All practice questions

Question 361Question →

A landscape architect designs a square courtyard with a side length of $4x - 3$ meters. A square garden bed with a side length of $2x - 1$ meters is placed in one of the corners of the courtyard. The remaining area of the courtyard is paved. If the paved area, in square meters, is represented by the polynomial $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, what is the value of $b$ ?

Show answer & explanation

Answer: -20

Answer

The value of

b

is

-20

.

The remaining paved area is the difference between the area of the courtyard and the garden bed:

(16x^2 - 24x + 9) - (4x^2 - 4x + 1) = 12x^2 - 20x + 8

. The coefficient of the linear

x

term is

-20

.

Step-by-Step Solution

1

Find the polynomial representing the total area of the courtyard.

A_{\text{total}} = (4x - 3)^2 = 16x^2 - 24x + 9

square meters

The area of a square is the square of its side length. We expand

(4x - 3)^2

using the identity

(p - q)^2 = p^2 - 2pq + q^2

.

2

Find the polynomial representing the area of the garden bed.

A_{\text{garden}} = (2x - 1)^2 = 4x^2 - 4x + 1

square meters

The garden bed is also a square, so we expand

(2x - 1)^2

using the binomial squaring identity.

3

Subtract the garden bed area from the total area to find the paved area.

A_{\text{paved}} = 12x^2 - 20x + 8

square meters

The paved area is the difference between the two areas. We distribute the negative sign to all terms of the subtracted polynomial:

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

, and then combine like terms.

4

Identify the coefficient of the

x

term,

b

.

b = -20

Comparing

12x^2 - 20x + 8

to

ax^2 + bx + c

, the coefficient of the linear

x

term is

-20

.

Key Concept

Operations on Polynomials

Alternative Method

To find only the coefficient of the linear term, expand and subtract only the linear terms from both binomial expansions:

2(4x)(-3) - 2(2x)(-1) = -24x - (-4x) = -20x

. The coefficient

b

is therefore

-20

.

Estimated Time:1m 30s

Question 362Question →

In the figure, line $L_1$ is parallel to line $L_2$ . Vertex $A$ of $\triangle ABC$ lies on $L_1$ , and vertices $B$ and $C$ lie on $L_2$ . Side $AB$ is perpendicular to $L_2$ . Point $D$ lies on $L_2$ such that $C$ is between $B$ and $D$ . If the measure of the exterior angle $\angle ACD$ is $132^\circ$ , what is the measure, in degrees, of $\angle BAC$ ?

Show answer & explanation

Answer: 42

Answer

42

The correct answer is

42

. Because

AB

is perpendicular to

L_2

,

\angle ABC

is

90^\circ

. The exterior angle

\angle ACD

is given as

132^\circ

, which means the adjacent interior angle

\angle ACB

must be supplementary to it:

180^\circ - 132^\circ = 48^\circ

. Since the interior angles of a triangle always sum to

180^\circ

, the remaining angle

\angle BAC

is

180^\circ - (90^\circ + 48^\circ) = 42^\circ

. Alternatively, applying the Exterior Angle Theorem, the exterior angle is equal to the sum of the two remote interior angles:

\angle ACD = \angle ABC + \angle BAC

, so

132^\circ = 90^\circ + \angle BAC

, which simplifies to

\angle BAC = 42^\circ

.

Step-by-Step Solution

1

Determine the measure of interior angle

\angle ABC

.

\angle ABC = 90^\circ

Since side

AB

is perpendicular to

L_2

, the angle it makes with

L_2

at vertex

B

is

90^\circ

.

2

Find the measure of interior angle

\angle ACB

.

\angle ACB = 48^\circ

The interior angle

\angle ACB

and the exterior angle

\angle ACD

form a linear pair along line

L_2

, making them supplementary:

\angle ACB = 180^\circ - 132^\circ = 48^\circ

.

3

Calculate the measure of

\angle BAC

using the angle sum of a triangle.

42^\circ

The interior angles of

\triangle ABC

sum to

180^\circ

. Subtracting the known angles gives

\angle BAC = 180^\circ - (90^\circ + 48^\circ) = 42^\circ

.

Key Concept

Triangle Angle Sum Theorem and Supplementary Angle Relationships

Question 363Question →

In the standard $(x, y)$ coordinate plane, a line segment has endpoints at $M(1, -3)$ and $N(5, 5)$ . If a second line is perpendicular to segment $MN$ at its midpoint, what is the $y$ -intercept of this second line?

Show answer & explanation

Answer: 2.5

Answer

The y-intercept of the second line is 2.5.

The slope of segment

MN

is

2

, meaning the perpendicular line must have a slope of

-0.5

. Since this perpendicular line bisects the segment, it must pass through the midpoint

(3, 1)

. Substituting these values into the slope-intercept form

y = mx + b

yields

1 = -0.5(3) + b

, which simplifies to

b = 2.5

.

Step-by-Step Solution

1

Calculate the coordinates of the midpoint of segment

MN

.

The midpoint is

(3, 1)

.

The perpendicular bisector must pass through the midpoint of the bisected segment.

2

Calculate the slope of segment

MN

.

The slope of

MN

is

2

.

The slope is needed to find the perpendicular slope.

3

Calculate the slope of the perpendicular line.

The perpendicular slope is

-0.5

.

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

4

Find the

y

-intercept (

b

) using the slope

-0.5

and the point

(3, 1)

.

The

y

-intercept is

2.5

.

Substituting the coordinates of the midpoint and the perpendicular slope into the equation

y = mx + b

gives

1 = -0.5(3) + b

, which simplifies to

b = 2.5

.

Key Concept

Finding the equation and y-intercept of a perpendicular bisector using midpoint and negative reciprocal slope.

Question 364Question →

On a coordinate grid, a line segment connects the points $A(1, 3)$ and $B(9, 15)$ . The midpoint of this segment is $M$ , and the midpoint of the segment connecting $A$ and $M$ is $C$ . What is the $y$ -coordinate of point $C$ ?

Show answer & explanation

Answer: 6

Answer

The

y

-coordinate of point

C

is 6.

Applying the midpoint formula to the endpoints

A(1, 3)

and

B(9, 15)

yields

M(5, 9)

. Applying the midpoint formula again to

A(1, 3)

and

M(5, 9)

yields

C(3, 6)

. The

y

-coordinate of point

C

is therefore 6.

Step-by-Step Solution

1

Calculate the coordinates of the midpoint

M

of segment

AB

.

M = (5, 9)

The midpoint formula is defined as

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

.

2

Calculate the coordinates of the midpoint

C

of segment

AM

.

C = (3, 6)

Apply the midpoint formula using the coordinates of

A(1, 3)

and the newly found midpoint

M(5, 9)

.

3

Identify the

y

-coordinate of point

C

.

6

For the point

C(3, 6)

, the

x

-coordinate is 3 and the

y

-coordinate is 6.

Key Concept

Midpoint Formula

Estimated Time:1m 15s

Question 365Question →

A toy rocket is launched upward from a platform. Its height $h$ , in meters, above the ground $t$ seconds after launch is modeled by the function $h(t) = -4.9t^2 + 7.35t + 12.25$ . According to this model, how many seconds after launch does the rocket strike the ground?

Show answer & explanation

Answer: 2.5

Answer

The rocket strikes the ground

2.5

seconds after launch.

The correct answer of

2.5

is determined by setting the height

h(t)

to

0

and solving the resulting quadratic equation using the quadratic formula. Since time must be non-negative in this physical context, the negative solution of

-1

is discarded, leaving

2.5

seconds as the time when the rocket strikes the ground.

Step-by-Step Solution

1

Set the height function

h(t)

to

0

.

-4.9t^2 + 7.35t + 12.25 = 0

The rocket strikes the ground when its height above the ground is

0

meters.

2

Apply the quadratic formula

t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

.

t = \frac{-7.35 \pm \sqrt{(7.35)^2 - 4(-4.9)(12.25)}}{2(-4.9)}

This formula provides the solutions to any quadratic equation of the form

at^2 + bt + c = 0

.

3

Calculate the discriminant and its square root.

b^2 - 4ac = 294.1225

and

\sqrt{294.1225} = 17.15

Evaluating the term under the radical simplifies the quadratic formula expression.

4

Evaluate the two possible values for

t

.

t = -1

or

t = 2.5

Solving the simplified expression gives the two mathematical roots of the quadratic equation.

5

Choose the physically valid solution.

t = 2.5

Time must be positive in this scenario, so the negative solution

t = -1

is discarded.

Key Concept

Solving a quadratic equation with decimal coefficients using the quadratic formula in a real-world motion context.

Question 366Question →

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , the measure of $\angle A$ is $60^\circ$ , and the length of $AC$ is $16$ units. Point $D$ lies on side $BC$ such that the measure of $\angle ADB$ is $45^\circ$ . What is the length of segment $CD$ , rounded to the nearest tenth?

Show answer & explanation

Answer: 5.9

Answer

The length of segment CD is approximately 5.9 units.

By recognizing that triangle ABC is a 30-60-90 right triangle, the shorter leg AB is found to be 8 (half of the hypotenuse 16), and the longer leg BC is 8*sqrt(3). Because triangle ABD is a 45-45-90 right triangle, the leg BD is equal to the leg AB, which is 8. Subtracting BD from BC yields CD = 8*sqrt(3) - 8, which is approximately 5.9.

Step-by-Step Solution

1

Determine the type of triangle ABC

Triangle ABC is a 30-60-90 special right triangle.

The triangle has a right angle (90 degrees) at B and an angle of 60 degrees at A, which leaves 30 degrees for angle C.

2

Calculate the lengths of sides AB and BC

AB = 8 units and BC = 8*sqrt(3) units.

In a 30-60-90 triangle with hypotenuse AC = 16, the side opposite 30 degrees (AB) is half the hypotenuse, and the side opposite 60 degrees (BC) is the shorter leg multiplied by sqrt(3).

3

Determine the type of triangle ABD

Triangle ABD is a 45-45-90 special right triangle.

Since D lies on BC, angle ABD is a right angle (90 degrees). Given that angle ADB is 45 degrees, the remaining angle BAD must also be 45 degrees.

4

Calculate the length of side BD

BD = 8 units.

In a 45-45-90 right triangle, the two legs opposite the 45-degree angles are equal in length, so BD = AB.

5

Calculate the length of segment CD and round to the nearest tenth

CD ≈ 5.9 units.

Since D lies on side BC, CD = BC - BD = 8*sqrt(3) - 8 ≈ 8(1.732) - 8 = 13.856 - 8 = 5.856, which rounds to 5.9.

Key Concept

Applying properties of 30-60-90 and 45-45-90 special right triangles to find segment lengths within nested figures.

Question 367Question →

What is the maximum value of $x$ that satisfies the inequality $\frac{2 - 5x}{3} \ge 2x + 8$ ?

Show answer & explanation

Answer: -2

Answer

-2

Solving the inequality by isolating

x

leads to

x \le -2

, meaning that any value of

x

less than or equal to

-2

satisfies the inequality. Therefore, the maximum possible value is

-2

.

Step-by-Step Solution

1

Multiply both sides of the inequality by 3 to clear the denominator.

2 - 5x \ge 6x + 24

Multiplying by a positive number eliminates the fraction without changing the inequality direction.

2

Subtract

6x

from both sides of the inequality.

2 - 11x \ge 24

This gathers the variable terms on the left side of the inequality.

3

Subtract 2 from both sides of the inequality.

-11x \ge 22

This isolates the variable term on the left side.

4

Divide both sides by

-11

and reverse the inequality sign.

x \le -2

Dividing by a negative number requires reversing the direction of the inequality sign. The resulting inequality defines the upper bound for

x

.

Key Concept

Solving linear inequalities and applying the sign-flip rule when multiplying or dividing by a negative number.

Question 368Question →

A quadratic equation is defined by $x^2 - bx + 18 = 0$ , where $b$ is a positive constant. If the difference between the two real solutions of this equation is 3, what is the value of $b$ ?

Show answer & explanation

Answer: 9

Answer

The value of the positive constant

b

is 9.

By applying the quadratic formula, the roots of the equation are

x = \frac{b \pm \sqrt{b^2 - 72}}{2}

. The difference between these roots is

\sqrt{b^2 - 72}

. Setting this equal to 3 gives

\sqrt{b^2 - 72} = 3

. Squaring both sides yields

b^2 - 72 = 9

, which simplifies to

b^2 = 81

. Taking the positive root since

b

is a positive constant gives

b = 9

.

Step-by-Step Solution

1

Express the roots of the quadratic equation

x^2 - bx + 18 = 0

using the quadratic formula.

The roots are

x = \frac{b \pm \sqrt{b^2 - 4(1)(18)}}{2} = \frac{b \pm \sqrt{b^2 - 72}}{2}

.

This provides a formulaic representation of the two solutions in terms of the unknown parameter

b

.

2

Subtract the smaller root from the larger root to represent the difference between the solutions, and set this expression equal to 3.

Difference

= \frac{b + \sqrt{b^2 - 72}}{2} - \frac{b - \sqrt{b^2 - 72}}{2} = \sqrt{b^2 - 72} = 3

.

The problem specifies that the difference between the two real solutions is 3.

3

Square both sides of the equation to eliminate the radical, and solve for the positive constant

b

.

b^2 - 72 = 9 \Rightarrow b^2 = 81 \Rightarrow b = 9

(since

b

is positive).

Squaring both sides allows us to isolate

b^2

and find the value of

b

that satisfies the initial condition.

Key Concept

Solving for quadratic coefficients using the difference of roots derived from the quadratic formula.

Alternative Method

Use Vieta's formulas. Let the roots be

r_1

and

r_2

. We know that

r_1 + r_2 = b

and

r_1 r_2 = 18

. We are given that the difference between the roots is 3, so

|r_1 - r_2| = 3

. We can use the algebraic identity

(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1 r_2

. Substituting the known values gives

3^2 = b^2 - 4(18)

, which simplifies to

9 = b^2 - 72

, leading to

b^2 = 81

. Since

b > 0

, we find

b = 9

.

Estimated Time:1m 30s

Question 369Question →

A convex polygon has $n$ sides. The interior angles of the polygon form an arithmetic progression with a common difference of $d^\circ$ , where $d$ is a positive integer. If the smallest interior angle of the polygon measures $100^\circ$ , what is the maximum possible value of $n$ ?

Show answer & explanation

Answer: 8

Answer

The maximum possible value of

n

is 8.

The correct answer is 8 because we set the sum of the interior angles

(n-2) \times 180^\circ

equal to the sum of the arithmetic progression

\frac{n}{2}[200 + (n-1)d]

. Solving for

(n-1)d

gives

(n-1)d = 160 - \frac{720}{n}

. Since the polygon is convex, the largest angle

100 + (n-1)d

must be strictly less than

180^\circ

, which means

(n-1)d < 80^\circ

. Substituting the expression yields

160 - \frac{720}{n} < 80

, which simplifies to

n < 9

. Since

n

must be an integer, the maximum possible value of

n

is 8. For

n = 8

, the common difference

d = 10

is a positive integer, satisfying all conditions.

Step-by-Step Solution

1

Express the sum of the interior angles using the polygon angle sum formula and the arithmetic progression formula.

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

. The sum of the angles in arithmetic progression with first term

100^\circ

and common difference

d^\circ

is

\frac{n}{2}[2(100) + (n-1)d] = 100n + \frac{n(n-1)d}{2}

.

This sets up the system relating the geometry of the polygon to the algebraic progression of its angles.

2

Equate the two expressions and solve for the quantity

(n-1)d

.

100n + \frac{n(n-1)d}{2} = 180n - 360 \implies \frac{n(n-1)d}{2} = 80n - 360 \implies (n-1)d = 160 - \frac{720}{n}

.

This isolates the quantity

(n-1)d

, which represents the difference between the largest and smallest angles.

3

Apply the convexity constraint that every interior angle must be strictly less than

180^\circ

.

The largest angle is the last term of the progression:

100 + (n-1)d

. For the polygon to be convex, this angle must be strictly less than

180^\circ

. Therefore,

100 + (n-1)d < 180 \implies (n-1)d < 80

. Substituting

(n-1)d = 160 - \frac{720}{n}

gives

160 - \frac{720}{n} < 80 \implies 80 < \frac{720}{n} \implies n < 9

.

A convex polygon cannot have any interior angles greater than or equal to

180^\circ

.

4

Identify the maximum integer value of

n

and verify that a positive integer common difference

d

exists.

Since

n < 9

and

n

must be an integer, the maximum possible value is

n = 8

. For

n = 8

, we calculate

(8-1)d = 160 - \frac{720}{8} \implies 7d = 70 \implies d = 10

. Since

d = 10

is a positive integer, the solution is valid.

This ensures the result satisfies all constraints, including that the common difference is a positive integer.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and all interior angles of a convex polygon must be strictly less than

180^\circ

.

Question 370Question →

In the standard $(x, y)$ coordinate system, a line given by the equation $y = 2x + 4$ intersects a parabola given by the equation $y = x^2 - 2x - 1$ at exactly two points. What is the sum of the $y$ -coordinates of these two points of intersection?

Show answer & explanation

Answer: 16

Answer

16

The system of equations is solved by setting the equations equal to each other, yielding the quadratic equation

x^2 - 4x - 5 = 0

. Solving for

x

gives

x = 5

and

x = -1

. Substituting these values into the linear equation gives the y-coordinates

14

and

2

. The sum of these y-coordinates is

14 + 2 = 16

.

Step-by-Step Solution

1

Equate the linear and quadratic equations to find the x-values of the intersection points.

x^2 - 2x - 1 = 2x + 4

At the points of intersection, the y-values of both equations must be equal.

2

Rearrange the equation to standard quadratic form.

x^2 - 4x - 5 = 0

Grouping all terms on one side allows the quadratic equation to be solved.

3

Factor the quadratic equation to find the x-coordinates.

(x - 5)(x + 1) = 0

, so

x = 5

or

x = -1

The numbers that multiply to

-5

and add up to

-4

are

-5

and

1

.

4

Substitute the x-coordinates into the linear equation to find the y-coordinates.

For

x = 5

:

y = 2(5) + 4 = 14

. For

x = -1

:

y = 2(-1) + 4 = 2

.

Evaluating the linear equation is simpler than evaluating the quadratic equation.

5

Find the sum of the y-coordinates.

14 + 2 = 16

The question asks for the sum of the y-coordinates of the two points of intersection.

Key Concept

Solving a system of linear and quadratic equations by substitution.

Alternative Method

Instead of solving for the individual intersection points, Vieta's formulas can be applied. The x-coordinates satisfy

x^2 - 4x - 5 = 0

, so their sum is

x_1 + x_2 = 4

. Since the points lie on the line

y = 2x + 4

, the sum of the y-coordinates is

y_1 + y_2 = (2x_1 + 4) + (2x_2 + 4) = 2(x_1 + x_2) + 8 = 2(4) + 8 = 16

.

Estimated Time:1m 30s

Question 371Question →

An irregular convex octagon has five interior angles that each measure $144^\circ$ . The remaining three interior angles have measures in the ratio $3:4:5$ . What is the measure, in degrees, of the largest interior angle of this octagon?

Show answer & explanation

Answer: 150

Answer

The measure of the largest interior angle of the octagon is

150^\circ

.

The total sum of the interior angles of an octagon is

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

. The sum of the five given angles is

5 \times 144^\circ = 720^\circ

, leaving a sum of

1080^\circ - 720^\circ = 360^\circ

for the remaining three angles. Since these three angles are in the ratio

3:4:5

, we set

3x + 4x + 5x = 360^\circ

, yielding

12x = 360^\circ

and

x = 30^\circ

. The largest of these three angles is

5 \times 30^\circ = 150^\circ

. Comparing

150^\circ

to the other angles of the octagon (which are

144^\circ

,

90^\circ

, and

120^\circ

), the largest interior angle is

150^\circ

.

Step-by-Step Solution

1

Calculate the total sum of the interior angles of the octagon.

The total sum is

1080^\circ

.

The sum of the interior angles of any convex polygon with

n

sides is given by

(n - 2) \times 180^\circ

. For an octagon,

n = 8

, so the sum is

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

.

2

Calculate the sum of the five given congruent angles.

The sum of these five angles is

720^\circ

.

Since five angles each measure

144^\circ

, their combined sum is

5 \times 144^\circ = 720^\circ

.

3

Determine the sum of the remaining three interior angles.

The sum of the remaining angles is

360^\circ

.

Subtracting the sum of the five congruent angles from the total sum of the octagon's interior angles yields the sum of the remaining three angles:

1080^\circ - 720^\circ = 360^\circ

.

4

Use the ratio

3:4:5

to find the measures of the remaining three angles.

The measures of the three angles are

90^\circ

,

120^\circ

, and

150^\circ

.

Let the measures of the remaining three angles be

3x

,

4x

, and

5x

. Their sum is

3x + 4x + 5x = 12x = 360^\circ

, which gives

x = 30^\circ

. The largest of these three angles is

5 \times 30^\circ = 150^\circ

.

5

Compare the measures of all interior angles of the octagon to find the largest one.

The largest angle is

150^\circ

.

The octagon's interior angles consist of five angles of

144^\circ

, and three angles of

90^\circ

,

120^\circ

, and

150^\circ

. Comparing these values,

150^\circ

is the largest measure.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and ratios can be used to partition a total sum into specific parts.

Estimated Time:2m 0s

Question 372Question →

In $\triangle ABC$ , the measure of $\angle A$ is $40^\circ$ . The measure of $\angle B$ is three times the measure of $\angle C$ . What is the measure, in degrees, of $\angle B$ ?

Show answer & explanation

Answer: 105

Answer

The measure of angle B is 105 degrees.

The sum of the interior angles in any triangle is

180^\circ

. By setting the measure of

\angle C

to

x

and the measure of

\angle B

to

3x

, we can write the equation

40 + 3x + x = 180

. Solving for

x

yields

4x = 140

, which simplifies to

x = 35

. The measure of

\angle B

is

3x

, which is

3 \times 35 = 105^\circ

.

Step-by-Step Solution

1

Set up the equation using the fact that the sum of angles in a triangle is 180 degrees.

m\angle A + m\angle B + m\angle C = 180^\circ

The angles of any triangle in plane geometry sum to 180 degrees.

2

Represent the unknown angles algebraically.

Let

m\angle C = x

, then

m\angle B = 3x

. Substitute

m\angle A = 40^\circ

.

This allows solving for the unknown angles with a single-variable equation.

3

Solve the equation for

x

.

40 + 4x = 180 \implies 4x = 140 \implies x = 35

To find the measure of angle C.

4

Calculate the measure of angle B.

m\angle B = 3(35) = 105^\circ

Angle B is three times angle C, and we need to find the measure of angle B.

Key Concept

The sum of the interior angles of a triangle is always 180 degrees.

Question 373Question →

An irregular convex hexagon has two interior angles that are right angles. The remaining four interior angles have measures in the ratio $4:5:5:6$ . What is the measure, in degrees, of the largest interior angle of this hexagon?

Show answer & explanation

Answer: 162

Answer

The correct answer is

162

degrees. The sum of the interior angles of a hexagon is

720^\circ

. Subtracting the two right angles (

180^\circ

total) leaves a remaining sum of

540^\circ

. The ratio of the remaining four angles is

4:5:5:6

, which can be represented as

4x, 5x, 5x, 6x

, summing to

20x = 540

. Solving for the multiplier gives

x = 27

. The largest angle is

6(27) = 162^\circ

, which is also greater than the two

90^\circ

angles.

The sum of the interior angles of a hexagon is

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

. Subtracting the two right angles (

180^\circ

) gives a remaining sum of

540^\circ

for the other four angles. Let these four angles be

4x, 5x, 5x,

and

6x

. Their sum is

20x = 540

, which solves to

x = 27

. The largest angle is

6x = 6(27) = 162^\circ

, which is also larger than the two

90^\circ

angles.

Step-by-Step Solution

1

Calculate the sum of all interior angles of a convex hexagon.

The sum is

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

.

The sum of the interior angles of any convex

n

-gon is given by

(n-2) \times 180^\circ

.

2

Subtract the sum of the two right angles from the total sum.

The remaining sum is

720^\circ - 180^\circ = 540^\circ

.

Two right angles contribute

90^\circ + 90^\circ = 180^\circ

to the total.

3

Set up a linear equation representing the ratio of the remaining four angles.

The equation is

4x + 5x + 5x + 6x = 540

, which simplifies to

20x = 540

, yielding

x = 27

.

The angles are proportional to the parts of the ratio, and their sum must equal the remaining

540^\circ

.

4

Calculate the largest angle from the ratio and compare with the right angles.

The largest angle is

6 \times 27 = 162^\circ

.

The largest term in the ratio is

6

, and the resulting angle

162^\circ

is larger than both

90^\circ

and the other calculated angles (

108^\circ

and

135^\circ

).

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and ratio relationships can be solved using algebraic multipliers.

Question 374Question →

For all real values of $n$ that satisfy the inequality $\frac{3 - 2n}{4} < \frac{3n - 22}{5}$ , what is the smallest possible integer value of $n$ ?

Show answer & explanation

Answer: 5

Answer

The smallest integer value of

n

that satisfies the inequality is 5.

Solving the inequality yields

n > \frac{103}{22} \approx 4.68

. The smallest integer greater than

4.68

is

5

.

Step-by-Step Solution

1

Multiply both sides by 20 to clear the denominators.

5(3 - 2n) < 4(3n - 22)

To eliminate the fractions and simplify the inequality.

2

Distribute the constants on both sides.

15 - 10n < 12n - 88

To remove the parentheses.

3

Subtract

12n

from both sides.

15 - 22n < -88

To group the terms containing the variable on the left side.

4

Subtract 15 from both sides.

-22n < -103

To isolate the variable term on the left side.

5

Divide both sides by

-22

and reverse the inequality sign.

n > \frac{103}{22}

Dividing by a negative number reverses the direction of the inequality sign.

6

Convert the fraction to a decimal to identify the boundary.

n > 4.68

To find the smallest integer value that satisfies this condition.

7

Identify the smallest integer greater than

4.68

.

5

The smallest integer greater than

4.68

is

5

.

Key Concept

Solving linear inequalities by applying inverse operations and reversing the inequality sign when multiplying or dividing by a negative number.

Estimated Time:1m 30s

Question 375Question →

An irregular convex decagon (10-sided polygon) has four interior angles that each measure $150^\circ$ . The remaining six interior angles are congruent to each other. What is the degree measure of each of these remaining six angles?

Show answer & explanation

Answer: 140

Answer

The measure of each of the remaining six interior angles is

140^\circ

.

The total sum of the interior angles of a 10-sided convex polygon is

(10-2) \times 180^\circ = 1,440^\circ

. Subtracting the sum of the four angles that each measure

150^\circ

(

4 \times 150^\circ = 600^\circ

) leaves

840^\circ

for the remaining six angles. Since these remaining six angles are congruent, each measures

840^\circ \div 6 = 140^\circ

.

Step-by-Step Solution

1

Calculate the total sum of the interior angles of a convex decagon.

1,440^\circ

The interior angle sum of a polygon with

n

sides is

(n-2) \times 180^\circ

, and a decagon has 10 sides.

2

Find the sum of the four angles measuring

150^\circ

.

600^\circ

Multiply the number of angles by their given degree measure.

3

Determine the sum of the remaining six congruent angles.

840^\circ

Subtract the sum of the four known angles from the total interior angle sum of the decagon.

4

Divide the remaining sum by the number of congruent angles.

140^\circ

Since the remaining six angles are equal in measure, dividing their sum by 6 yields the measure of each individual angle.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

.

Question 376Question →

In the standard $(x, y)$ coordinate plane, square $PQRS$ has adjacent vertices at $P(1, 2)$ and $Q(4, 6)$ . The line containing the side $QR$ has a y-intercept of $b$ . What is the value of $b$ ?

Show answer & explanation

Answer: 9

Answer

The correct answer is 9.

The slope of segment

PQ

is

\frac{6 - 2}{4 - 1} = \frac{4}{3}

. Since adjacent sides of a square are perpendicular, the line containing

QR

is perpendicular to

PQ

and passes through

Q(4, 6)

. The slope of this perpendicular line is the negative reciprocal of

\frac{4}{3}

, which is

-\frac{3}{4}

. Using the slope-intercept form

y = mx + b

with point

Q(4, 6)

, we substitute the values:

6 = -\frac{3}{4}(4) + b \Rightarrow 6 = -3 + b \Rightarrow b = 9

.

Step-by-Step Solution

1

Calculate the slope of side

PQ

using the coordinates of

P(1, 2)

and

Q(4, 6)

.

The slope of

PQ

is

m_{PQ} = \frac{6 - 2}{4 - 1} = \frac{4}{3}

.

To determine the direction of side

PQ

so we can find the perpendicular slope for

QR

.

2

Find the slope of the line containing side

QR

.

The slope of

QR

is

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

.

Because adjacent sides of a square are perpendicular, the slope of

QR

is the negative reciprocal of the slope of

PQ

.

3

Find the equation of the line containing

QR

using the slope-intercept form and the coordinates of vertex

Q(4, 6)

.

Substituting the slope

m = -\frac{3}{4}

and point

(4, 6)

into

y = mx + b

gives

6 = -\frac{3}{4}(4) + b

, which simplifies to

6 = -3 + b

, so

b = 9

.

The line containing side

QR

must pass through vertex

Q

, which allows us to determine the y-intercept

b

.

Key Concept

The slopes of perpendicular lines are negative reciprocals of each other:

m_1 \cdot m_2 = -1

.

Question 377Question →

In the standard $(x, y)$ coordinate plane, a trapezoid has vertices at $A(-5, -4)$ , $B(7, -4)$ , $C(7, 1)$ , and $D(-5, 6)$ . What is the perimeter of this trapezoid?

Show answer & explanation

Answer: 40

Answer

The perimeter of the trapezoid is 40.

The perimeter of the trapezoid is found by summing the lengths of its four sides. Two sides are vertical (AD of length 10 and BC of length 5), one side is horizontal (AB of length 12), and the final side is skewed (CD of length 13, found using the distance formula). Summing these gives 10 + 5 + 12 + 13 = 40.

Step-by-Step Solution

1

Calculate the lengths of the vertical sides AD and BC

AD = 10, BC = 5

Since vertices A(-5, -4) and D(-5, 6) share the same x-coordinate of -5, the side AD is vertical. Its length is the difference in their y-coordinates: 6 - (-4) = 10. Similarly, vertices B(7, -4) and C(7, 1) share the same x-coordinate of 7, so the side BC is vertical. Its length is the difference in their y-coordinates: 1 - (-4) = 5.

2

Calculate the length of the horizontal side AB

AB = 12

Since vertices A(-5, -4) and B(7, -4) share the same y-coordinate of -4, the side AB is horizontal. Its length is the difference in their x-coordinates: 7 - (-5) = 12.

3

Calculate the length of the skewed side CD using the distance formula

CD = 13

Using the coordinates of C(7, 1) and D(-5, 6), the distance is calculated as: CD = \sqrt{(-5 - 7)^2 + (6 - 1)^2} = \sqrt{(-12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13.

4

Sum the lengths of all sides to find the perimeter

Perimeter = 40

The perimeter of a polygon is the sum of the lengths of all its sides: Perimeter = AB + BC + CD + DA = 12 + 5 + 13 + 10 = 40.

Key Concept

Calculating side lengths of a geometric figure on a coordinate plane to find its perimeter

Question 378Question →

In the standard $(x, y)$ coordinate plane, the line $L_1$ is parallel to the line $y = 3x - 7$ . The line $L_2$ is perpendicular to $L_1$ and passes through the point $(6, 5)$ . If the $y$ -intercept of $L_2$ is $(0, c)$ , what is the value of $c$ ?

Show answer & explanation

Answer: 7

Answer

The value of

c

is

7

.

Since the line

L_1

is parallel to

y = 3x - 7

, its slope is

3

. The line

L_2

is perpendicular to

L_1

, so its slope is the negative reciprocal of

3

, which is

-\frac{1}{3}

. Using the point-slope formula with the point

(6, 5)

yields the equation of

L_2

:

y - 5 = -\frac{1}{3}(x - 6)

, which simplifies to

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

. The

y

-intercept of this line is

(0, 7)

, which gives

c = 7

.

Step-by-Step Solution

1

Determine the slope of line

L_1

.

Slope of

L_1

is

3

.

Parallel lines have equal slopes, and the given reference line

y = 3x - 7

has a slope of

3

.

2

Determine the slope of line

L_2

.

Slope of

L_2

is

-\frac{1}{3}

.

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

3

Find the equation of line

L_2

.

The equation is

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

.

Use the point-slope form

y - y_1 = m(x - x_1)

with point

(6, 5)

and slope

-\frac{1}{3}

.

4

Identify the

y

-intercept constant

c

.

c = 7

.

The equation of

L_2

is in slope-intercept form

y = mx + b

, meaning the

y

-intercept is

(0, 7)

.

Key Concept

The relationship between the slopes of parallel lines (which are equal) and perpendicular lines (which are negative reciprocals).

Question 379Question →

An equilateral triangle $ABC$ has a side length of $8$ inches. Point $D$ lies on side $BC$ such that the distance from $B$ to $D$ is $3$ inches. What is the length, in inches, of the segment $AD$ ?

Show answer & explanation

Answer: 7

Answer

The length of segment

AD

is

7

inches.

Dropping altitude

AM

from

A

to

BC

divides the equilateral triangle into two

30^\circ-60^\circ-90^\circ

right triangles. Since

M

is the midpoint of

BC

,

BM = 4

inches. In

\triangle ABM

, the hypotenuse is

8

and the shorter leg is

4

, so the altitude

AM = 4\sqrt{3}

inches. Since

BD = 3

inches, the segment

DM

has length

BM - BD = 4 - 3 = 1

inch. Applying the Pythagorean Theorem to right triangle

\triangle ADM

gives

AD^2 = AM^2 + DM^2 = (4\sqrt{3})^2 + 1^2 = 48 + 1 = 49

, which simplifies to

AD = 7

inches.

Step-by-Step Solution

1

Find the midpoint of side

BC

by dropping altitude

AM

.

BM = 4

inches

In an equilateral triangle, the altitude to a side bisects that side.

2

Calculate the length of the altitude

AM

.

AM = 4\sqrt{3}

inches

The altitude forms a

30^\circ-60^\circ-90^\circ

triangle with the hypotenuse of

8

inches, making the altitude length equal to

8 \times \frac{\sqrt{3}}{2} = 4\sqrt{3}

.

3

Determine the length of the segment

DM

.

DM = 1

inch

Since

D

is

3

inches from

B

and

M

is

4

inches from

B

, the remaining distance is

4 - 3 = 1

.

4

Apply the Pythagorean Theorem on right triangle

\triangle ADM

to find

AD

.

AD = 7

inches

The hypotenuse squared is the sum of the squares of the legs:

AD^2 = (4\sqrt{3})^2 + 1^2 = 48 + 1 = 49

, which gives

AD = 7

.

Key Concept

Using the altitude of an equilateral triangle to create special right triangles and applying the Pythagorean Theorem.

Question 380Question →

On a coordinate map of a harbor, a lighthouse is located at $L(-4, 9)$ and a dock is located at $D(8, -7)$ . A buoy is positioned at the midpoint of the straight-line segment connecting the lighthouse and the dock. If a boat is anchored at $B(-1, 5)$ , what is the distance, in coordinate units, between the boat and the buoy?

Show answer & explanation

Answer: 5

Answer

The distance between the boat and the buoy is 5 units.

To find the distance between the boat and the buoy, first determine the location of the buoy. Since the buoy is at the midpoint of the segment connecting the lighthouse at

L(-4, 9)

and the dock at

D(8, -7)

, we use the midpoint formula:

M = \left(\frac{-4 + 8}{2}, \frac{9 + (-7)}{2}\right) = (2, 1)

. Next, find the distance between the boat at

B(-1, 5)

and the buoy at

M(2, 1)

using the distance formula:

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

.

Step-by-Step Solution

1

Find the coordinates of the buoy, which is the midpoint

M(x_m, y_m)

of the segment connecting the lighthouse

L(-4, 9)

and the dock

D(8, -7)

.

M(2, 1)

The midpoint formula is given by

x_m = \frac{x_1 + x_2}{2}

and

y_m = \frac{y_1 + y_2}{2}

. Substituting the coordinates of

L

and

D

, we get

x_m = \frac{-4 + 8}{2} = 2

and

y_m = \frac{9 + (-7)}{2} = 1

.

2

Calculate the distance

d

between the boat at

B(-1, 5)

and the buoy at

M(2, 1)

.

5

units

The distance formula is given by

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

. Substituting the coordinates of

B

and

M

yields

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

.

Key Concept

Applying the midpoint and distance formulas sequentially to solve a coordinate geometry word problem.

Answer

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