All practice questions

Question 341Question →

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 342Question →

A circle in the standard $(x, y)$ coordinate plane is described by the equation $x^2 + y^2 = 25$ . A line is described by the equation $3x - 4y = c$ , where $c$ is a positive constant. If the system of these two equations has exactly one real solution for $(x, y)$ , what is the value of $c$ ?

Show answer & explanation

Answer: 25

Answer

The value of

c

is

25

.

The correct answer is

25

. The equation

x^2 + y^2 = 25

represents a circle centered at

(0, 0)

with a radius of

5

. For the system of equations to have exactly one real solution, the line

3x - 4y = c

must be tangent to the circle. The perpendicular distance from the center

(0,0)

to the line

3x - 4y - c = 0

is given by

\frac{|3(0) - 4(0) - c|}{\sqrt{3^2 + (-4)^2}} = \frac{|c|}{5}

. Setting this distance equal to the radius of the circle yields

\frac{|c|}{5} = 5

, which gives

|c| = 25

. Since

c

is specified as a positive constant,

c

must be

25

.

Step-by-Step Solution

1

Find the center and radius of the circle.

Center is

(0, 0)

and radius is

r = 5

.

The circle equation

x^2 + y^2 = 25

is in the standard form

x^2 + y^2 = r^2

centered at the origin with radius

r = \sqrt{25} = 5

.

2

Set up the condition for tangency (exactly one real solution).

The perpendicular distance from the center

(0,0)

to the line

3x - 4y - c = 0

must equal the radius

5

.

A line intersects a circle at exactly one point if and only if the line is tangent to the circle.

3

Apply the point-to-line distance formula.

Distance

d = \frac{|3(0) - 4(0) - c|}{\sqrt{3^2 + (-4)^2}} = \frac{|c|}{5}

.

The formula for the distance from

(x_0, y_0)

to the line

Ax + By + C = 0

is

d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}

.

4

Solve for the positive constant

c

.

c = 25

Setting the distance

\frac{|c|}{5}

equal to the radius

5

gives

|c| = 25

. Since

c

is a positive constant,

c = 25

.

Key Concept

Determining conditions for tangency in a system of linear and circular equations.

Alternative Method

Instead of using the geometric distance formula, the system can be solved algebraically by substitution. Express

y

in terms of

x

from the linear equation:

y = \frac{3x - c}{4}

. Substitute this expression into the circle's equation:

x^2 + \left(\frac{3x - c}{4}\right)^2 = 25

. Expand the terms and multiply by

16

to clear the denominator:

16x^2 + 9x^2 - 6cx + c^2 = 400

, which simplifies to the quadratic equation

25x^2 - 6cx + (c^2 - 400) = 0

. For the system to have exactly one solution, this quadratic equation must have a discriminant equal to zero. Calculate the discriminant:

D = (-6c)^2 - 4(25)(c^2 - 400) = 36c^2 - 100c^2 + 40000 = -64c^2 + 40000 = 0

. Solving for

c

yields

64c^2 = 40000 \implies c^2 = 625

. Since

c

is a positive constant,

c = 25

.

Estimated Time:1m 30s

Question 343Question →

One of the solutions to the quadratic equation $0.5x^2 + bx - 6 = 0$ , where $b$ is a constant, is $x = 3$ . What is the value of the other solution?

Show answer & explanation

Answer: -4

Answer

The other solution to the quadratic equation is -4.

Substituting the given solution

x = 3

into the equation yields

0.5(3)^2 + 3b - 6 = 0

. Simplifying this expression gives

4.5 + 3b - 6 = 0

, which leads to

3b = 1.5

and thus

b = 0.5

. With

b = 0.5

, the quadratic equation becomes

0.5x^2 + 0.5x - 6 = 0

. Multiplying the entire equation by 2 to obtain integer coefficients results in

x^2 + x - 12 = 0

. This quadratic factors into

(x - 3)(x + 4) = 0

, which gives the solutions

x = 3

and

x = -4

. Therefore, the other solution is

-4

. Alternatively, using Vieta's formulas, the product of the roots of a quadratic equation

ax^2 + bx + c = 0

is equal to

c/a

. Here, the product of the roots is

-6 / 0.5 = -12

. Since one root is

3

, the other root must be

-12 / 3 = -4

.

Step-by-Step Solution

1

Substitute the given solution

x = 3

into the quadratic equation to find the value of

b

.

b = 0.5

Since

x = 3

is a solution, it must satisfy the equation, allowing us to solve for the unknown coefficient

b

.

2

Rewrite the equation using

b = 0.5

and simplify by multiplying all terms by 2.

x^2 + x - 12 = 0

Multiplying the equation by 2 eliminates the decimal coefficients, making the quadratic expression easier to factor.

3

Factor the quadratic equation to determine the roots.

x = 3

or

x = -4

The equation factors into

(x - 3)(x + 4) = 0

. Solving for

x

yields the given root of 3 and the second root of -4.

Key Concept

Solving quadratic equations by utilizing a known solution to determine unknown coefficients, and applying factoring techniques or root relationships to find the remaining solution.

Estimated Time:1m 30s

Question 344Question →

In $\triangle ABC$ , the measure of exterior angle $\angle ACD$ is $135^\circ$ , where $D$ lies on the extension of side $BC$ past $C$ . If the measure of interior angle $\angle A$ is $25^\circ$ greater than the measure of interior angle $\angle B$ , what is the measure, in degrees, of $\angle B$ ?

Show answer & explanation

Answer: 55

Answer

55

According to the Exterior Angle Theorem, the measure of exterior angle

\angle ACD

is equal to the sum of the measures of its remote interior angles,

\angle A

and

\angle B

. This gives the equation

\text{m}\angle A + \text{m}\angle B = 135^\circ

. Using the information that

\text{m}\angle A = \text{m}\angle B + 25^\circ

, we substitute this expression into the equation to get

(\text{m}\angle B + 25^\circ) + \text{m}\angle B = 135^\circ

. Simplifying this equation gives

2\text{m}\angle B + 25 = 135

, which simplifies to

2\text{m}\angle B = 110

, and dividing by 2 yields

\text{m}\angle B = 55^\circ

.

Step-by-Step Solution

1

Apply the Exterior Angle Theorem to express the relation between the exterior angle and the two remote interior angles.

\text{m}\angle A + \text{m}\angle B = 135^\circ

The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

2

Substitute the relationship between the interior angles into the equation.

(\text{m}\angle B + 25^\circ) + \text{m}\angle B = 135^\circ

The problem states that the measure of interior angle

\angle A

is

25^\circ

greater than the measure of interior angle

\angle B

.

3

Solve the algebraic equation for the measure of interior angle

\angle B

.

\text{m}\angle B = 55^\circ

Combining like terms gives

2\text{m}\angle B + 25 = 135

. Subtracting 25 from both sides gives

2\text{m}\angle B = 110

. Dividing by 2 yields

\text{m}\angle B = 55^\circ

.

Key Concept

Exterior Angle Theorem and remote interior angles relation

Question 345Question →

A convex polygon has $n$ sides. The sum of the measures of its interior angles is $3$ times the sum of the measures of its exterior angles. What is the value of $n$ ?

Show answer & explanation

Answer: 8

Answer

8

The sum of the interior angle measures of an

n

-sided convex polygon is

(n-2) \times 180^\circ

. The sum of the exterior angle measures is always

360^\circ

. According to the problem, the sum of the interior angles is

3

times the sum of the exterior angles, which can be written as the equation

(n-2) \times 180 = 3 \times 360

. Simplifying the right side gives

(n-2) \times 180 = 1080

. Dividing both sides by

180

results in

n - 2 = 6

. Adding

2

to both sides yields

n = 8

.

Step-by-Step Solution

1

Use the formula for the sum of the interior angle measures of a convex polygon with

n

sides.

The sum of the interior angles is

(n-2) \times 180^\circ

.

By the polygon interior angle sum theorem, the sum of the interior angles of any convex

n

-gon is

(n-2) \times 180^\circ

.

2

Identify the sum of the exterior angle measures of a convex polygon.

The sum of the exterior angles is

360^\circ

.

The sum of the exterior angles of any convex polygon is always constant and equals

360^\circ

, regardless of the number of sides.

3

Set up the equation relating the two sums as given in the problem statement.

(n-2) \times 180^\circ = 3 \times 360^\circ

The problem states that the sum of the interior angles is

3

times the sum of the exterior angles.

4

Solve the equation for

n

.

n = 8

Divide both sides by

180^\circ

to get

n - 2 = 6

, then add

2

to both sides to find

n = 8

.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and the sum of its exterior angles is

360^\circ

.

Question 346Question →

What is the greatest integer value of $k$ that satisfies the inequality $8 - 3(2k - 5) \geq 4(k + 6)$ ?

Show answer & explanation

Answer: -1

Answer

The greatest integer value of

k

that satisfies the inequality is

-1

.

Solving the inequality step-by-step yields

k \leq -0.1

. The greatest integer less than or equal to

-0.1

is

-1

.

Step-by-Step Solution

1

Distribute the coefficients to remove parentheses

8 - 6k + 15 \geq 4k + 24

Expanding the terms makes it possible to combine like terms on each side of the inequality.

2

Combine like terms on the left side

23 - 6k \geq 4k + 24

Simplifying the constant values on the left side (

8 + 15 = 23

) simplifies the expression.

3

Subtract

4k

from both sides

23 - 10k \geq 24

This groups all the variable terms on the left-hand side.

4

Subtract

23

from both sides

-10k \geq 1

This isolates the variable term on the left-hand side.

5

Divide by

-10

and flip the inequality sign

k \leq -0.1

Dividing both sides by a negative number requires reversing the direction of the inequality sign.

6

Identify the greatest integer satisfying the inequality

k = -1

The largest integer that is less than or equal to

-0.1

is

-1

.

Key Concept

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

Estimated Time:1m 30s

Question 347Question →

A regular decagon has $10$ sides of equal length. What is the measure, in degrees, of one exterior angle of this decagon?

Show answer & explanation

Answer: 36

Answer

The correct answer is 36.

The sum of the exterior angles of any convex polygon is always

360^\circ

. A regular decagon has

10

congruent sides and therefore

10

congruent exterior angles. Dividing

360^\circ

by

10

yields

36^\circ

for each exterior angle.

Step-by-Step Solution

1

Determine the number of exterior angles in a regular decagon.

A decagon has 10 sides, so it has 10 exterior angles.

A polygon has the same number of exterior angles as its number of sides.

2

State the sum of the exterior angles for a convex polygon.

The sum of the exterior angles is 360 degrees.

The exterior angles of any convex polygon always sum to 360 degrees regardless of the number of sides.

3

Calculate the measure of one exterior angle.

360 / 10 = 36

Since the decagon is regular, all of its exterior angles are equal in measure, so we divide the total sum by the number of angles.

Key Concept

The sum of the exterior angles of any convex polygon is

360^\circ

. For a regular polygon with

n

sides, the measure of each exterior angle is

\frac{360^\circ}{n}

.

Question 348Question →

A business analyst models a company's weekly net profit, $N(x)$ , in dollars, as the difference between its weekly revenue, $R(x) = (2x + 5)(30 - x)$ , and its weekly production cost, $C(x) = (x - 4)^2 + 150$ , where $x$ represents the number of items sold. When $N(x)$ is simplified and written in standard form as $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are integers, what is the value of the coefficient $b$ ?

Show answer & explanation

Answer: 63

Answer

The coefficient

b

is

63

.

Expanding the revenue polynomial gives

R(x) = -2x^2 + 55x + 150

and the cost polynomial gives

C(x) = x^2 - 8x + 166

. Subtracting them gives

N(x) = (-2x^2 + 55x + 150) - (x^2 - 8x + 166) = -3x^2 + 63x - 16

. The coefficient of the

x

term, which corresponds to

b

, is

63

.

Step-by-Step Solution

1

Expand the revenue expression

R(x) = (2x + 5)(30 - x)

using polynomial multiplication.

R(x) = -2x^2 + 55x + 150

To find the polynomial representing revenue in standard form.

2

Expand the cost expression

C(x) = (x - 4)^2 + 150

using binomial squaring.

C(x) = x^2 - 8x + 166

To find the polynomial representing cost in standard form.

3

Subtract the cost polynomial from the revenue polynomial:

N(x) = R(x) - C(x)

, distributing the negative sign through all terms of the cost polynomial.

N(x) = -3x^2 + 63x - 16

To determine the net profit polynomial

N(x)

in standard form.

4

Identify the coefficient

b

of the

x

term in the standard form

ax^2 + bx + c

.

b = 63

To answer the specific question asking for the coefficient of the middle term.

Key Concept

Polynomial subtraction and expansion of algebraic expressions.

Question 349Question →

An irregular convex polygon has $n$ sides. The measures of its interior angles, in degrees, are all distinct integers. If all of the interior angles are obtuse, what is the maximum possible value of $n$ ?

Show answer & explanation

Answer: 26

Answer

The maximum possible value of

n

is 26.

For a convex

26

-gon, the sum of the interior angles is

24 \times 180^\circ = 4320^\circ

. We can choose

26

distinct integer angles from the range

[91^\circ, 179^\circ]

that sum to exactly

4320^\circ

because the maximum possible sum of

26

distinct integers in this range is

4329^\circ

, which is greater than

4320^\circ

. For

n = 27

, the sum of the interior angles must be

25 \times 180^\circ = 4500^\circ

, but the maximum possible sum of

27

distinct integers in the range is only

4482^\circ

, which is less than

4500^\circ

. Therefore,

26

is the maximum value of

n

.

Step-by-Step Solution

1

Determine the set of possible angle measures.

The angles must be integers in the range

[91^\circ, 179^\circ]

.

Interior angles of a convex polygon must be less than

180^\circ

. Since they are obtuse and distinct integers, they must be strictly greater than

90^\circ

, giving the range

[91, 179]

.

2

Write the sum of the interior angles of a convex

n

-gon.

Sum

= (n - 2) \times 180^\circ

.

This is the standard formula for the sum of the interior angles of any convex

n

-gon.

3

Find the maximum possible sum of

n

distinct angles in the range

[91, 179]

.

Maximum Sum

= 179n - \frac{n(n - 1)}{2}

.

The maximum sum is achieved by selecting the largest

n

integers from the set:

179, 178, \dots, 179 - (n - 1)

.

4

Set up the inequality and simplify.

n^2 + n - 720 \le 0

.

Since the sum of the angles must be less than or equal to the maximum possible sum, we have

(n-2) \times 180 \le 179n - \frac{n(n-1)}{2}

. Multiplying by 2 and simplifying yields the quadratic inequality.

5

Solve the quadratic inequality for the largest integer

n

.

n = 26

.

Evaluating the quadratic expression for consecutive integers: for

n = 26

,

26^2 + 26 - 720 = -18 \le 0

; for

n = 27

,

27^2 + 27 - 720 = 36 > 0

. Thus, 26 is the maximum possible integer value.

Key Concept

Sum of interior angles of a convex polygon combined with algebraic optimization.

Question 350Question →

A convex polygon has $n$ sides. The sum of the measures of its interior angles is $6$ times the sum of the measures of its exterior angles (one at each vertex). What is the value of $n$ ?

Show answer & explanation

Answer: 14

Answer

The number of sides,

n

, of the convex polygon is 14.

The sum of the interior angles of a convex polygon with

n

sides is given by the formula

(n-2) \times 180^\circ

, and the sum of its exterior angles is always

360^\circ

. According to the problem, the sum of the interior angles is

6

times the sum of the exterior angles, yielding the equation

(n - 2) \times 180 = 6 \times 360

. Dividing both sides of the equation by

180

gives

n - 2 = 12

. Adding

2

to both sides results in

n = 14

.

Step-by-Step Solution

1

State the sum of interior and exterior angles formulas.

Interior sum =

(n-2) \times 180^\circ

, Exterior sum =

360^\circ

.

To represent the geometric properties of the polygon algebraically.

2

Set up the equation based on the given ratio.

(n-2) \times 180 = 6 \times 360

.

The problem states the interior sum is 6 times the exterior sum.

3

Solve the equation for

n

.

n = 14

.

Divide by 180 to get

n - 2 = 12

, then add 2 to both sides.

Key Concept

The sum of the interior angles of an

n

-sided convex polygon is

(n-2) \times 180^\circ

, and the sum of the exterior angles (one per vertex) is always

360^\circ

.

Question 351Question →

In the quadratic equation $2x^2 - 11x + c = 0$ , where $c$ is a constant, the ratio of the two real solutions is $3:8$ . What is the value of $c$ ?

Show answer & explanation

Answer: 12

Answer

The value of the constant

c

is

12

.

The correct answer is

12

. By representing the roots in the ratio of

3:8

as

3r

and

8r

, Vieta's formula for the sum of roots (

-\frac{b}{a}

) gives

3r + 8r = -\frac{-11}{2} \implies 11r = 5.5 \implies r = 0.5

. The actual roots are therefore

1.5

and

4

. Using Vieta's formula for the product of roots (

\frac{c}{a}

) gives

(1.5)(4) = \frac{c}{2} \implies 6 = \frac{c}{2} \implies c = 12

.

Step-by-Step Solution

1

Represent the roots using the given ratio.

Let the two roots of the quadratic equation be

3r

and

8r

.

The ratio of the two solutions is specified as

3:8

.

2

Apply Vieta's formula for the sum of roots to find the ratio multiplier

r

.

3r + 8r = -\frac{-11}{2} \implies 11r = 5.5 \implies r = 0.5

.

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is given by

-\frac{b}{a}

.

3

Determine the numerical values of the two roots.

The roots are

3(0.5) = 1.5

and

8(0.5) = 4

.

Substitute the value of

r = 0.5

back into the expressions for the roots.

4

Apply Vieta's formula for the product of roots to solve for the constant

c

.

(1.5)(4) = \frac{c}{2} \implies 6 = \frac{c}{2} \implies c = 12

.

For a quadratic equation

ax^2 + bx + c = 0

, the product of the roots is given by

\frac{c}{a}

.

Key Concept

Vieta's formulas and the relationship between the roots and coefficients of a quadratic equation

Alternative Method

Alternatively, you can express the roots using the quadratic formula:

x = \frac{11 \pm \sqrt{121 - 8c}}{4}

. Since the ratio of the smaller root to the larger root is

3:8

, we set up the equation:

\frac{11 - \sqrt{121 - 8c}}{11 + \sqrt{121 - 8c}} = \frac{3}{8}

. Cross-multiplying gives

8(11 - \sqrt{121 - 8c}) = 3(11 + \sqrt{121 - 8c}) \implies 88 - 8\sqrt{121 - 8c} = 33 + 3\sqrt{121 - 8c} \implies 55 = 11\sqrt{121 - 8c} \implies 5 = \sqrt{121 - 8c} \implies 25 = 121 - 8c \implies 8c = 96 \implies c = 12

.

Estimated Time:1m 30s

Question 352Question →

For all real values of $x$ that satisfy the inequality $5 - \frac{3x - 2}{4} \ge \frac{x + 6}{2}$ , the solution set is represented by $x \le b$ . What is the value of $b$ ?

Show answer & explanation

Answer: 2

Answer

The value of

b

is 2.

To solve

5 - \frac{3x - 2}{4} \ge \frac{x + 6}{2}

, multiply all terms by 4 to clear the denominators, resulting in

20 - (3x - 2) \ge 2(x + 6)

. Carefully distribute the negative sign to obtain

20 - 3x + 2 \ge 2x + 12

. Simplifying the left side gives

22 - 3x \ge 2x + 12

. Moving the variable terms to one side yields

10 \ge 5x

, which simplifies to

x \le 2

. Thus, the upper bound value

b

is 2.

Step-by-Step Solution

1

Multiply both sides of the inequality by the least common denominator, which is 4.

20 - (3x - 2) \ge 2(x + 6)

Multiplying all terms by the common denominator eliminates fractions and simplifies the inequality.

2

Distribute the negative sign to the numerator terms on the left and distribute the 2 on the right.

20 - 3x + 2 \ge 2x + 12

Distributing the negative sign across

(3x - 2)

changes it to

-3x + 2

. Distributing 2 across

(x + 6)

yields

2x + 12

.

3

Combine like terms on the left side of the inequality.

22 - 3x \ge 2x + 12

Combining the constant terms

20

and

2

simplifies the expression to

22

.

4

Isolate the variable terms by adding

3x

and subtracting 12 from both sides.

10 \ge 5x

Grouping variables on one side and constants on the other allows us to solve for

x

.

5

Divide both sides by 5.

2 \ge x

(or

x \le 2

)

Dividing by a positive number isolates the variable without changing the direction of the inequality sign.

Key Concept

Solving linear inequalities involving fractions and distributing negative coefficients.

Alternative Method

We can write the inequality by separating each fraction term first:

5 - \frac{3}{4}x + \frac{2}{4} \ge \frac{1}{2}x + \frac{6}{2}

. This simplifies to

5.5 - 0.75x \ge 0.5x + 3

. Subtracting

0.5x

and

5.5

from both sides gives

-1.25x \ge -2.5

. Dividing by

-1.25

and reversing the inequality sign gives

x \le 2

.

Estimated Time:1m 30s

Question 353Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 = 25$ and a line is defined by the equation $3x + 4y = 15$ . The line intersects the circle at two points, $A$ and $B$ . What is the distance between point $A$ and point $B$ ?

Show answer & explanation

Answer: 8

Answer

The distance between the two intersection points is 8.

The correct answer is 8. We can find the intersection points by substituting

y = \frac{15 - 3x}{4}

into the circle's equation

x^2 + y^2 = 25

, yielding the quadratic equation

5x^2 - 18x - 35 = 0

. Solving this gives

x = 5

and

x = -1.4

, with corresponding

y

-coordinates

y = 0

and

y = 4.8

. The distance between

(5, 0)

and

(-1.4, 4.8)

is

\sqrt{(-1.4 - 5)^2 + (4.8 - 0)^2} = \sqrt{6.4^2 + 4.8^2} = \sqrt{64} = 8

. Alternatively, we can use geometry: the distance from the center of the circle

(0,0)

to the line

3x + 4y - 15 = 0

is

d = \frac{|3(0) + 4(0) - 15|}{\sqrt{3^2 + 4^2}} = 3

. Since the radius of the circle is

r = 5

, the right triangle formed by the radius, the perpendicular segment, and half the chord has a half-chord length of

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

. Thus, the total chord length is

2 \times 4 = 8

.

Step-by-Step Solution

1

Express the linear equation in terms of one variable

y = \frac{15 - 3x}{4}

This allows for substitution into the equation of the circle.

2

Substitute the expression into the circle's equation and simplify

x^2 + \left(\frac{15 - 3x}{4}\right)^2 = 25 \Rightarrow 16x^2 + (225 - 90x + 9x^2) = 400 \Rightarrow 25x^2 - 90x - 175 = 0 \Rightarrow 5x^2 - 18x - 35 = 0

To create a single quadratic equation in terms of

x

representing the intersection points.

3

Solve the quadratic equation for

x

(5x + 7)(x - 5) = 0 \Rightarrow x = 5

or

x = -1.4

To find the

x

-coordinates of the intersection points.

4

Calculate the corresponding

y

-coordinates

For

x = 5

,

y = 0

, giving point

A(5, 0)

. For

x = -1.4

,

y = 4.8

, giving point

B(-1.4, 4.8)

.

To determine the exact coordinates of both intersection points.

5

Apply the distance formula to find the length of the segment

AB

d = \sqrt{(-1.4 - 5)^2 + (4.8 - 0)^2} = \sqrt{(-6.4)^2 + 4.8^2} = \sqrt{40.96 + 23.04} = \sqrt{64} = 8

To compute the final distance between the two intersection points.

Key Concept

Solving systems of linear and quadratic equations to determine intersection points and calculating the distance between coordinates.

Question 354Question →

A regular hexagon $ABCDEF$ has a side length of $8$ inches. Point $M$ lies on side $CD$ such that the length of segment $CM$ is $2$ inches. What is the length, in inches, of segment $AM$ ?

Show answer & explanation

Answer: 14

Answer

14

The correct answer is 14 because

\triangle ACM

is a right triangle with legs

AC = 8\sqrt{3}

and

CM = 2

. Applying the Pythagorean Theorem yields

AM^2 = (8\sqrt{3})^2 + 2^2 = 192 + 4 = 196

, so

AM = \sqrt{196} = 14

.

Step-by-Step Solution

1

Find the properties of the regular hexagon and the diagonal

AC

.

The interior angle at vertex

B

is

120^\circ

. Since

AB = BC = 8

, the triangle

\triangle ABC

is an isosceles triangle with angles

\angle BAC = \angle BCA = 30^\circ

. Using the properties of

30^\circ

-

60^\circ

-

90^\circ

triangles, the diagonal length is

AC = 8\sqrt{3}

.

To find the length of the leg

AC

for the right triangle

\triangle ACM

.

2

Determine the angle

\angle ACD

to show

\triangle ACM

is a right triangle.

Since the interior angle

\angle BCD = 120^\circ

and

\angle BCA = 30^\circ

, the remaining angle is

\angle ACD = 120^\circ - 30^\circ = 90^\circ

. Thus,

\triangle ACM

is a right triangle with the right angle at vertex

C

.

To establish the right-angle relationship between the legs

AC

and

CM

.

3

Apply the Pythagorean Theorem to calculate the hypotenuse

AM

.

AM^2 = AC^2 + CM^2 = (8\sqrt{3})^2 + 2^2 = 192 + 4 = 196

. Taking the square root gives

AM = 14

.

To find the final length of segment

AM

.

Key Concept

Using properties of regular hexagons, special right triangles, and the Pythagorean Theorem to find lengths in multi-step plane geometry configurations.

Question 355Question →

In the standard $(x, y)$ coordinate plane, line $L_1$ passes through the points $(2, 5)$ and $(6, -3)$ . A second line, $L_2$ , is perpendicular to $L_1$ and intersects $L_1$ at its $y$ -intercept. What is the $x$ -coordinate of the $x$ -intercept of $L_2$ ?

Show answer & explanation

Answer: -18

Answer

The

x

-coordinate of the

x

-intercept of

L_2

is

-18

.

First, the slope of

L_1

is calculated as

-2

using the slope formula. Substituting one of the points into the slope-intercept form gives the

y

-intercept of

L_1

as

(0, 9)

. Since

L_2

is perpendicular to

L_1

, its slope is the negative reciprocal of

-2

, which is

\frac{1}{2}

. Since

L_2

shares the

y

-intercept

(0, 9)

, its equation is

y = \frac{1}{2}x + 9

. Setting

y = 0

to find the

x

-intercept gives

x = -18

.

Step-by-Step Solution

1

Calculate the slope of line

L_1

.

The slope of

L_1

is

-2

.

Using the slope formula

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

with the given points

(2, 5)

and

(6, -3)

yields

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

.

2

Find the

y

-intercept of

L_1

.

The

y

-intercept is

(0, 9)

.

Substituting

m = -2

and the coordinates of

(2, 5)

into the slope-intercept equation

y = mx + b

gives

5 = -2(2) + b

, which simplifies to

b = 9

.

3

Find the slope of the perpendicular line,

L_2

.

The slope of

L_2

is

\frac{1}{2}

.

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

-2

is

\frac{1}{2}

.

4

Determine the equation of

L_2

and calculate its

x

-intercept.

The

x

-coordinate of the

x

-intercept of

L_2

is

-18

.

Since

L_2

passes through the

y

-intercept

(0, 9)

, its equation is

y = \frac{1}{2}x + 9

. Setting

y = 0

to find the

x

-intercept yields

0 = \frac{1}{2}x + 9

, which solves to

x = -18

.

Key Concept

Determining the equation and intercepts of a line perpendicular to a given line that passes through a specific shared point.

Estimated Time:1m 30s

Question 356Question →

In the standard $(x, y)$ coordinate plane, a line segment has endpoints $A(-3, k)$ and $B(5, 3)$ . If the midpoint of segment $AB$ lies on the $x$ -axis, what is the length of segment $AB$ ?

Show answer & explanation

Answer: 10

Answer

The length of segment

AB

is 10.

The midpoint of segment

AB

with endpoints

A(-3, k)

and

B(5, 3)

is

\left(1, \frac{k + 3}{2}\right)

. Since the midpoint lies on the

x

-axis, its

y

-coordinate must be

0

. Solving

\frac{k + 3}{2} = 0

gives

k = -3

. This means the endpoints are

A(-3, -3)

and

B(5, 3)

. The distance between these two points is

\sqrt{(5 - (-3))^2 + (3 - (-3))^2} = \sqrt{8^2 + 6^2} = \sqrt{100} = 10

.

Step-by-Step Solution

1

Set up the equation for the

y

-coordinate of the midpoint.

\frac{k + 3}{2} = 0

The midpoint of a segment with endpoints

(x_1, y_1)

and

(x_2, y_2)

has a

y

-coordinate of

\frac{y_1 + y_2}{2}

. Since the midpoint lies on the

x

-axis, its

y

-coordinate must be

0

.

2

Solve the equation for

k

.

k = -3

Multiply both sides of the equation by

2

to get

k + 3 = 0

, then subtract

3

from both sides.

3

Substitute

k = -3

to find the coordinates of point

A

.

A(-3, -3)

This provides both complete endpoints,

A(-3, -3)

and

B(5, 3)

, which are needed to find the distance.

4

Apply the distance formula to find the length of segment

AB

.

AB = 10

The distance formula is

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

. Substituting the coordinates gives

AB = \sqrt{(5 - (-3))^2 + (3 - (-3))^2} = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10

.

Key Concept

Distance and Midpoint Formulas

Question 357Question →

In isosceles trapezoid $ABCD$ , the parallel bases are $AB$ and $CD$ . If the measure of interior angle $A$ is $70^\circ$ , what is the measure, in degrees, of interior angle $C$ ?

Show answer & explanation

Answer: 110

Answer

The measure of interior angle

C

is

110

degrees.

Since the trapezoid is isosceles with bases

AB

and

CD

, the base angles

\angle A

and

\angle B

are congruent, so

\angle B = 70^\circ

. The consecutive interior angles along the leg

BC

are supplementary because

AB \parallel CD

, which means

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

. Solving for

\angle C

gives

180^\circ - 70^\circ = 110^\circ

.

Step-by-Step Solution

1

Find the measure of angle

B

using the properties of an isosceles trapezoid.

\angle B = 70^\circ

In an isosceles trapezoid, the angles sharing a base are congruent. Since

AB

is a base,

\angle A = \angle B = 70^\circ

.

2

Calculate the measure of angle

C

using the parallel lines property.

\angle C = 110^\circ

Because the bases

AB

and

CD

are parallel, the consecutive interior angles

\angle B

and

\angle C

must sum to

180^\circ

. Therefore,

\angle C = 180^\circ - 70^\circ = 110^\circ

.

Key Concept

Properties of an isosceles trapezoid

Alternative Method

Since the sum of interior angles in any quadrilateral is

360^\circ

, and in an isosceles trapezoid the base angles are equal (

\angle A = \angle B = 70^\circ

and

\angle C = \angle D

), we can write

70^\circ + 70^\circ + \angle C + \angle D = 360^\circ

. Since

\angle C = \angle D

, this simplifies to

140^\circ + 2\angle C = 360^\circ \implies 2\angle C = 220^\circ \implies \angle C = 110^\circ

.

Estimated Time:45s

Question 358Question →

The sum of the measures of all but one of the interior angles of a convex polygon is $2010^\circ$ . What is the measure, in degrees, of the remaining interior angle?

Show answer & explanation

Answer: 150

Answer

The measure of the remaining interior angle is 150 degrees.

The sum of the interior angles of any convex polygon with

n

sides is a multiple of

180^\circ

given by

(n-2) \times 180^\circ

. Because the polygon is convex, the measure of the remaining angle must be strictly less than

180^\circ

. Thus, the total sum of all interior angles must be the smallest multiple of

180^\circ

that is strictly greater than the given sum of

2010^\circ

. Since

11 \times 180^\circ = 1980^\circ

(which is less than

2010^\circ

), the total sum must be at least

12 \times 180^\circ = 2160^\circ

. Subtracting the given sum of the other angles from this total gives

2160^\circ - 2010^\circ = 150^\circ

. Since

150^\circ

is less than

180^\circ

, this is a mathematically valid remaining angle for a convex polygon.

Step-by-Step Solution

1

Set up the inequality for the sum of the interior angles.

The total sum

S = (n-2) \times 180^\circ

must satisfy

2010^\circ < S < 2010^\circ + 180^\circ

, which simplifies to

2010^\circ < S < 2190^\circ

.

Since the polygon is convex, the remaining interior angle must have a measure strictly between

0^\circ

and

180^\circ

.

2

Determine the value of

n-2

by finding the unique integer multiple.

Dividing the inequality by

180^\circ

gives

11.17 < n-2 < 12.17

. Since

n

must be an integer,

n-2 = 12

, which means the polygon has

n = 14

sides.

The number of sides of a polygon must be a whole number, so

n-2

must be an integer.

3

Calculate the measure of the remaining interior angle.

x = (12 \times 180^\circ) - 2010^\circ = 2160^\circ - 2010^\circ = 150^\circ

.

Subtract the sum of the other interior angles from the total sum of the interior angles of a 14-gon.

Key Concept

The sum of the interior angles of a convex polygon with

n

sides is

(n-2) \times 180^\circ

, and each interior angle of a convex polygon must measure strictly less than

180^\circ

.

Question 359Question →

A convex polygon has $n$ sides. The sum of the measures of all but one of its interior angles is $2020^\circ$ . What is the measure, in degrees, of the remaining interior angle?

Show answer & explanation

Answer: 140

Answer

The measure of the remaining interior angle is

140^\circ

.

For a convex polygon with

n

sides, the sum of all interior angles is

(n-2) \times 180^\circ

. If we represent the remaining interior angle as

x

, then the sum of all interior angles can be written as

2020^\circ + x

. Because the polygon is convex, the measure of the remaining interior angle must satisfy the inequality

0^\circ < x < 180^\circ

. Substituting this into the sum expression gives the inequality

2020^\circ < (n-2) \times 180^\circ < 2200^\circ

. Dividing by

180^\circ

, we find

11.22 < n-2 < 12.22

. Since

n-2

must be an integer,

n-2

must equal

12

, which means the total sum of the interior angles is

12 \times 180^\circ = 2160^\circ

. The remaining angle is found by subtracting

2020^\circ

from

2160^\circ

, resulting in

140^\circ

.

Step-by-Step Solution

1

Use the polygon interior angle sum formula for an

n

-sided polygon.

The sum of all interior angles is

(n-2) \times 180^\circ

.

This formula connects the number of sides of a convex polygon to the total sum of its interior angles.

2

Set up an equation containing the sum of all but one angle (

2020^\circ

) and the remaining angle (

x

).

(n-2) \times 180^\circ = 2020^\circ + x

The total sum of all interior angles is equal to the sum of the

n-1

known angles plus the remaining angle.

3

Apply the convexity constraint

0^\circ < x < 180^\circ

to construct an inequality for the total sum of the interior angles.

2020^\circ < (n-2) \times 180^\circ < 2200^\circ

Since the remaining angle must be strictly between

0^\circ

and

180^\circ

for a convex polygon, adding

2020^\circ

gives the boundaries for the total sum.

4

Divide the inequality by

180^\circ

to isolate the term

n-2

.

11.22 < n-2 < 12.22

This determines the numerical boundaries for the integer value of

n-2

.

5

Find the unique integer value for

n-2

and calculate the exact total sum of the interior angles.

n-2 = 12

, which gives a total sum of

12 \times 180^\circ = 2160^\circ

.

Because

n

must be an integer,

n-2

must be an integer. The only integer in the interval

(11.22, 12.22)

is

12

.

6

Subtract the sum of the other angles from the total sum of the interior angles to solve for

x

.

x = 2160^\circ - 2020^\circ = 140^\circ

.

This yields the exact value of the remaining interior angle.

Key Concept

The sum of the interior angles of an

n

-sided convex polygon is

(n-2) \times 180^\circ

, where each interior angle is strictly between

0^\circ

and

180^\circ

.

Question 360Question →

In the standard $(x, y)$ coordinate plane, a rhombus $ABCD$ has vertices $A(1, 2)$ and $C(7, 10)$ . The length of diagonal $BD$ is half the length of diagonal $AC$ . If the $x$ -coordinate of vertex $B$ is greater than the $x$ -coordinate of vertex $D$ , what is the $y$ -coordinate of vertex $B$ ?

Show answer & explanation

Answer: 4.5

Answer

The

y

-coordinate of vertex

B

is

4.5

.

By using the geometric properties of a rhombus, we know that its diagonals bisect each other perpendicularly. The midpoint of diagonal

AC

is calculated as

M(4, 6)

and its length is

10

. Consequently, the perpendicular diagonal

BD

must pass through

M(4, 6)

with a slope of

-\frac{3}{4}

(the negative reciprocal of the slope of

AC

, which is

\frac{4}{3}

). Since the length of

BD

is half the length of

AC

, the length of

BD

is

5

, meaning vertices

B

and

D

are each a distance of

2.5

units away from

M(4, 6)

. Solving for points along the line

y - 6 = -0.75(x - 4)

at this distance gives

(6, 4.5)

and

(2, 7.5)

. The condition that the

x

-coordinate of

B

is greater than the

x

-coordinate of

D

uniquely determines

B

to be

(6, 4.5)

, yielding a

y

-coordinate of

4.5

.

Step-by-Step Solution

1

Calculate the midpoint

M

and the length of diagonal

AC

.

M = (4, 6)

and

AC = 10

.

The diagonals of a rhombus bisect each other at their midpoint and their lengths determine the proportions of the shape.

2

Find the slope and length of diagonal

BD

.

Slope of

BD

is

-\frac{3}{4}

, and length is

5

.

Diagonals of a rhombus are perpendicular, meaning their slopes are negative reciprocals (

m_1 \cdot m_2 = -1

). The problem specifies that the length of

BD

is half of

AC

(

10 \div 2 = 5

).

3

Set up equations to find coordinates of

B(x, y)

and

D(x, y)

that are at distance

2.5

from

M(4, 6)

along the line of diagonal

BD

.

(x - 4)^2 + (y - 6)^2 = 6.25

and

y - 6 = -0.75(x - 4)

.

Since the diagonals bisect each other, the distance from the midpoint

M

to each of the remaining vertices

B

and

D

is half the length of diagonal

BD

(

5 \div 2 = 2.5

).

4

Solve the system of equations for the coordinates.

P_1(6, 4.5)

and

P_2(2, 7.5)

.

Substituting

y-6

into the distance equation yields

(x-4)^2 + 0.5625(x-4)^2 = 6.25

, which simplifies to

1.5625(x-4)^2 = 6.25 \implies (x-4)^2 = 4 \implies x - 4 = \pm 2

. Thus,

x_1 = 6

(giving

y_1 = 4.5

) and

x_2 = 2

(giving

y_2 = 7.5

).

5

Identify vertex

B

using the given coordinate condition.

B = (6, 4.5)

, so the

y

-coordinate is

4.5

.

The problem states that the

x

-coordinate of

B

is greater than the

x

-coordinate of

D

. Comparing the two solutions, the one with the larger

x

-value (

6 > 2

) must belong to vertex

B

.

Key Concept

Rhombus Diagonal Properties in the Coordinate Plane

Alternative Method

Alternatively, since the diagonals of a rhombus divide it into four congruent right triangles, we can determine the side length of the rhombus. The legs of these right triangles are half the diagonal lengths:

5

and

2.5

. By the Pythagorean theorem, the square of the side length is

5^2 + 2.5^2 = 31.25

. We can set up distance equations from

B(x, y)

to

A(1, 2)

and

C(7, 10)

:

(x-1)^2 + (y-2)^2 = 31.25

and

(x-7)^2 + (y-10)^2 = 31.25

. Subtracting the second equation from the first simplifies to the linear relation

y = -0.75x + 9

, which can then be substituted back into one of the quadratic equations to find

x = 6

or

x = 2

, yielding

y = 4.5

or

y = 7.5

.

Estimated Time:3m 0s

Answer

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