All practice questions

Question 2781Question →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 - 8x + 6y - 11 = 0$ is a circle. If the center of the circle is $(h, k)$ and the radius is $r$ , what is the value of $h + k + r$ ?

Show answer & explanation

Answer: 7

Answer

7

To find

h + k + r

, we convert the general equation

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

to standard form

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

by completing the square. Grouping terms gives

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

. Adding 16 and 9 to both sides to complete the square yields

(x^2 - 8x + 16) + (y^2 + 6y + 9) = 11 + 16 + 9

, which simplifies to

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

. Comparing this to the standard form gives center

(h, k) = (4, -3)

and radius

r = \sqrt{36} = 6

. The sum of these values is

4 + (-3) + 6 = 7

.

Step-by-Step Solution

1

Group the variable terms and move the constant term to the right side of the equation.

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

To arrange the equation for completing the square.

2

Complete the square for both variables by adding the square of half the coefficient of the linear terms to both sides.

(x^2 - 8x + 16) + (y^2 + 6y + 9) = 11 + 16 + 9

This creates factorable perfect square trinomials on the left side while maintaining equality.

3

Factor the perfect square trinomials and add the constants on the right side.

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

To write the equation in the standard form of a circle:

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

.

4

Determine the coordinates of the center

(h, k)

and the radius

r

from the standard form.

Center coordinates

h = 4

,

k = -3

, and radius

r = 6

Comparing

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

to standard form reveals

h = 4

,

k = -3

, and

r = \sqrt{36} = 6

.

5

Sum the values of

h

,

k

, and

r

together.

4 + (-3) + 6 = 7

To find the final requested value.

Key Concept

Converting the general equation of a circle into standard form by completing the square to identify its center and radius.

Question 2782Question →

What is the area, in square units, of the circle in the $xy$ -plane represented by the equation $x^2 + y^2 + 6x - 4y - 12 = 0$ ?

5\pi

12\pi

\pi

25\pi

Show answer & explanation

Answer: $25\pi$

Answer

The correct answer is

25\pi

.

To find the area of the circle, the equation is rewritten in the standard form

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

. Rearranging the terms gives

(x^2 + 6x) + (y^2 - 4y) = 12

. Completing the square by adding 9 and 4 to both sides yields

(x + 3)^2 + (y - 2)^2 = 12 + 9 + 4 = 25

. In this standard form, the squared radius of the circle is

r^2 = 25

. Because the area of a circle is calculated as

A = \pi r^2

, the area is

25\pi

.

Step-by-Step Solution

1

Group the

x

-terms and

y

-terms, and move the constant term to the right side of the equation.

(x^2 + 6x) + (y^2 - 4y) = 12

This prepares the equation for completing the square for both variables.

2

Complete the square for the

x

-terms by adding

(6/2)^2 = 9

to both sides, and for the

y

-terms by adding

(-4/2)^2 = 4

to both sides.

(x^2 + 6x + 9) + (y^2 - 4y + 4) = 12 + 9 + 4

, which simplifies to

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

.

This expresses the equation in the standard form of a circle,

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

, where the center is

(h, k)

and the radius is

r

.

3

Identify the squared radius

r^2

from the standard form and calculate the area of the circle.

r^2 = 25

, so the area of the circle is

\pi r^2 = 25\pi

.

The area of a circle is calculated using the formula

A = \pi r^2

, so we directly substitute the value of

r^2

as 25.

Key Concept

Converting the general equation of a circle to standard form to find its radius and calculate its area.

Question 2783Question →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 - 12x + 10y + k = 0$ , where $k$ is a constant, is a circle. If this circle is tangent to the $y$ -axis, what is the value of $k$ ?

Show answer & explanation

Answer: 25

Answer

The value of

k

is 25.

Completing the square for the equation

x^2 + y^2 - 12x + 10y + k = 0

yields standard form

(x - 6)^2 + (y + 5)^2 = 61 - k

. The center of the circle is

(6, -5)

and the radius squared is

r^2 = 61 - k

. Since the circle is tangent to the

y

-axis (the line

x = 0

), its radius must be equal to the horizontal distance from the center to the

y

-axis, which is the absolute value of the

x

-coordinate of the center,

|6| = 6

. Therefore, the radius is

6

, and the radius squared is

r^2 = 36

. Setting

61 - k = 36

gives

k = 25

.

Step-by-Step Solution

1

Complete the square for

x

and

y

variables in the equation.

(x - 6)^2 + (y + 5)^2 = 61 - k

Completing the square allows the general form equation of the circle to be written in standard form, where the center and radius are easily identifiable.

2

Determine the center and the expression for the radius squared from the standard form.

Center is

(6, -5)

and

r^2 = 61 - k

In the standard circle equation

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

, the center is

(h, k_c)

and the right side represents the radius squared.

3

Relate the tangency condition to the radius of the circle.

r = 6

A circle tangent to the

y

-axis has its radius equal to the distance from its center to the line

x = 0

, which is the absolute value of the center's

x

-coordinate.

4

Calculate the value of the constant

k

.

k = 25

Since

r = 6

, then

r^2 = 36

. Setting

61 - k = 36

and solving for

k

gives

k = 25

.

Key Concept

Completing the square to find the standard form of a circle's equation and applying geometric tangency properties to determine coordinates or constants.

Question 2784Question →

In right triangle $PQR$ , the measure of angle $PQR$ is $90^\circ$ . Point $S$ lies on side $PQ$ . If the length of side $QR$ is $15$ , the length of segment $SR$ is $17$ , and the length of segment $PS$ is $12$ , what is the length of side $PR$ ?

Show answer & explanation

Answer: 25

Answer

The length of side

PR

is

25

.

First, the Pythagorean theorem is applied to the smaller right triangle

SQR

to solve for the unknown leg

QS

:

QS = \sqrt{SR^2 - QR^2} = \sqrt{17^2 - 15^2} = 8

. Next, the total length of leg

PQ

is the sum of its parts,

PS

and

QS

, which gives

PQ = 12 + 8 = 20

. Finally, the Pythagorean theorem is applied to the main right triangle

PQR

to find the hypotenuse

PR

:

PR = \sqrt{PQ^2 + QR^2} = \sqrt{20^2 + 15^2} = \sqrt{400 + 225} = \sqrt{625} = 25

.

Step-by-Step Solution

1

Use the Pythagorean theorem on the right triangle

SQR

to find the length of segment

QS

.

QS = 8

Triangle

SQR

shares the right angle at

Q

with triangle

PQR

. Thus,

QS^2 + QR^2 = SR^2 \implies QS^2 + 15^2 = 17^2 \implies QS = 8

.

2

Determine the length of side

PQ

.

PQ = 20

Since

S

lies on side

PQ

, the length of

PQ

is the sum of

PS

and

QS

:

PQ = 12 + 8 = 20

.

3

Use the Pythagorean theorem on right triangle

PQR

to calculate the length of the hypotenuse

PR

.

PR = 25

PQ^2 + QR^2 = PR^2 \implies 20^2 + 15^2 = PR^2 \implies 400 + 225 = 625 \implies PR = 25

.

Key Concept

Multi-step application of the Pythagorean theorem in nested right triangles

Question 2785Question →

An isosceles trapezoid has a shorter base of length $10$ , a longer base of length $22$ , and a height of $8$ . What is the perimeter of the trapezoid?

Show answer & explanation

Answer: 52

Answer

The perimeter of the trapezoid is

52

.

To find the perimeter of the isosceles trapezoid, we first determine the length of the two congruent slanted sides. By drawing altitudes from the endpoints of the shorter top base to the longer bottom base, we create two right triangles at the sides of the trapezoid. The vertical leg of each right triangle is equal to the height of the trapezoid (

8

). The horizontal leg of each right triangle is equal to half the difference between the two base lengths, which is

\frac{22 - 10}{2} = 6

. Applying the Pythagorean theorem, the length of each slanted side is

\sqrt{6^2 + 8^2} = 10

. Finally, the perimeter is the sum of all four sides:

10 + 22 + 10 + 10 = 52

.

Step-by-Step Solution

1

Calculate the horizontal base of the right triangles formed by drawing heights from the top base to the bottom base.

6

An isosceles trapezoid is symmetrical. Drawing vertical lines representing the height of

8

from the endpoints of the shorter base of length

10

down to the longer base of length

22

divides the longer base into a middle segment of length

10

and two equal end segments. The length of each end segment is

\frac{22 - 10}{2} = 6

.

2

Use the Pythagorean theorem to find the length of the slanted legs of the trapezoid.

10

Each slanted leg is the hypotenuse of a right triangle with legs of length

6

and

8

. According to the Pythagorean theorem, the hypotenuse length is

\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10

.

3

Find the perimeter of the trapezoid by adding the lengths of all four sides.

52

The perimeter is the sum of the shorter base (

10

), the longer base (

22

), and the two slanted legs (each of length

10

). Therefore, the perimeter is

10 + 22 + 10 + 10 = 52

.

Key Concept

Applying the Pythagorean theorem to find missing side lengths in composite geometric figures

Question 2786Question →

A $25$ -foot ladder is leaning against a vertical wall such that the base of the ladder is $7$ feet away from the base of the wall. If the top of the ladder slides down the wall by $4$ feet, how many feet does the base of the ladder slide outward along the ground?

4

8

9

15

Show answer & explanation

Answer: $8$

Answer

The base of the ladder slides outward by

8

feet.

The correct answer is

8

feet. By applying the Pythagorean theorem, the initial height of the ladder is found to be

\sqrt{25^2 - 7^2} = 24

feet. When the ladder slides down

4

feet, its new height is

20

feet. Using the Pythagorean theorem again, the new base distance is

\sqrt{25^2 - 20^2} = 15

feet. The difference between the new base distance and the initial base distance is

15 - 7 = 8

feet.

Step-by-Step Solution

1

Find the initial height of the top of the ladder on the wall using the Pythagorean theorem.

\text{Initial Height} = \sqrt{25^2 - 7^2} = \sqrt{625 - 49} = \sqrt{576} = 24

feet

The wall, the ground, and the ladder form a right triangle where the ladder is the hypotenuse (

25

feet) and the base distance is one leg (

7

feet).

2

Calculate the new height of the top of the ladder after it slides down.

\text{New Height} = 24 - 4 = 20

feet

The top of the ladder slides down by

4

feet from its initial height of

24

feet.

3

Calculate the new distance of the base of the ladder from the wall using the Pythagorean theorem.

\text{New Base Distance} = \sqrt{25^2 - 20^2} = \sqrt{625 - 400} = \sqrt{225} = 15

feet

The ladder length (hypotenuse) remains

25

feet, and the new height (one leg) is

20

feet.

4

Find the distance the base of the ladder slid outward by subtracting the initial base distance from the new base distance.

\text{Outward Slide} = 15 - 7 = 8

feet

The base was initially

7

feet from the wall and is now

15

feet from the wall.

Key Concept

Applying the Pythagorean theorem to relate side lengths in changing right triangle configurations.

Estimated Time:1m 30s

Question 2787Question →

In the $xy$ -plane, a circle is defined by the equation $x^2 + y^2 + kx - 8y = 9$ , where $k$ is a constant. If the center of the circle lies on the line $y = 2x$ , what is the value of $k$ ?

-16

-4

-2

4

Show answer & explanation

Answer: -4

Answer

-4

The correct answer is the value that satisfies the relation between the circle's center coordinates and the given line. By completing the square on the circle's equation, we rewrite it as

(x + k/2)^2 + (y - 4)^2 = 25 + k^2/4

. This reveals the center of the circle is at the point (-k/2, 4). Substituting these coordinates into the equation of the line, which is y = 2x, gives the equation 4 = 2(-k/2). Simplifying this relation gives 4 = -k, which yields k = -4.

Step-by-Step Solution

1

Group the x and y terms of the circle equation.

(x^2 + kx) + (y^2 - 8y) = 9

Grouping terms helps in completing the square for x and y separately.

2

Complete the square for both the x and y quadratic expressions.

(x + \frac{k}{2})^2 - \frac{k^2}{4} + (y - 4)^2 - 16 = 9

, which simplifies to

(x + \frac{k}{2})^2 + (y - 4)^2 = 25 + \frac{k^2}{4}

This puts the equation into the standard circle form

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

.

3

Identify the coordinates of the center of the circle.

Center:

(h, j) = (-\frac{k}{2}, 4)

In standard form, the terms

(x - h)^2

and

(y - j)^2

correspond to a center at

(h, j)

.

4

Substitute the center coordinates into the line equation

y = 2x

to solve for

k

.

4 = 2(-\frac{k}{2}) \implies 4 = -k \implies k = -4

Since the center lies on the line, the coordinates of the center must satisfy the line's equation.

Key Concept

Completing the square to find the center of a circle in the coordinate plane and using coordinate geometry relations.

Alternative Method

Instead of completing the square fully, recall that for any circle equation of the form

x^2 + y^2 + Dx + Ey + F = 0

, the center coordinates

(h, j)

are given by

(-D/2, -E/2)

. In the given equation,

D = k

and

E = -8

. Therefore, the center is

(-\frac{k}{2}, 4)

. Substituting these coordinates directly into the line equation

y = 2x

yields

4 = 2(-\frac{k}{2})

, which simplifies to

k = -4

.

Estimated Time:1m 30s

Question 2788Question →

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . If $\sin A = \frac{5}{13}$ and the perimeter of the triangle is $60$ , what is the length of the shortest side of triangle $ABC$ ?

5

10

12

24

Show answer & explanation

Answer: 10

Answer

10

The trigonometric ratio

\sin A = \frac{5}{13}

gives the ratio of the side opposite to angle

A

to the hypotenuse. Using the Pythagorean theorem, the adjacent side corresponds to the 5-12-13 right triangle ratio. The perimeter in terms of the ratio factor

k

is

5k + 12k + 13k = 30k

. Setting

30k = 60

yields

k = 2

. The shortest side of the triangle corresponds to the smallest ratio value, which is

5k

. Multiplying

5

by the scale factor

2

yields a length of

10

.

Step-by-Step Solution

1

Use the definition of sine to set up the ratio of the side lengths.

Since

\sin A = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{5}{13}

, let the side opposite to angle

A

(

BC

) be

5k

and the hypotenuse (

AB

) be

13k

, where

k

is a positive constant.

This relates the given trigonometric ratio to the side lengths of the right triangle.

2

Find the third side using the Pythagorean theorem or Pythagorean triples.

Using the Pythagorean theorem,

AC^2 + BC^2 = AB^2

, we get

AC^2 + (5k)^2 = (13k)^2 \implies AC^2 + 25k^2 = 169k^2 \implies AC^2 = 144k^2 \implies AC = 12k

. This matches the 5-12-13 Pythagorean triple.

To find the perimeter, all three side lengths of the right triangle are needed.

3

Set up an equation for the perimeter and solve for

k

.

The perimeter is the sum of all sides:

5k + 12k + 13k = 30k

. We are given that the perimeter is

60

. Thus,

30k = 60 \implies k = 2

.

Using the given perimeter helps determine the exact side lengths.

4

Determine the length of the shortest side.

The side lengths are

BC = 5(2) = 10

,

AC = 12(2) = 24

, and

AB = 13(2) = 26

. The shortest side is

BC

, which has a length of

10

.

The question asks for the length of the shortest side.

Key Concept

Using trigonometric ratios and the Pythagorean theorem to find side lengths of right triangles from perimeter.

Estimated Time:1m 30s

Question 2789Question →

In the $xy$ -plane, triangle $ABC$ is a right triangle with the right angle at $B$ . The vertices of the triangle are $A(3, 11)$ , $B(15, 20)$ , and $C(27, y)$ , where $y$ is a constant. If the length of the hypotenuse $AC$ is 25 and $y < 20$ , what is the value of $y$ ?

Show answer & explanation

Answer: 4

Answer

The value of

y

is 4.

Applying the distance formula to vertices

A(3, 11)

and

B(15, 20)

yields a length of 15 for side

AB

. Since triangle

ABC

has its right angle at

B

, the hypotenuse is

AC = 25

. Using the Pythagorean theorem (

AB^2 + BC^2 = AC^2

), we find that

BC = 20

. Setting the distance between

B(15, 20)

and

C(27, y)

equal to 20 gives

(27-15)^2 + (y-20)^2 = 20^2

. This simplifies to

(y-20)^2 = 256

, which has solutions

y = 36

or

y = 4

. Given that

y < 20

, the value of

y

must be 4.

Step-by-Step Solution

1

Calculate the length of segment

AB

using the distance formula.

AB = 15

To find one of the leg lengths of the right triangle.

2

Apply the Pythagorean theorem to find the length of leg

BC

.

BC = 20

To determine the length of the other leg of the right triangle.

3

Use the distance formula for

BC

and the condition

y < 20

to solve for

y

.

y = 4

To find the vertical coordinate of vertex

C

.

Key Concept

Using the Pythagorean theorem in coordinate geometry to solve for unknown coordinates.

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