All practice questions

Question 2401Question →

The interior angles of a quadrilateral are in the ratio $2:3:4:6$ . What is the degree measure of the largest interior angle of the quadrilateral?

4

48

72

144

240

Show answer & explanation

Answer: 144

Answer

144

The sum of the interior angles of a quadrilateral is

360^\circ

. Given the ratio

2:3:4:6

, the sum of the parts is

2 + 3 + 4 + 6 = 15

. The value of one part is

360^\circ \div 15 = 24^\circ

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

6

. Therefore, the largest angle measure is

6 \times 24^\circ = 144^\circ

.

Step-by-Step Solution

1

Determine the sum of the interior angles of a quadrilateral.

The sum of the interior angles of any quadrilateral is

360^\circ

.

This is a fundamental property of quadrilaterals, which can also be derived using the formula

(n - 2) \times 180^\circ

with

n = 4

.

2

Calculate the total number of parts in the given ratio.

The sum of the ratio parts is

2 + 3 + 4 + 6 = 15

.

Adding the parts of the ratio allows us to find the size of a single share of the total angle sum.

3

Find the measure of one part of the ratio.

One part is equal to

360^\circ \div 15 = 24^\circ

.

Dividing the total sum of the angles by the sum of the ratio parts determines the angle measure per ratio unit.

4

Multiply the largest ratio part by the value of one part to find the largest angle.

The largest angle is

6 \times 24^\circ = 144^\circ

.

The largest interior angle corresponds to the largest number in the ratio, which is

6

.

Key Concept

Using ratios to find angle measures in a polygon.

Estimated Time:45s

Question 2402Question →

A line in the standard $(x, y)$ coordinate plane has a slope of $\frac{3}{4}$ and contains the points $P(a, 3)$ and $Q(5, b)$ . If the distance between $P$ and $Q$ is $10$ units, what is one possible value of $a + b$ ?

-6

2

6

12

22

Show answer & explanation

Answer: $6$

Answer

The correct answer is

6

.

The correct answer is

6

. Using the slope definition

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

, we write the slope equation as

\frac{b - 3}{5 - a} = \frac{3}{4}

, which gives

b - 3 = \frac{3}{4}(5 - a)

. Substituting this expression into the distance formula

d = \sqrt{(5 - a)^2 + (b - 3)^2} = 10

yields

10 = \sqrt{(5 - a)^2 + \left(\frac{3}{4}(5 - a)\right)^2} = \frac{5}{4}|5 - a|

. Solving for the horizontal change gives

|5 - a| = 8

, which means

5 - a = \pm 8

. If

5 - a = 8

, then

a = -3

and

b - 3 = 6 \Rightarrow b = 9

. The sum of these coordinates is

-3 + 9 = 6

.

Step-by-Step Solution

1

Set up the slope equation using the definition of slope:

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

.

\frac{b - 3}{5 - a} = \frac{3}{4}

, which simplifies to

b - 3 = \frac{3}{4}(5 - a)

.

This establishes a relationship between the coordinate differences of points

P

and

Q

based on the given slope.

2

Set up the distance equation using the distance formula:

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

.

10 = \sqrt{(5 - a)^2 + (b - 3)^2}

.

This uses the given distance of

10

units to constrain the coordinate differences.

3

Substitute the slope relationship

b - 3 = \frac{3}{4}(5 - a)

into the distance equation and solve for

(5 - a)

.

10 = \sqrt{(5 - a)^2 + \left(\frac{3}{4}(5 - a)\right)^2} \Rightarrow 10 = \sqrt{(5 - a)^2 \left(1 + \frac{9}{16}\right)} \Rightarrow 10 = \frac{5}{4}|5 - a| \Rightarrow |5 - a| = 8

. Thus,

5 - a = 8

or

5 - a = -8

.

Substituting one variable simplifies the system to a single quadratic equation in terms of the horizontal change.

4

Solve for the two possible values of

a

and calculate the corresponding values of

b

.

Case 1: If

5 - a = 8

, then

a = -3

. Substituting this back gives

b - 3 = \frac{3}{4}(8) = 6 \Rightarrow b = 9

. The sum is

a + b = -3 + 9 = 6

. Case 2: If

5 - a = -8

, then

a = 13

. Substituting this back gives

b - 3 = \frac{3}{4}(-8) = -6 \Rightarrow b = -3

. The sum is

a + b = 13 + (-3) = 10

.

Evaluating both branches of the absolute value equation yields the two valid coordinate combinations that satisfy both the slope and distance criteria.

Key Concept

Calculating the slope of a line and using it in conjunction with the distance formula to find unknown coordinates on the coordinate plane.

Question 2403Question →

In $\triangle ABC$ , the measures of the interior angles $\angle A$ , $\angle B$ , and $\angle C$ are in the ratio $3:4:5$ , respectively. What is the measure of the largest exterior angle of $\triangle ABC$ ?

45^\circ

75^\circ

105^\circ

108^\circ

135^\circ

Show answer & explanation

Answer: $135^\circ$

Answer

135 degrees

The interior angles of a triangle sum to

180^\circ

. Since the angles are in the ratio

3:4:5

, they can be represented as

3x

,

4x

, and

5x

. Adding these gives

12x = 180^\circ

, which simplifies to

x = 15^\circ

. The interior angles are therefore

45^\circ

,

60^\circ

, and

75^\circ

. Because an exterior angle is supplementary to its adjacent interior angle, the largest exterior angle is paired with the smallest interior angle:

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

. Alternatively, the exterior angle is equal to the sum of the two non-adjacent interior angles:

60^\circ + 75^\circ = 135^\circ

.

Step-by-Step Solution

1

Represent the interior angles algebraically using the given ratio.

Let the measures of the interior angles be

3x

,

4x

, and

5x

.

The ratio of the angles is

3:4:5

, so their measures must be multiples of these ratio numbers by the same factor

x

.

2

Set up and solve an equation for

x

using the triangle angle sum theorem.

3x + 4x + 5x = 180^\circ \implies 12x = 180^\circ \implies x = 15^\circ

.

The sum of the interior angles of any triangle is always

180^\circ

.

3

Determine the measures of the three interior angles.

The angles measure

3(15^\circ) = 45^\circ

,

4(15^\circ) = 60^\circ

, and

5(15^\circ) = 75^\circ

.

Multiplying the value of

x

by each term of the ratio gives the individual interior angle measures.

4

Find the largest exterior angle of the triangle.

The largest exterior angle is supplementary to the smallest interior angle:

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

.

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

180^\circ

. The smallest interior angle will yield the largest exterior angle.

Key Concept

Triangle Angle Sum Theorem and Exterior Angle Relationships

Alternative Method

The exterior angle at any vertex of a triangle is equal to the sum of the measures of the two opposite interior angles. The two largest interior angles are

60^\circ

and

75^\circ

. Therefore, the largest exterior angle is the sum of these two angles:

60^\circ + 75^\circ = 135^\circ

.

Estimated Time:1m 0s

Question 2404Question →

In right triangle $ABC$ with $\angle B = 90^\circ$ and $\angle A = 30^\circ$ , the hypotenuse $AC$ has a length of $12$ centimeters. An altitude $BD$ is drawn from vertex $B$ to hypotenuse $AC$ . From point $D$ , a perpendicular segment $DE$ is drawn to side $AB$ , with point $E$ lying on $AB$ . What is the length, in centimeters, of segment $EC$ ?

3\sqrt{7}

\frac{3\sqrt{22}}{2}

\frac{3\sqrt{19}}{2}

\frac{3\sqrt{13}}{2}

\sqrt{39}

Show answer & explanation

Answer: $\frac{3\sqrt{19}}{2}$

Answer

The length of segment EC is \frac{3\sqrt{19}}{2} centimeters.

To find the length of segment EC, we can construct the right triangle EBC with a right angle at B. By using the properties of 30-60-90 right triangles, we determine the side lengths of the triangles in the figure: first finding BC = 6 and AB = 6\sqrt{3} in triangle ABC; then finding AD = 9 in triangle ABD; then finding AE = \frac{9\sqrt{3}}{2} in triangle ADE; and finally finding EB = AB - AE = \frac{3\sqrt{3}}{2}. Applying the Pythagorean theorem to right triangle EBC yields EC = \sqrt{(\frac{3\sqrt{3}}{2})^2 + 6^2} = \frac{3\sqrt{19}}{2}.

Step-by-Step Solution

1

Determine the side lengths of the main right triangle ABC. Since angle A = 30 degrees and angle B = 90 degrees, triangle ABC is a 30-60-90 right triangle. With hypotenuse AC = 12, the leg opposite the 30-degree angle is BC = \frac{12}{2} = 6, and the leg opposite the 60-degree angle is AB = 6\sqrt{3}.

BC = 6 and AB = 6\sqrt{3}

Knowing the side lengths of triangle ABC is necessary to find the dimensions of the smaller inscribed triangles.

2

Find the length of segment AD in right triangle ABD. Altitude BD is perpendicular to AC, making triangle ABD a right triangle with right angle ADB. Since angle A = 30 degrees, triangle ABD is also a 30-60-90 right triangle with hypotenuse AB = 6\sqrt{3}. The side adjacent to the 30-degree angle, AD, is given by AB \times \cos(30^\circ) = 6\sqrt{3} \times \frac{\sqrt{3}}{2} = 9.

AD = 9

Determining AD allows us to analyze the smaller right triangle ADE built on it.

3

Find the lengths of segments AE and EB. In right triangle ADE (where DE is perpendicular to AB), the hypotenuse is AD = 9 and angle A = 30 degrees. The side adjacent to the 30-degree angle is AE = AD \times \cos(30^\circ) = 9 \times \frac{\sqrt{3}}{2} = \frac{9\sqrt{3}}{2}. Segment EB is then found by subtracting AE from AB: EB = AB - AE = 6\sqrt{3} - \frac{9\sqrt{3}}{2} = \frac{3\sqrt{3}}{2}.

EB = \frac{3\sqrt{3}}{2}

We need the length of segment EB to apply the Pythagorean theorem in the final right triangle EBC.

4

Apply the Pythagorean theorem to right triangle EBC. Since line segment AB is perpendicular to BC, angle EBC is a right angle. In right triangle EBC, the legs are EB = \frac{3\sqrt{3}}{2} and BC = 6. The hypotenuse EC is calculated as EC = \sqrt{EB^2 + BC^2} = \sqrt{(\frac{3\sqrt{3}}{2})^2 + 6^2} = \sqrt{\frac{27}{4} + 36} = \sqrt{\frac{171}{4}} = \frac{3\sqrt{19}}{2}.

EC = \frac{3\sqrt{19}}{2}

Applying the Pythagorean theorem to the legs EB and BC gives the length of the hypotenuse EC.

Key Concept

Applying 30-60-90 right triangle properties and the Pythagorean theorem across multiple connected geometric figures.

Estimated Time:3m 0s

Question 2405Question →

In the standard $(x, y)$ coordinate plane, a circle is defined by the equation $x^2 + y^2 - 10x - 4y = 0$ . The line $y = x$ intersects the circle at the origin $(0, 0)$ and at a second point $P$ . What is the $x$ -coordinate of point $P$ ?

Show answer & explanation

Answer: 7

Answer

The correct answer is 7.

To find the points of intersection between the circle and the line, we substitute the line's equation

y = x

into the circle's equation

x^2 + y^2 - 10x - 4y = 0

. This substitution yields

2x^2 - 14x = 0

. Factoring out

2x

gives

2x(x - 7) = 0

, which has solutions

x = 0

and

x = 7

. The solution

x = 0

corresponds to the origin

(0, 0)

. Therefore, the

x

-coordinate of the second point

P

is

7

.

Step-by-Step Solution

1

Substitute

y = x

into the equation of the circle.

x^2 + x^2 - 10x - 4x = 0

Since the line is

y = x

, any point of intersection must satisfy both equations.

2

Combine like terms to simplify the quadratic equation.

2x^2 - 14x = 0

Grouping the quadratic and linear terms yields a single equation in terms of

x

.

3

Factor the quadratic equation.

2x(x - 7) = 0

Factoring allows us to identify the individual roots of the equation.

4

Solve for

x

and identify the value for point

P

.

x = 7

The solution

x = 0

corresponds to the origin. Thus, the

x

-coordinate of the second point

P

must be

7

.

Key Concept

Finding points of intersection between a line and a circle in the coordinate plane.

Question 2406Question →

A vertical flagpole casts a horizontal shadow on the ground. The distance from the top of the flagpole to the tip of the shadow is $20$ feet. If the length of the shadow is $16$ feet, what is the height, in feet, of the flagpole?

12

16

25.6

36

4

Show answer & explanation

Answer: 12

Answer

12

The flagpole, ground, and line from the top of the pole to the tip of the shadow form a right triangle. The diagonal distance of

20

feet represents the hypotenuse, and the shadow length of

16

feet represents one of the legs. Using the Pythagorean theorem (

a^2 + b^2 = c^2

), we set up the equation

a^2 + 16^2 = 20^2

, which simplifies to

a^2 + 256 = 400

. Subtracting

256

from both sides gives

a^2 = 144

. Taking the square root of both sides gives the height of the flagpole as

12

feet.

Step-by-Step Solution

1

Identify the parts of the right triangle formed by the flagpole, ground, and the line from the top of the flagpole to the shadow's tip.

The hypotenuse (

c

) is

20

feet, and one leg (

b

) is

16

feet.

The flagpole is vertical and the ground is horizontal, forming a right angle. The distance from the top of the pole to the tip of the shadow is the diagonal (hypotenuse).

2

Apply the Pythagorean theorem

a^2 + b^2 = c^2

to find the missing leg (

a

).

a^2 + 16^2 = 20^2

, which simplifies to

a^2 + 256 = 400

.

The Pythagorean theorem relates the lengths of the sides of a right triangle.

3

Solve for the unknown height

a

by subtracting and taking the square root.

a^2 = 144 \implies a = 12

feet.

Isolating

a^2

gives

144

, and taking the square root of

144

gives the height of the flagpole.

Key Concept

Pythagorean Theorem

Question 2407Question →

A rectangular painting is $2$ feet wide and $6$ feet long. A wooden frame of uniform width $x$ feet is placed around the painting. If the total area of the painting and the frame is $21$ square feet, what is the width of the frame, in feet?

1.5

1.0

0.5

4.5

8.0

Show answer & explanation

Answer: 0.5

Answer

The correct width of the frame is 0.5 feet.

The correct answer is 0.5 feet. To find the width of the frame, we define the total dimensions of the framed painting as

(2 + 2x)

and

(6 + 2x)

. Setting their product equal to the total area of 21 square feet gives the quadratic equation

4x^2 + 16x - 9 = 0

. Solving this using the quadratic formula yields

x = 0.5

and

x = -4.5

. Since a physical width must be positive, the width of the frame is 0.5 feet.

Step-by-Step Solution

1

Set up the equation for the total area. The painting's dimensions are

2

feet by

6

feet. Adding a frame of uniform width

x

on all sides increases both the width and the length by

2x

.

The total dimensions are

(2 + 2x)

and

(6 + 2x)

, and the total area is given by the equation:

(2 + 2x)(6 + 2x) = 21

.

To find the width of the frame, we must relate the final total area to the dimensions of the painting plus the frame.

2

Expand the equation and write it in standard quadratic form

ax^2 + bx + c = 0

.

12 + 4x + 12x + 4x^2 = 21 \implies 4x^2 + 16x + 12 = 21 \implies 4x^2 + 16x - 9 = 0

.

Standard form is required to apply the quadratic formula or to factor the quadratic expression.

3

Solve the quadratic equation

4x^2 + 16x - 9 = 0

using the quadratic formula:

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

.

x = \frac{-16 \pm \sqrt{16^2 - 4(4)(-9)}}{2(4)} = \frac{-16 \pm \sqrt{256 + 144}}{8} = \frac{-16 \pm \sqrt{400}}{8} = \frac{-16 \pm 20}{8}

.

This formula provides the solutions to any quadratic equation.

4

Calculate the two possible values for

x

and select the physically meaningful one.

x = \frac{-16 + 20}{8} = 0.5

or

x = \frac{-16 - 20}{8} = -4.5

. Since width must be positive, the only valid solution is

x = 0.5

.

A physical measurement like width cannot be negative.

Key Concept

Quadratic Equations and the Quadratic Formula

Question 2408Question →

In the standard $(x, y)$ coordinate plane, point $P(2, -3)$ is an endpoint of line segment $PQ$ , and $M(-1, 5)$ is the midpoint of $PQ$ . What is the distance from endpoint $Q$ to the point $(0, 10)$ ?

2

\sqrt{7}

5

\sqrt{26}

7

Show answer & explanation

Answer: 5

Answer

5

To solve this problem, first determine the coordinates of endpoint

Q(x, y)

using the midpoint formula. Since

M(-1, 5)

is the midpoint of segment

PQ

and

P(2, -3)

is one endpoint, the horizontal coordinate satisfies

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

, which yields

x = -4

. The vertical coordinate satisfies

5 = \frac{-3 + y}{2}

, which yields

y = 13

. Thus, the coordinates of

Q

are

(-4, 13)

. Next, calculate the distance between

Q(-4, 13)

and the target point

(0, 10)

using the distance formula:

d = \sqrt{(0 - (-4))^2 + (10 - 13)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = 5

. This matches the correct value.

Step-by-Step Solution

1

Set up equations using the midpoint formula to determine the coordinates of endpoint

Q(x_Q, y_Q)

.

-1 = \frac{2 + x_Q}{2}

and

5 = \frac{-3 + y_Q}{2}

The coordinates of a midpoint are the averages of the coordinates of the two endpoints.

2

Solve the midpoint equations for the coordinates of

Q

.

x_Q = -4

and

y_Q = 13

, so

Q = (-4, 13)

Multiplying by two and subtracting the coordinates of the first endpoint isolates the variables.

3

Apply the distance formula to find the distance between

Q(-4, 13)

and the target point

(0, 10)

.

d = \sqrt{(0 - (-4))^2 + (10 - 13)^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

The distance formula calculates the straight line distance between two points in a coordinate plane.

Key Concept

Using the midpoint formula to find a missing endpoint and applying the distance formula to find the distance between two coordinate points.

Estimated Time:1m 30s

Question 2409Question →

A botanist models the growth of a rare seedling. The number of weeks $w$ that the seedling has been growing satisfies the linear equation:

\frac{1}{4}(3w - 8) + 0.6 = \frac{1}{5}(2w + 7)

If the seedling's growth continues to follow this model, what is the value of 3 less than 5 times the number of weeks the seedling has been growing?

-37

-17

22

25

37

Show answer & explanation

Answer: 37

Answer

37

Solving the given equation for

w

yields

w = 8

. The question asks for the value of 3 less than 5 times the number of weeks, which translates to the expression

5w - 3

. Substituting

w = 8

into the expression results in

5(8) - 3 = 37

.

Step-by-Step Solution

1

Multiply both sides of the equation by the least common multiple of the denominators, which is 20.

5(3w - 8) + 12 = 4(2w + 7)

This clears the fractions and simplifies the equation to integer coefficients.

2

Distribute the coefficients across the terms inside the parentheses.

15w - 40 + 12 = 8w + 28

This removes the parentheses so terms can be grouped.

3

Combine the constant terms on the left side of the equation.

15w - 28 = 8w + 28

Combining

-40

and

12

simplifies the expression on the left.

4

Isolate the variable

w

by moving the variable terms to the left side and constant terms to the right side.

7w = 56

Subtracting

8w

and adding

28

to both sides groups like terms together.

5

Solve for

w

by dividing both sides by 7.

w = 8

This isolates the variable

w

to find the number of weeks.

6

Translate '3 less than 5 times the number of weeks' into an algebraic expression and evaluate it for

w = 8

.

5w - 3 = 5(8) - 3 = 37

This translates the verbal question into mathematical terms and calculates the final value.

Key Concept

Solving multi-step linear equations containing fractions and decimals, and translating verbal expressions into algebraic terms.

Alternative Method

Instead of clearing the fractions first, one could convert the fractions to decimals:

0.25(3w - 8) + 0.6 = 0.2(2w + 7)

. Distribute to get

0.75w - 2 + 0.6 = 0.4w + 1.4

, which simplifies to

0.75w - 1.4 = 0.4w + 1.4

. Subtracting

0.4w

and adding

1.4

to both sides yields

0.35w = 2.8

, which gives

w = \frac{2.8}{0.35} = 8

. Then, compute

5(8) - 3 = 37

.

Estimated Time:1m 30s

Question 2410Question →

In the standard $(x, y)$ coordinate plane, point $P$ is rotated $90^\circ$ counterclockwise about the origin, and then translated $3$ units to the left and $5$ units down. If the coordinates of the resulting image point $P'$ are $(-6, 2)$ , what are the coordinates of the original point $P$ ?

(-3, 9)

(-7, -3)

(7, -3)

(7, 3)

(3, -9)

Show answer & explanation

Answer: $(7, 3)$

Answer

(7, 3)

The correct answer is

(7, 3)

. Working backward from the final image point

(-6, 2)

, we first reverse the translation of

3

units left and

5

units down by translating the point

3

units right and

5

units up, yielding

(-6+3, 2+5) = (-3, 7)

. We then reverse the

90^\circ

counterclockwise rotation by performing a

90^\circ

clockwise rotation about the origin. The rule for a

90^\circ

clockwise rotation is

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

, which maps

(-3, 7)

to

(7, 3)

.

Step-by-Step Solution

1

Identify the sequence of transformations and set up the backward process.

The forward process is: Point

P \rightarrow

Rotate

90^\circ

counterclockwise

\rightarrow

Translate left

3

and down

5 \rightarrow P'(-6, 2)

. To find the original point

P

, we must reverse these steps starting from

P'

: Translate right

3

and up

5 \rightarrow

Rotate

90^\circ

clockwise

\rightarrow

Point

P

.

Working backward requires reversing both the order and the direction of each transformation in the sequence.

2

Apply the reverse translation to the image point

P'(-6, 2)

.

Translating

3

units to the right adds

3

to the

x

-coordinate:

-6 + 3 = -3

. Translating

5

units up adds

5

to the

y

-coordinate:

2 + 5 = 7

. This gives the intermediate point

P_1(-3, 7)

.

Reversing a translation of

3

units left and

5

units down requires adding

3

to the

x

-coordinate and

5

to the

y

-coordinate.

3

Apply the reverse rotation to the intermediate point

P_1(-3, 7)

.

The reverse of a

90^\circ

counterclockwise rotation is a

90^\circ

clockwise rotation. The coordinate rule for a

90^\circ

clockwise rotation about the origin is

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

. Applying this rule to

P_1(-3, 7)

yields

(7, -(-3)) = (7, 3)

.

Rotating a point

90^\circ

clockwise about the origin swaps the coordinates and negates the new

y

-coordinate.

Key Concept

Reversing composite transformations in the coordinate plane

Question 2411Question →

In a right triangle, the measure of one of the acute angles is $30^\circ$ . If the side opposite this $30^\circ$ angle has a length of $6.5$ inches, what is the length, in inches, of the hypotenuse?

Show answer & explanation

Answer: 13

Answer

The length of the hypotenuse is

13

inches.

In a

30^\circ

-

60^\circ

-

90^\circ

special right triangle, the length of the hypotenuse is exactly twice the length of the shorter leg (the side opposite the

30^\circ

angle). Given that the shorter leg has a length of

6.5

inches, the hypotenuse has a length of

2 \times 6.5 = 13

inches.

Step-by-Step Solution

1

Determine the relationship between the given side and the hypotenuse using special right triangle properties.

The triangle is a

30^\circ

-

60^\circ

-

90^\circ

right triangle, meaning the hypotenuse is twice the length of the shorter leg.

By geometric theorem, the sides of a

30^\circ

-

60^\circ

-

90^\circ

triangle are in the ratio

1 : \sqrt{3} : 2

, with the shortest side opposite the

30^\circ

angle and the longest side being the hypotenuse.

2

Multiply the length of the side opposite the

30^\circ

angle by 2.

6.5 \text{ inches} \times 2 = 13 \text{ inches}

Doubling the shorter leg length of

6.5

inches gives the length of the hypotenuse.

Key Concept

In a

30^\circ

-

60^\circ

-

90^\circ

special right triangle, the length of the hypotenuse is always twice the length of the shorter leg (the side opposite the

30^\circ

angle).

Question 2412Question →

Line $A$ passes through the points $(-3, 4)$ and $(5, 8)$ in a coordinate plane. Line $B$ is perpendicular to Line $A$ and passes through the point $(2, -1)$ . What is the $y$ -coordinate of the point where Line $B$ crosses the $y$ -axis?

Show answer & explanation

Answer: 3

Answer

The correct answer is 3.

The correct answer is 3. Finding the slope of the first line gives 0.5. The negative reciprocal of 0.5 is -2, which is the slope of the perpendicular line. Setting up the slope equation between the point (2, -1) and the y-intercept (0, 3) confirms the y-coordinate is indeed 3.

Step-by-Step Solution

1

Calculate the slope of Line A.

The slope of Line A is 0.5.

Using the slope formula

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

for points

(-3, 4)

and

(5, 8)

gives

m_A = \frac{8 - 4}{5 - (-3)} = \frac{4}{8} = 0.5

.

2

Find the slope of Line B.

The slope of Line B is -2.

Because Line B is perpendicular to Line A, its slope must be the negative reciprocal of the slope of Line A:

m_B = -\frac{1}{0.5} = -2

.

3

Determine the y-intercept of Line B.

The y-coordinate of the y-intercept is 3.

Line B passes through

(2, -1)

and

(0, b)

. Using the slope formula:

\frac{b - (-1)}{0 - 2} = -2 \implies \frac{b + 1}{-2} = -2 \implies b + 1 = 4 \implies b = 3

.

Key Concept

The slope of a line measures its steepness, calculated as the change in

y

divided by the change in

x

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

Alternative Method

Instead of using the slope formula directly to solve for the y-intercept, write the equation of Line B in point-slope form:

y - y_1 = m(x - x_1)

. Substituting the point

(2, -1)

and slope

-2

gives

y - (-1) = -2(x - 2)

, which simplifies to

y + 1 = -2x + 4

, or

y = -2x + 3

. To find the y-intercept, set

x = 0

, which yields

y = 3

.

Estimated Time:1m 30s

Question 2413Question →

In $\triangle ABC$ , point $D$ lies on side $BC$ such that $AD = BD$ . If the measure of $\angle ADC$ is $112^\circ$ and the measure of $\angle BAC$ is $85^\circ$ , what is the measure of $\angle C$ , in degrees?

Show answer & explanation

Answer: 39

Answer

The measure of

\angle C

is

39^\circ

.

The measure of

\angle C

is found by first calculating the interior angle

\angle ADB = 180^\circ - 112^\circ = 68^\circ

since

BDC

forms a straight line. Because

AD = BD

,

\triangle ABD

is isosceles with

\angle B = \angle BAD

. Using the angle sum of

\triangle ABD

, we have

2(\angle B) + 68^\circ = 180^\circ

, which yields

\angle B = 56^\circ

. Finally, using the angle sum of

\triangle ABC

, we calculate

\angle C = 180^\circ - (85^\circ + 56^\circ) = 39^\circ

.

Step-by-Step Solution

1

Find the measure of

\angle ADB

using the linear pair relationship with

\angle ADC

.

\angle ADB = 68^\circ

Angles on a straight line add up to

180^\circ

. Since

D

lies on

BC

,

\angle ADB + \angle ADC = 180^\circ

.

2

Calculate the measure of

\angle B

using the properties of the isosceles triangle

ABD

.

\angle B = 56^\circ

Since

AD = BD

, the base angles opposite to these sides are equal:

\angle BAD = \angle B

. The sum of angles in

\triangle ABD

is

180^\circ

, so

2(\angle B) + 68^\circ = 180^\circ

.

3

Find the measure of

\angle C

using the triangle angle sum theorem on the large triangle

ABC

.

\angle C = 39^\circ

The sum of the angles in

\triangle ABC

is

180^\circ

, meaning

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

. Substituting the known values gives

85^\circ + 56^\circ + \angle C = 180^\circ

.

Key Concept

Using the Isosceles Triangle Theorem, the Triangle Angle Sum Theorem, and linear pairs to trace unknown angles in a geometric figure.

Practice More

Try finding the missing angles when a transversal cuts two parallel lines that form a triangle with a third intersecting line.

Alternative Method

Instead of finding

\angle B

first and then solving for

\angle C

in

\triangle ABC

, we can find the angle

\angle DAC

first. Since

\angle ADC = 112^\circ

is an exterior angle to

\triangle ABD

, we have

\angle ADC = \angle B + \angle BAD

. Since

\angle B = \angle BAD

, we get

2(\angle BAD) = 112^\circ \implies \angle BAD = 56^\circ

. Because

\angle BAC = 85^\circ

, we have

\angle DAC = 85^\circ - 56^\circ = 29^\circ

. Now looking at

\triangle ADC

, we can solve for

\angle C

directly:

\angle C = 180^\circ - (112^\circ + 29^\circ) = 39^\circ

.

Estimated Time:1m 30s

Question 2414Question →

In the standard $(x, y)$ coordinate plane, a line passes through the point $(3, 1)$ and has a slope of $\frac{2}{5}$ . If the line intersects the $y$ -axis at the point $(0, b)$ , what is the value of $b$ ?

-\frac{13}{2}

-\frac{4}{5}

\frac{11}{5}

-\frac{1}{5}

\frac{5}{4}

Show answer & explanation

Answer: $-\frac{1}{5}$

Answer

The value of the

y

-coordinate of the

y

-intercept is

-\frac{1}{5}

.

To find the

y

-intercept of the line, we can use the point-slope equation of a line:

y - y_1 = m(x - x_1)

, where

m

is the slope and

(x_1, y_1)

is a point on the line. Substituting the given point

(3, 1)

and slope

\frac{2}{5}

yields

y - 1 = \frac{2}{5}(x - 3)

. Since the

y

-intercept occurs at

x = 0

, we substitute

0

for

x

to find the corresponding

y

-value:

y - 1 = \frac{2}{5}(0 - 3)

, which simplifies to

y - 1 = -\frac{6}{5}

. Adding

1

to both sides gives

y = -\frac{1}{5}

. Therefore, the value of the

y

-coordinate of the

y

-intercept is

-\frac{1}{5}

.

Step-by-Step Solution

1

Identify the point-slope form of a linear equation.

y - y_1 = m(x - x_1)

This formula allows us to write the equation of a line when we know its slope (

m

) and a point on the line (

(x_1, y_1)

).

2

Substitute the given slope

m = \frac{2}{5}

and point

(3, 1)

into the formula.

y - 1 = \frac{2}{5}(x - 3)

This establishes the equation for the specific line described in the problem.

3

Substitute

x = 0

to find the

y

-intercept of the line.

y - 1 = \frac{2}{5}(0 - 3) \implies y - 1 = -\frac{6}{5}

The

y

-intercept of any graph is the point where it crosses the

y

-axis, which always has an

x

-coordinate of

0

.

4

Solve for

y

to determine the value of

b

.

y = 1 - \frac{6}{5} = -\frac{1}{5}

, so

b = -\frac{1}{5}

Isolating

y

gives the

y

-coordinate of the

y

-intercept, which is

b

since the point is defined as

(0, b)

.

Key Concept

Writing and evaluating linear equations in point-slope form to find intercepts.

Question 2415Question →

In the standard $(x, y)$ coordinate plane, the equation $x^2 + y^2 + 8x - 12y + c = 0$ defines a circle. If this circle is tangent to the $y$ -axis, what is the value of the constant $c$ ?

16

36

48

52

68

Show answer & explanation

Answer: $36$

Answer

The value of the constant

c

is

36

.

By completing the square on

x^2 + y^2 + 8x - 12y + c = 0

, we get the standard form equation

(x+4)^2 + (y-6)^2 = 52-c

. This shows the circle's center is at

(-4, 6)

and its radius squared is

r^2 = 52-c

. Because the circle is tangent to the

y

-axis, the radius is the horizontal distance from the center to the

y

-axis, which is

4

units. Therefore, the radius squared is

16

. Equating the two expressions for the radius squared gives

52-c = 16

, which solves to

c = 36

.

Step-by-Step Solution

1

Group the

x

and

y

terms and move the constant to the right side of the equation.

(x^2 + 8x) + (y^2 - 12y) = -c

To prepare the equation for completing the square.

2

Complete the square for both the

x

and

y

groups by adding the square of half the coefficient of the linear terms to both sides.

(x^2 + 8x + 16) + (y^2 - 12y + 36) = -c + 16 + 36

, which simplifies to

(x+4)^2 + (y-6)^2 = 52-c

.

To rewrite the equation in standard circle form

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

.

3

Determine the radius of the circle based on the geometric condition of being tangent to the

y

-axis.

The center of the circle is

(-4, 6)

. Since the circle is tangent to the

y

-axis (the line

x=0

), the radius

r

is the horizontal distance from the center to the

y

-axis, which is

|-4| = 4

.

A circle tangent to a vertical line has its radius equal to the horizontal distance from its center to that line.

4

Set the radius squared equal to the right side of the standard equation and solve for

c

.

r^2 = 4^2 = 16

. Setting

52 - c = 16

gives

c = 36

.

In standard form, the right-hand side represents

r^2

.

Key Concept

Converting the general form of a circle equation to standard form

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

by completing the square, and using the geometric definition of tangency to determine the radius.

Question 2416Question →

An equilateral triangle $ABC$ has a side length of $12$ inches. An altitude $AD$ is drawn from vertex $A$ to the side $BC$ . A point $P$ lies on the segment $AD$ such that $\triangle BPC$ is a right triangle with a right angle at $P$ . What is the length, in inches, of the segment $AP$ ?

6\sqrt{3} - 3\sqrt{2}

6\sqrt{3} - 6\sqrt{2}

6\sqrt{3} - 6

6\sqrt{3} + 6

12\sqrt{3} - 6

Show answer & explanation

Answer: $6\sqrt{3} - 6$

Answer

The correct answer is

6\sqrt{3} - 6

inches.

The correct answer is

6\sqrt{3} - 6

. Since the side length of the equilateral triangle is

12

, the altitude

AD

splits it into two

30^\circ

-

60^\circ

-

90^\circ

right triangles with base

BD = 6

and altitude

AD = 6\sqrt{3}

. The right triangle

\triangle BPC

has

\angle BPC = 90^\circ

and

PB = PC

, making it an isosceles right triangle. The altitude

PD

splits

\triangle BPC

into two

45^\circ

-

45^\circ

-

90^\circ

right triangles, so

PD = BD = 6

. The length of

AP

is found by subtracting

PD

from

AD

, yielding

6\sqrt{3} - 6

.

Step-by-Step Solution

1

Find the length of the altitude

AD

using the properties of the

30^\circ

-

60^\circ

-

90^\circ

triangle

\triangle ABD

.

The length of

AD

is

6\sqrt{3}

inches.

Since

\triangle ABC

is equilateral with side length

12

inches, the altitude

AD

bisects the base

BC

, making

BD = 6

inches. The altitude splits the equilateral triangle into two

30^\circ

-

60^\circ

-

90^\circ

right triangles. The length of the longer leg is the shorter leg multiplied by

\sqrt{3}

, which gives

AD = 6\sqrt{3}

.

2

Find the length of the segment

PD

using the properties of the

45^\circ

-

45^\circ

-

90^\circ

triangle

\triangle PDB

.

The length of

PD

is

6

inches.

Since

P

lies on the altitude

AD

, which is the axis of symmetry,

\triangle BPC

is an isosceles right triangle with

\angle BPC = 90^\circ

. The altitude

PD

is perpendicular to

BC

and bisects

\angle BPC

, forming two

45^\circ

-

45^\circ

-

90^\circ

right triangles:

\triangle PDB

and

\triangle PDC

. In a

45^\circ

-

45^\circ

-

90^\circ

triangle, the two legs are congruent, so

PD = BD = 6

.

3

Subtract the length of

PD

from the length of

AD

to find the length of segment

AP

.

AP = 6\sqrt{3} - 6

inches.

Since point

P

lies on segment

AD

, the length of

AP

is the difference between the total altitude

AD

and the segment

PD

.

Key Concept

Properties of special right triangles (30-60-90 and 45-45-90) and their multi-step application in geometry.

Estimated Time:2m 0s

Question 2417Question →

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 2418Question →

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 2419Question →

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 2420Question →

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.

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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Common Mistakes

Alternative Method

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Practice More

Alternative Method

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