC

, the leg

EC

is adjacent to the

30^\circ

angle, so its length is the hypotenuse

DC

multiplied by

\cos(30^\circ)

or

\frac{\sqrt{3}}{2}

. Thus,

EC = 8\sqrt{3} \times \frac{\sqrt{3}}{2} = 12

.

6

Calculate the length of segment

AE

.

AE = 8

cm

Since point

E

lies on segment

AC

, we can find

AE

by subtracting

EC

from

AC

. Thus,

AE = AC - EC = 20 - 12 = 8

.

Key Concept

Using the properties of

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

special right triangles to find missing lengths in composite geometric configurations.

Question 328Question →

A convex pentagon has four interior angles that measure $80^\circ$ , $110^\circ$ , $120^\circ$ , and $130^\circ$ . What is the degree measure of the fifth interior angle?

Show answer & explanation

Answer: 100

Answer

The degree measure of the fifth interior angle is

100^\circ

.

The sum of the interior angles of a pentagon (

n = 5

) is

(5 - 2) \times 180^\circ = 540^\circ

. The sum of the four given angles is

80^\circ + 110^\circ + 120^\circ + 130^\circ = 440^\circ

. The measure of the fifth angle is the difference between these two values:

540^\circ - 440^\circ = 100^\circ

.

Step-by-Step Solution

1

Find the sum of the interior angles of a convex pentagon.

The sum of the interior angles is

540^\circ

.

The sum of the interior angles of any

n

-sided convex polygon is calculated using the formula

(n - 2) \times 180^\circ

. For a pentagon,

n = 5

, which gives

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

.

2

Sum the measures of the four given interior angles.

The sum of the four given angles is

440^\circ

.

Adding the given measures:

80^\circ + 110^\circ + 120^\circ + 130^\circ = 440^\circ

.

3

Subtract the sum of the four given angles from the total sum of the interior angles.

The measure of the fifth interior angle is

100^\circ

.

Subtracting the sum of the known angles from the total pentagon interior angle sum yields

540^\circ - 440^\circ = 100^\circ

.

Key Concept

The sum of the interior angles of an

n

-sided convex polygon is

(n-2) \times 180^\circ

.

Question 329Question →

A right triangle has legs of length $5$ inches and $12$ inches. What is the length, in inches, of the hypotenuse of this triangle?

Show answer & explanation

Answer: 13

Answer

The length of the hypotenuse is 13 inches.

The Pythagorean theorem states that in any right triangle with legs

a

and

b

and hypotenuse

c

,

a^2 + b^2 = c^2

. Substituting the given values:

5^2 + 12^2 = 25 + 144 = 169

. Taking the square root of

169

gives the hypotenuse length of

13

.

Step-by-Step Solution

1

Identify the lengths of the two legs.

a = 5

,

b = 12

These are the given side lengths perpendicular to each other.

2

Apply the Pythagorean theorem

a^2 + b^2 = c^2

.

5^2 + 12^2 = 25 + 144 = 169

To find the square of the hypotenuse.

3

Solve for the hypotenuse

c

by taking the square root.

c = \sqrt{169} = 13

To find the side length of the hypotenuse.

Key Concept

Pythagorean Theorem

Question 330Question →

A line is defined by the equation $y = 2x + k$ , where $k$ is a constant. A parabola is defined by the equation $y = x^2 - 4x + 14$ . If the line and the parabola intersect at exactly one point in the standard $(x, y)$ coordinate plane, what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of the constant

k

must be 5.

To find the intersection of the line and the parabola, set their equations equal to each other:

x^2 - 4x + 14 = 2x + k

. Rearranging this into standard quadratic form gives

x^2 - 6x + (14 - k) = 0

. For the system to have exactly one solution, the discriminant of this quadratic equation must be zero. The discriminant is

b^2 - 4ac = (-6)^2 - 4(1)(14 - k) = 36 - 56 + 4k = 4k - 20

. Setting

4k - 20 = 0

yields

k = 5

.

Step-by-Step Solution

1

Equate the linear and quadratic expressions to find their intersection.

x^2 - 4x + 14 = 2x + k

Setting the two equations equal to each other allows us to find the

x

-coordinates of any intersection points.

2

Rearrange the terms to write the equation in standard quadratic form,

ax^2 + bx + c = 0

.

x^2 - 6x + (14 - k) = 0

Subtracting

2x

and

k

from both sides simplifies the equation into a quadratic form where

a = 1

,

b = -6

, and

c = 14 - k

.

3

Apply the condition for exactly one intersection point by setting the discriminant to zero.

(-6)^2 - 4(1)(14 - k) = 0

A quadratic equation has exactly one real root if and only if its discriminant,

b^2 - 4ac

, is equal to zero.

4

Solve the linear equation for

k

.

k = 5

Expanding the equation yields

36 - 56 + 4k = 0

, which simplifies to

4k - 20 = 0

, leading to

k = 5

.

Key Concept

Determining the condition for a linear equation to be tangent to a quadratic equation by setting the discriminant of their intersection equation to zero.

Question 331Question →

What is the smallest integer value of $y$ that satisfies the inequality $5(2 - y) < 3(y - 6)$ ?

Show answer & explanation

Answer: 4

Answer

The smallest integer value of

y

that satisfies the inequality is

4

.

Evaluating the inequality leads to

y > 3.5

. The smallest integer greater than

3.5

is

4

. Substituting

y = 4

into the original inequality gives

5(2 - 4) < 3(4 - 6) \implies -10 < -6

, which is true. Substituting the next smallest integer,

3

, gives

-5 < -9

, which is false.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses on both sides.

10 - 5y < 3y - 18

To clear the parentheses and simplify the terms.

2

Subtract

3y

and

10

from both sides of the inequality to group the variable terms on one side and constants on the other.

-8y < -28

To isolate the variable term.

3

Divide both sides by

-8

and reverse the inequality sign because of division by a negative number.

y > 3.5

To solve the inequality for

y

.

4

Identify the smallest integer that satisfies the inequality

y > 3.5

.

4

To find the smallest integer value greater than

3.5

.

Key Concept

Solving multi-step linear inequalities involving distribution and division by a negative number.

Alternative Method

Instead of subtracting variables to the left, we can add

5y

to both sides to keep the variable coefficient positive:

10 - 5y < 3y - 18 \implies 10 < 8y - 18 \implies 28 < 8y \implies y > 3.5

. This avoids the need to divide by a negative number and flip the sign, reducing the risk of a sign-flip error.

Estimated Time:1m 0s

Question 332Question →

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 333Question →

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 334Question →

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 335Question →

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 336Question →

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 337Question →

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 338Question →

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 339Question →

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 340Question →

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)

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

Answer

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Key Concept

Common Mistakes

Alternative Method

Answer

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Key Concept

Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Common Mistakes

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Key Concept

Common Mistakes