Tüm alıştırma soruları

Soru 2261Soru →

In the standard $(x,y)$ coordinate plane, a rectangle has vertices at $(-2, -3)$ , $(4, -3)$ , $(4, 2)$ , and $(-2, 2)$ . What is the area, in square units, of this rectangle?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The area of the rectangle is 30 square units.

The area of a rectangle is the product of its length and width. By finding the difference between the x-coordinates of the horizontal vertices (

4 - (-2) = 6

) and the difference between the y-coordinates of the vertical vertices (

2 - (-3) = 5

), we find the dimensions to be 6 and 5. Multiplying these gives

6 \times 5 = 30

.

Adım Adım Çözüm

1

Determine the length of the horizontal sides of the rectangle.

The horizontal sides have a length of 6 units.

The horizontal sides connect vertices with the same y-coordinates, such as

(-2, -3)

and

(4, -3)

. The distance is the difference in their x-coordinates:

4 - (-2) = 6

.

2

Determine the length of the vertical sides of the rectangle.

The vertical sides have a length of 5 units.

The vertical sides connect vertices with the same x-coordinates, such as

(4, -3)

and

(4, 2)

. The distance is the difference in their y-coordinates:

2 - (-3) = 5

.

3

Calculate the area of the rectangle.

The area of the rectangle is 30 square units.

The area of a rectangle is found by multiplying its length by its width:

\text{Area} = 6 \times 5 = 30

.

Anahtar Kavram

Finding the area of a rectangle on the coordinate plane by calculating the lengths of its horizontal and vertical sides.

Soru 2262Soru →

On a coordinate grid, line $d$ is represented by the equation $y = -\frac{3}{4}x + 5$ . Line $e$ is perpendicular to line $d$ and passes through the point $(1, 2)$ . Which of the following is the equation of line $e$ ?

y = -\frac{4}{3}x + \frac{10}{3}

y = -\frac{3}{4}x + \frac{11}{4}

y = \frac{4}{3}x + \frac{2}{3}

y = \frac{4}{3}x + 1

y = \frac{3}{4}x + \frac{5}{4}

Cevabı ve açıklamayı göster

Cevap: $y = \frac{4}{3}x + \frac{2}{3}$

Cevap

The equation of line

e

is

y = \frac{4}{3}x + \frac{2}{3}

.

The correct equation has a slope of

\frac{4}{3}

and a y-intercept of

\frac{2}{3}

. The slope of the given line is

-\frac{3}{4}

, meaning any line perpendicular to it must have a slope that is the negative reciprocal, which is

\frac{4}{3}

. Using the slope-intercept form

y = mx + b

with the point

(1, 2)

allows us to solve for

b

by calculating

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

, which simplifies to

b = 2 - \frac{4}{3} = \frac{2}{3}

. Writing this together in slope-intercept form yields

y = \frac{4}{3}x + \frac{2}{3}

.

Adım Adım Çözüm

1

Identify the slope of the given line

d

.

The slope of line

d

is

-\frac{3}{4}

.

The equation

y = -\frac{3}{4}x + 5

is in slope-intercept form (

y = mx + b

), where the coefficient of

x

represents the slope.

2

Determine the slope of the perpendicular line

e

.

The slope of line

e

is

\frac{4}{3}

.

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

-\frac{3}{4}

is

\frac{4}{3}

.

3

Substitute the perpendicular slope and the given point

(1, 2)

into the slope-intercept equation to solve for the y-intercept

b

.

b = \frac{2}{3}

Plugging the values into

y = mx + b

gives

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

. Solving for

b

requires subtracting

\frac{4}{3}

from

2

, which yields

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

.

4

Write the final equation of line

e

in slope-intercept form.

y = \frac{4}{3}x + \frac{2}{3}

Substituting the slope

m = \frac{4}{3}

and y-intercept

b = \frac{2}{3}

into the standard slope-intercept form equation.

Anahtar Kavram

Finding the equation of a line perpendicular to a given line through a given point using negative reciprocal slopes.

Tahmini Süre:1m 0s

Soru 2263Soru →

The length $L$ , in centimeters, of a copper rod at a temperature of $T$ degrees Celsius can be modeled by the linear equation $L = 150.04 + 0.012T$ . If the length of the rod is measured to be $150.40$ centimeters, what is its temperature in degrees Celsius?

Cevabı ve açıklamayı göster

Cevap: 30

Cevap

The temperature of the rod is 30 degrees Celsius.

The correct answer is 30. By substituting the given length of

150.40

centimeters into the equation for

L

, we get

150.40 = 150.04 + 0.012T

. Subtracting

150.04

from both sides gives

0.36 = 0.012T

. Finally, dividing both sides by

0.012

yields the temperature

T = 30

degrees Celsius.

Adım Adım Çözüm

1

Substitute the measured length

L = 150.40

into the equation.

150.40 = 150.04 + 0.012T

To set up the equation with the given value for length.

2

Subtract 150.04 from both sides of the equation.

0.36 = 0.012T

To isolate the variable term containing

T

.

3

Divide both sides of the equation by 0.012.

T = 30

To find the temperature

T

.

Anahtar Kavram

Solving a multi-step linear equation involving decimals

Soru 2264Soru →

A straight hiking trail ascends a hill at a constant incline. On a coordinate grid where the units represent meters, the path of the trail is a straight line starting at the coordinate point $(10, 150)$ and ending at the coordinate point $(90, 190)$ . What is the slope of this trail?

Cevabı ve açıklamayı göster

Cevap: 0.5

Cevap

The correct answer is

0.5

(or

\frac{1}{2}

).

The slope of a line is defined as the change in the

y

-coordinates divided by the change in the

x

-coordinates:

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

. By substituting the coordinates of the start of the trail

(10, 150)

and the end of the trail

(90, 190)

, we calculate

\frac{190 - 150}{90 - 10} = \frac{40}{80} = 0.5

.

Adım Adım Çözüm

1

Identify the coordinates from the problem statement.

(x_1, y_1) = (10, 150)

and

(x_2, y_2) = (90, 190)

To calculate the slope between two points, we first need to define their coordinates.

2

Recall the formula for the slope of a line passing through two points.

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

Slope represents the vertical change (rise) divided by the horizontal change (run).

3

Substitute the coordinate values into the slope formula and simplify.

m = \frac{190 - 150}{90 - 10} = \frac{40}{80} = 0.5

Plugging the values into the formula yields the constant rate of change (slope) of the trail.

Anahtar Kavram

Slope of a Line

Soru 2265Soru →

A system of equations consists of the linear equation $y = x - 2$ and the quadratic equation $y = (x - 3)^2 - 5$ . If $(x_1, y_1)$ and $(x_2, y_2)$ are the two distinct real solutions to this system, what is the value of the product $y_1 y_2$ ?

-10

-4

3

4

6

Cevabı ve açıklamayı göster

Cevap: -4

Cevap

The product of the two

y

-coordinates of the intersection points is

-4

.

Substituting the linear expression for

y

into the quadratic equation yields the quadratic equation

x^2 - 7x + 6 = 0

. Factoring this equation gives the

x

-coordinates

6

and

1

. Substituting these values back into the linear equation

y = x - 2

yields the corresponding

y

-coordinates

4

and

-1

. The product of these

y

-coordinates is

4 \times (-1) = -4

.

Adım Adım Çözüm

1

Equate the two expressions for

y

to solve for the

x

-coordinates of the intersection points.

x - 2 = (x - 3)^2 - 5

At the points of intersection, the

y

-values of both equations are equal.

2

Expand the squared binomial and simplify the equation into standard quadratic form.

x^2 - 7x + 6 = 0

Expanding

(x - 3)^2

yields

x^2 - 6x + 9

. Substituting this back gives

x - 2 = x^2 - 6x + 4

. Subtracting

x

and adding

2

to both sides yields the standard quadratic equation.

3

Factor the quadratic equation to find the solutions for

x

.

x = 6

or

x = 1

The equation factors as

(x - 6)(x - 1) = 0

, giving the two

x

-values.

4

Substitute each

x

-value back into the linear equation

y = x - 2

to find the corresponding

y

-coordinates.

y_1 = 4

and

y_2 = -1

For

x = 6

,

y = 6 - 2 = 4

. For

x = 1

,

y = 1 - 2 = -1

.

5

Calculate the product of the two

y

-coordinates.

y_1 y_2 = 4 \times (-1) = -4

The question asks for the product of the two

y

-coordinates.

Anahtar Kavram

Solving a system of linear and quadratic equations using substitution and factoring.

Alternatif Yöntem

Alternatively, you can expand the quadratic equation first to

y = x^2 - 6x + 4

and set it equal to the linear equation

y = x - 2

. Solving for

x

yields

x^2 - 7x + 6 = 0

, from which you can find the coordinates and calculate their product.

Tahmini Süre:1m 30s

Soru 2266Soru →

A triangle has side lengths of $7$ , $12$ , and $2x + 1$ . If $x$ is an integer, how many possible values of $x$ exist?

5

6

7

8

13

Cevabı ve açıklamayı göster

Cevap: $6$

Cevap

6

To form a valid triangle, the length of any side must be strictly less than the sum of the other two sides and strictly greater than the positive difference of the other two sides. Applying this to the side lengths

7

,

12

, and

2x+1

gives the inequality

12 - 7 < 2x + 1 < 12 + 7

, which simplifies to

5 < 2x + 1 < 19

. Subtracting

1

from all parts gives

4 < 2x < 18

, and dividing by

2

gives

2 < x < 9

. The integers in this open interval are

3, 4, 5, 6, 7,

and

8

, which counts to 6 possible values.

Adım Adım Çözüm

1

Apply the Triangle Inequality Theorem, which states that the sum of any two sides of a triangle must be strictly greater than the third side.

We obtain three inequalities: (1)

7 + 12 > 2x + 1

, (2)

7 + (2x + 1) > 12

, and (3)

12 + (2x + 1) > 7

.

To find the valid range for the unknown side length expression

2x + 1

.

2

Solve the three inequalities for

x

.

From (1),

18 > 2x \implies x < 9

. From (2),

2x + 8 > 12 \implies 2x > 4 \implies x > 2

. From (3),

2x + 13 > 7 \implies 2x > -6 \implies x > -3

. Combining the most restrictive bounds gives the interval

2 < x < 9

.

To isolate the variable

x

and establish its upper and lower bounds.

3

Identify and count all integers

x

that satisfy the inequality

2 < x < 9

.

The integers strictly between 2 and 9 are

3, 4, 5, 6, 7

, and

8

. There are 6 such integers.

To find the number of possible integer values for

x

as requested by the question.

Anahtar Kavram

Triangle Inequality Theorem

Soru 2267Soru →

A triangle in the standard $(x, y)$ coordinate plane has vertices at $A(1, 1)$ , $B(10, 16)$ , and $C(5, 9)$ . A line passes through the point $P(4, 7)$ on the side $AC$ and intersects the side $AB$ at a point $Q$ . If this line divides the triangle into two regions of equal area, what is the length of the line segment $PQ$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

5

The total area of the triangle is 6. To divide the triangle into two equal-area regions, each region must have an area of 3. Since point

P(4,7)

lies

\frac{3}{4}

of the way from

A

to

C

, the area of the sub-triangle

\triangle PBC

is only

\frac{1}{4} \times 6 = 1.5

. Thus, the dividing line must intersect side

AB

at a point

Q

to form

\triangle APQ

with an area of 3. Using the area ratio formula,

\text{Area}(\triangle APQ) = \frac{AP}{AC} \times \frac{AQ}{AB} \times \text{Area}(\triangle ABC) \implies 3 = \frac{3}{4} \times \frac{AQ}{AB} \times 6

, which gives

\frac{AQ}{AB} = \frac{2}{3}

. Using the section formula, the coordinates of

Q

are

A + \frac{2}{3}(B - A) = (1, 1) + \frac{2}{3}(9, 15) = (7, 11)

. Finally, the length of

PQ

is

\sqrt{(7-4)^2 + (11-7)^2} = \sqrt{3^2 + 4^2} = 5

.

Adım Adım Çözüm

1

Calculate the area of the entire triangle

ABC

using the Shoelace formula.

Area of

\triangle ABC = 6

.

Establishing the total area of the triangle is necessary to determine the target area of 3 for each of the two equal-area regions.

2

Determine which side of the triangle the dividing line intersects by comparing the area of

\triangle PBC

to the target area of 3.

The line must intersect side

AB

at a point

Q

.

Since

P(4,7)

lies

\frac{3}{4}

of the way along

AC

, the base

PC

is

\frac{1}{4}

of

AC

. The area of

\triangle PBC

is

\frac{1}{4} \times 6 = 1.5

. Since this is less than 3, the dividing line cannot intersect side

BC

and must intersect side

AB

instead.

3

Set up the area ratio equation for

\triangle APQ

to find the ratio

\frac{AQ}{AB}

.

\frac{AQ}{AB} = \frac{2}{3}

.

The area of

\triangle APQ

is given by

\text{Area}(\triangle APQ) = \frac{AP}{AC} \times \frac{AQ}{AB} \times \text{Area}(\triangle ABC) \implies 3 = \frac{3}{4} \times \frac{AQ}{AB} \times 6 \implies \frac{AQ}{AB} = \frac{2}{3}

.

4

Find the coordinates of

Q

using the section formula along segment

AB

from

A(1,1)

to

B(10,16)

.

Q(7, 11)

.

Applying

Q = A + \frac{2}{3}(B - A) = (1, 1) + \frac{2}{3}(9, 15) = (7, 11)

yields the exact coordinates of

Q

.

5

Calculate the length of segment

PQ

using the distance formula between

P(4,7)

and

Q(7,11)

.

PQ = 5

.

The question asks for the length of the segment

PQ

, which is the distance between these two points.

Anahtar Kavram

Using coordinate geometry formulas and area ratios to solve problems involving geometric figures on the coordinate plane.

Soru 2268Soru →

In the standard $(x, y)$ coordinate plane, the point $M$ is the midpoint of the line segment with endpoints $P(-4, -2)$ and $Q(4, 2)$ . A second line segment is drawn from $M$ to a point $R(x, y)$ such that the length of the segment $MR$ is $8$ . If the midpoint of the segment $MR$ lies on the line $3x - 4y + 12 = 0$ , what is the smallest possible value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: -8

Cevap

The smallest possible value of

x

is

-8

.

By finding the midpoint

M(0,0)

of

PQ

and writing the midpoint of

MR

as

\left(\frac{x}{2}, \frac{y}{2}\right)

, we substitute this into the line equation to find

y = \frac{3}{4}x + 6

. We then substitute this into the distance formula equation

x^2 + y^2 = 64

to get the quadratic equation

25x^2 + 144x - 448 = 0

, which yields the solutions

x = -8

and

x = 2.24

. The smallest possible value is

-8

.

Adım Adım Çözüm

1

Calculate the coordinates of the midpoint

M

of segment

PQ

.

M = (0, 0)

The midpoint formula states that the midpoint of a segment with endpoints

(x_1, y_1)

and

(x_2, y_2)

is

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

.

2

Express the midpoint

N

of segment

MR

in terms of

R(x, y)

.

N = \left(\frac{x}{2}, \frac{y}{2}\right)

The midpoint of

M(0, 0)

and

R(x, y)

is found by averaging their coordinates.

3

Substitute the coordinates of

N

into the equation of the line

3x - 4y + 12 = 0

.

y = \frac{3}{4}x + 6

Since the midpoint

N

lies on the line, its coordinates must satisfy the line's equation, which gives a linear relationship between

x

and

y

.

4

Set up the equation for the distance

MR = 8

.

x^2 + y^2 = 64

The distance formula between

M(0, 0)

and

R(x, y)

is

d = \sqrt{x^2 + y^2}

, and squaring both sides gives

x^2 + y^2 = d^2

.

5

Substitute

y = \frac{3}{4}x + 6

into the distance equation and solve the quadratic equation.

x = -8

and

x = 2.24

Substituting the linear relationship into the quadratic circle equation gives a single quadratic equation in terms of

x

, which can be solved using the quadratic formula.

6

Determine the smallest value of

x

from the two possible solutions.

-8

Comparing the two solutions,

-8

is smaller than

2.24

.

Anahtar Kavram

Distance and Midpoint Formulas

Alternatif Yöntem

Instead of solving algebraically, one can scale the line

3x - 4y + 12 = 0

by a factor of 2 centered at the origin

M(0,0)

to directly obtain the line equation on which

R

lies:

3x - 4y + 24 = 0

. Then, find the intersection of this line with the circle

x^2 + y^2 = 64

using substitution.

Tahmini Süre:2m 30s

Soru 2269Soru →

An online retail store charges a flat shipping fee of $10.00$ for any order, plus $5.00$ per pound of the package's weight. A customer places an order and uses a coupon that gives them a $20\%$ discount on the total cost (the flat shipping fee plus the weight cost). If the customer's total bill after the discount is $36.00$ , what would their total bill have been, in dollars, if they had not used the coupon?

30.00

38.00

42.50

43.20

45.00

Cevabı ve açıklamayı göster

Cevap: $45.00$

Cevap

The total bill without the coupon would have been

45.00

.

The correct answer represents the total cost before the coupon discount. Since a

20\%

discount was applied to the entire bill, the final bill of

36.00

represents

80\%

of the original total bill. Setting up the equation

0.80T = 36.00

and dividing

36.00

by

0.80

yields the original total bill of

45.00

.

Adım Adım Çözüm

1

Define the variable for the original total bill and set up the equation using the discount percentage.

Let

T

represent the total bill before the coupon discount. A

20\%

discount means the customer pays

100\% - 20\% = 80\%

of the original total bill. This gives the linear equation

0.80T = 36.00

.

Establishing a direct relationship between the original price and the discounted price allows for a direct solution without needing to calculate the weight of the package first.

2

Solve the equation for

T

by dividing the discounted cost by the remaining percentage factor.

T = \frac{36.00}{0.80} = 45.00

Dividing both sides of the equation by

0.80

isolates the variable

T

, representing the original total bill in dollars.

Anahtar Kavram

Translating percentage changes and linear cost models into algebraic equations.

Alternatif Yöntem

You can also solve this by finding the weight of the package first. Setting up the equation with the weight

w

gives

0.80(10.00 + 5.00w) = 36.00

. Dividing by

0.80

simplifies the expression to

10.00 + 5.00w = 45.00

, which is the original bill. Solving further gives

5.00w = 35.00

and

w = 7

pounds. Substituting

7

back into the original cost formula yields

10.00 + 5.00(7) = 45.00

.

Tahmini Süre:1m 30s

Soru 2270Soru →

In the standard $(x, y)$ coordinate plane, the points $(k, -3)$ , $(1, k)$ , and $(4, 9)$ lie on the same straight line, where $k$ is a constant. Which of the following is a possible value for the slope of this line?

-\frac{3}{4}

\frac{4}{3}

-\frac{4}{3}

4

\frac{22}{3}

Cevabı ve açıklamayı göster

Cevap: $-\frac{4}{3}$

Cevap

The correct answer is

-\frac{4}{3}

.

The correct answer is

-\frac{4}{3}

. If three points are collinear, the slope computed between any two pairs must be equal. Setting the slope between

(k, -3)

and

(1, k)

equal to the slope between

(1, k)

and

(4, 9)

gives the equation

\frac{k + 3}{1 - k} = \frac{9 - k}{3}

. Cross-multiplying yields

3(k + 3) = (9 - k)(1 - k)

, which simplifies to

k^2 - 13k = 0

. Solving this gives

k = 0

or

k = 13

. Substituting

k = 13

back into the coordinates gives the points

(13, -3)

,

(1, 13)

, and

(4, 9)

. The slope of the line passing through these points is

\frac{9 - 13}{4 - 1} = -\frac{4}{3}

.

Adım Adım Çözüm

1

Set up the collinearity condition using the slope formula

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

.

The slope of the line segment between

(k, -3)

and

(1, k)

must equal the slope of the line segment between

(1, k)

and

(4, 9)

.

Since all three points lie on the same straight line, the slope between any two pairs of points must be equal.

2

Write the algebraic expressions for the slopes and set them equal to each other.

\frac{k - (-3)}{1 - k} = \frac{9 - k}{4 - 1} \Rightarrow \frac{k + 3}{1 - k} = \frac{9 - k}{3}

This establishes a rational equation containing the variable

k

.

3

Cross-multiply to eliminate the fractions and simplify.

3(k + 3) = (9 - k)(1 - k) \Rightarrow 3k + 9 = 9 - 10k + k^2

Cross-multiplying allows us to convert the rational equation into a polynomial equation.

4

Rearrange the terms into standard quadratic form and solve for

k

.

k^2 - 13k = 0 \Rightarrow k(k - 13) = 0 \Rightarrow k = 0 \text{ or } k = 13

Setting the quadratic expression to zero allows us to find the two possible values for the constant

k

by factoring.

5

Calculate the slope of the line for each possible value of

k

.

If

k = 0

, the slope is

m = \frac{9 - 0}{4 - 1} = 3

. If

k = 13

, the slope is

m = \frac{9 - 13}{4 - 1} = -\frac{4}{3}

.

We must check which of the calculated slopes matches one of the given choices.

Anahtar Kavram

Finding the slope of a line and using the condition of collinearity to solve for missing coordinates.

Tahmini Süre:2m 0s

Soru 2271Soru →

A rectangle has vertices at $A(-1, -2)$ , $B(3, -2)$ , $C(3, 1)$ , and $D(-1, 1)$ in the standard $(x,y)$ coordinate plane. What is the perimeter of this rectangle?

7

10

11

12

14

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

14

The horizontal length of the rectangle is

3 - (-1) = 4

units, and the vertical length is

1 - (-2) = 3

units. Using the formula for the perimeter of a rectangle,

P = 2(\text{width} + \text{height})

, we get

P = 2(4 + 3) = 14

units.

Adım Adım Çözüm

1

Calculate the horizontal length (width) of the rectangle.

width = 4

Since vertices

A(-1, -2)

and

B(3, -2)

share the same y-coordinate, the horizontal distance between them is the difference in their x-coordinates:

3 - (-1) = 4

.

2

Calculate the vertical length (height) of the rectangle.

height = 3

Since vertices

B(3, -2)

and

C(3, 1)

share the same x-coordinate, the vertical distance between them is the difference in their y-coordinates:

1 - (-2) = 3

.

3

Use the perimeter formula for a rectangle to find the total perimeter.

perimeter = 14

The perimeter is given by

P = 2(\text{width} + \text{height}) = 2(4 + 3) = 2(7) = 14

.

Anahtar Kavram

Perimeter of geometric figures on the coordinate plane

Tahmini Süre:45s

Soru 2272Soru →

A graphic designer uses a coordinate plane to design a logo containing a triangular shape. The final image of the triangle has vertices at $(3, -1)$ , $(5, -3)$ , and $(2, -5)$ . The designer created this final image by performing a $90^\circ$ counterclockwise rotation of the original triangle about the origin, followed by a translation of $4$ units to the right and $3$ units down. What are the coordinates of the vertices of the original triangle?

(2, 1)

,

(0, -1)

, and

(-2, 2)

(-2, -1)

,

(0, 1)

, and

(2, -2)

(-4, -7)

,

(-6, -9)

, and

(-8, -6)

(5, 0)

,

(7, 2)

, and

(9, -1)

(-1, -2)

,

(1, 0)

, and

(-2, 2)

Cevabı ve açıklamayı göster

Cevap: $(2, 1)$ , $(0, -1)$ , and $(-2, 2)$

Cevap

The coordinates of the vertices of the original triangle are

(2, 1)

,

(0, -1)

, and

(-2, 2)

.

The correct answer is obtained by working backward from the final image. First, undo the translation of

4

units right and

3

units down by translating the final vertices

4

units left and

3

units up. This yields intermediate vertices

(-1, 2)

,

(1, 0)

, and

(-2, -2)

. Second, undo the

90^\circ

counterclockwise rotation by rotating these intermediate vertices

90^\circ

clockwise about the origin. The algebraic rule for a

90^\circ

clockwise rotation is

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

. Applying this rule to

(-1, 2)

,

(1, 0)

, and

(-2, -2)

gives the original coordinates

(2, 1)

,

(0, -1)

, and

(-2, 2)

.

Adım Adım Çözüm

1

Identify the two transformations in reverse order and define their inverse operations.

The final transformation was a translation of

4

units to the right and

3

units down, so the first step in working backward is a translation of

4

units to the left and

3

units up. The initial transformation was a

90^\circ

counterclockwise rotation about the origin, so the second step in working backward is a

90^\circ

clockwise rotation about the origin.

To find the pre-image, we must apply the inverse of each transformation in the reverse order of how they were originally applied.

2

Apply the inverse translation of

4

units left and

3

units up to the final image vertices:

(3, -1)

,

(5, -3)

, and

(2, -5)

.

The intermediate vertices are:

A'(3, -1) \rightarrow A_1(3 - 4, -1 + 3) = (-1, 2)

;

B'(5, -3) \rightarrow B_1(5 - 4, -3 + 3) = (1, 0)

;

C'(2, -5) \rightarrow C_1(2 - 4, -5 + 3) = (-2, -2)

.

Undoing a translation of

(+4, -3)

requires subtracting

4

from the

x

-coordinates and adding

3

to the

y

-coordinates.

3

Apply the inverse rotation, a

90^\circ

clockwise rotation about the origin, to the intermediate vertices.

The rule for a

90^\circ

clockwise rotation about the origin is

(x_1, y_1) \rightarrow (y_1, -x_1)

. Applying this to the intermediate vertices yields:

A_1(-1, 2) \rightarrow A(2, 1)

;

B_1(1, 0) \rightarrow B(0, -1)

;

C_1(-2, -2) \rightarrow C(-2, 2)

.

A

90^\circ

clockwise rotation is the inverse of a

90^\circ

counterclockwise rotation, which reverses the coordinates and changes the sign of the new

y

-coordinate.

Anahtar Kavram

Finding the pre-image of a figure under composite transformations in the coordinate plane by applying inverse transformations in reverse order.

Soru 2273Soru →

In the standard $(x, y)$ coordinate plane, triangle $PQR$ has vertices $P(2, 1)$ , $Q(5, 1)$ , and $R(2, 5)$ . The triangle is first rotated $90^\circ$ counterclockwise about the origin to form triangle $P'Q'R'$ . Next, triangle $P'Q'R'$ is reflected across the $y$ -axis to form triangle $P''Q''R''$ . What are the coordinates of the vertex $R''$ ?

(5, 2)

(-5, 2)

(5, -2)

(-5, -2)

(2, -5)

Cevabı ve açıklamayı göster

Cevap: $(5, 2)$

Cevap

(5, 2)

The correct answer is the coordinate pair

(5, 2)

. First, rotating the point

R(2, 5)

counterclockwise by

90^\circ

about the origin uses the transformation rule

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

, which maps

R(2, 5)

to

R'(-5, 2)

. Next, reflecting the point

R'(-5, 2)

across the

y

-axis uses the transformation rule

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

, which maps

R'(-5, 2)

to

R''(5, 2)

.

Adım Adım Çözüm

1

Apply the rotation of

90^\circ

counterclockwise about the origin to the vertex

R(2, 5)

.

The rule for a

90^\circ

counterclockwise rotation is

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

. Applying this to

R(2, 5)

yields

R'(-5, 2)

.

To find the coordinates after the first transformation step.

2

Apply the reflection across the

y

-axis to the intermediate point

R'(-5, 2)

.

The rule for reflection across the

y

-axis is

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

. Applying this to

R'(-5, 2)

yields

R''(5, 2)

.

To find the final coordinates after the second transformation step.

Anahtar Kavram

Applying composite transformations (rotation followed by reflection) to coordinates in the coordinate plane.

Soru 2274Soru →

A triangle has side lengths such that one side is $3$ units less than twice an integer $y$ , another side is $4$ units more than $y$ , and the third side is $11$ units. How many different triangles can be formed with these side lengths?

8

13

14

15

16

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

14

The correct answer is 14. Applying the Triangle Inequality Theorem, the sum of any two sides of the triangle must be strictly greater than the third side. This gives three inequalities: (2y - 3) + (y + 4) > 11, (2y - 3) + 11 > y + 4, and (y + 4) + 11 > 2y - 3. Solving these inequalities yields y > 10/3, y > -4, and y < 18. The common intersection is 10/3 < y < 18. Since y is an integer, y can range from 4 to 17 inclusive. There are 17 - 4 + 1 = 14 integers in this range.

Adım Adım Çözüm

1

Translate the verbal descriptions into algebraic expressions for the side lengths.

The three side lengths are represented as

2y - 3

,

y + 4

, and

11

.

To apply mathematical theorems, the side lengths must first be written in algebraic form.

2

Set up the three inequalities required by the Triangle Inequality Theorem, stating that the sum of any two sides must be strictly greater than the third side.

The inequalities are: 1)

(2y - 3) + (y + 4) > 11

, 2)

(2y - 3) + 11 > y + 4

, and 3)

(y + 4) + 11 > 2y - 3

.

The Triangle Inequality Theorem guarantees that three segment lengths can form a non-degenerate triangle.

3

Solve each of the three inequalities for

y

.

1)

3y + 1 > 11 \implies 3y > 10 \implies y > \frac{10}{3} \approx 3.33

.
2)

2y + 8 > y + 4 \implies y > -4

.
3)

y + 15 > 2y - 3 \implies 18 > y \implies y < 18

.

Solving the inequalities identifies the constraints on the variable

y

.

4

Find the intersection of all three solution intervals and identify the valid integer values for

y

.

The intersection of the intervals is

3.33 < y < 18

. Since

y

must be an integer,

y

can be any integer from

4

to

17

, inclusive:

\{4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17\}

.

The value of

y

must simultaneously satisfy all three inequality conditions.

5

Count the number of integers in the inclusive range

[4, 17]

.

The number of integers is

17 - 4 + 1 = 14

.

This gives the total number of distinct triangles that can be formed.

Anahtar Kavram

Triangle Inequality Theorem

Soru 2275Soru →

A line has a slope of $-3$ . If the sum of its $x$ -intercept and its $y$ -intercept is $12$ , what is the $y$ -intercept of the line?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The y-intercept of the line is 9.

The equation of the line is

y = -3x + b

, where

b

is the

y

-intercept. The

x

-intercept is found by setting

y=0

, which yields

x = \frac{b}{3}

. Setting their sum to

12

gives

\frac{b}{3} + b = 12

, which simplifies to

4b = 36

and results in

b = 9

.

Adım Adım Çözüm

1

Express the equation of the line using the slope-intercept form.

The equation of the line is

y = -3x + b

, where

b

is the

y

-intercept.

We are given that the slope of the line is

-3

.

2

Find the

x

-intercept of the line in terms of

b

.

Setting

y = 0

gives

0 = -3x + b \implies 3x = b \implies x = \frac{b}{3}

. Thus, the

x

-intercept is

\frac{b}{3}

.

The

x

-intercept of a line is the

x

-coordinate where the line crosses the

x

-axis, which occurs when

y = 0

.

3

Set up an equation using the given sum of the intercepts.

\frac{b}{3} + b = 12

The problem states that the sum of the

x

-intercept and the

y

-intercept is

12

.

4

Solve the linear equation for the

y

-intercept

b

.

Multiply the entire equation by

3

to clear the denominator:

b + 3b = 36 \implies 4b = 36 \implies b = 9

.

Solving this equation yields the value of the

y

-intercept.

Anahtar Kavram

Linear equations, slope-intercept form, and finding intercepts.

Alternatif Yöntem

Alternatively, you can write the equation of the line in intercept form:

\frac{x}{a} + \frac{y}{b} = 1

, where

a

is the

x

-intercept and

b

is the

y

-intercept. The slope of this line is

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

, which means

b = 3a

. Since the sum of the intercepts is

12

, we write

a + b = 12

. Substituting

b = 3a

into this sum gives

a + 3a = 12 \implies 4a = 12 \implies a = 3

. Therefore, the

y

-intercept

b

is

3(3) = 9

.

Tahmini Süre:1m 30s

Soru 2276Soru →

In the standard $(x, y)$ coordinate plane, line $p$ is perpendicular to the line represented by the equation $y = -0.2x + 9$ . What is the slope of line $p$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The slope of line

p

is

5

.

The given line is in slope-intercept form

y = mx + b

with a slope of

-0.2

, which can be written as

-\frac{1}{5}

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

-\frac{1}{5}

, which is

5

.

Adım Adım Çözüm

1

Identify the slope of the given line from its equation.

The slope of the line

y = -0.2x + 9

is

-0.2

(or

-\frac{1}{5}

).

The equation is in slope-intercept form

y = mx + b

, where the coefficient of

x

represents the slope

m

.

2

Calculate the slope of the perpendicular line.

The negative reciprocal of

-\frac{1}{5}

is

5

.

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

m_1 \cdot m_2 = -1

).

Anahtar Kavram

Perpendicular lines in the coordinate plane have slopes that are negative reciprocals of each other.

Tahmini Süre:45s

Soru 2277Soru →

If $x$ is a real number such that $\log_4(x - 3) = \frac{1}{2} + \log_4(2)$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The value of

x

is 7.

By moving the logarithmic terms to the same side and applying the quotient rule, the equation simplifies to

\log_4\left(\frac{x-3}{2}\right) = \frac{1}{2}

. Converting this to exponential form yields

\frac{x-3}{2} = 4^{1/2}

. Since

4^{1/2} = 2

, we have

\frac{x-3}{2} = 2

, which gives

x - 3 = 4

, or

x = 7

. Substituting

x = 7

back into the original equation shows it is a valid solution.

Adım Adım Çözüm

1

Group logarithmic terms on one side of the equation.

\log_4(x - 3) - \log_4(2) = \frac{1}{2}

Grouping the logarithms allows them to be combined using logarithmic properties.

2

Apply the quotient property of logarithms.

\log_4\left(\frac{x - 3}{2}\right) = \frac{1}{2}

The quotient property states that

\log_b(a) - \log_b(c) = \log_b(\frac{a}{c})

.

3

Convert the equation from logarithmic form to exponential form.

\frac{x - 3}{2} = 4^{1/2}

A logarithmic equation

\log_b(y) = z

is equivalent to

b^z = y

.

4

Evaluate the fractional exponent and solve the linear equation for

x

.

x = 7

Since

4^{1/2} = 2

, the equation becomes

\frac{x - 3}{2} = 2

. Multiplying both sides by 2 gives

x - 3 = 4

, so adding 3 to both sides yields

x = 7

.

Anahtar Kavram

Solving logarithmic equations using properties of logarithms

Tahmini Süre:1m 30s

Soru 2278Soru →

In $\triangle PQR$ , the measure of interior angle $\angle P$ is $40^\circ$ . If the measure of $\angle Q$ is twice the measure of $\angle P$ , what is the measure, in degrees, of the interior angle $\angle R$ ?

Cevabı ve açıklamayı göster

Cevap: 60

Cevap

The measure of interior angle

\angle R

is 60 degrees.

By definition, the interior angles of any triangle sum to 180 degrees. Given that angle P is 40 degrees, and angle Q is twice angle P (80 degrees), the sum of angles P and Q is 120 degrees. Subtracting this from 180 degrees gives a remaining measure of 60 degrees for angle R.

Adım Adım Çözüm

1

Calculate the measure of angle Q

m\angle Q = 80^\circ

The problem states that the measure of angle Q is twice the measure of angle P, which is 40 degrees.

2

Apply the Triangle Angle Sum Theorem

m\angle P + m\angle Q + m\angle R = 180^\circ

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

3

Solve for the measure of angle R

m\angle R = 60^\circ

Substitute the values of angles P and Q into the equation: 40 + 80 + m\angle R = 180, which simplifies to 120 + m\angle R = 180, and solving for m\angle R gives 60.

Anahtar Kavram

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

Soru 2279Soru →

For all non-zero real numbers $x$ and $y$ , the expression $\frac{(x^2 + y^{-1})^2}{x^4}$ is equivalent to which of the following?

1 + x^{-4}y^{-2}

1 + 2x^{-2}y + x^{-4}y^2

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

1 + 2x^2 y^{-1} + y^{-2}

y^{-2} + 2x^{-2}y^{-2} + x^{-4}y^{-2}

Cevabı ve açıklamayı göster

Cevap: $1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}$

Cevap

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

The correct expression is obtained by expanding the binomial in the numerator using the perfect square formula

(a + b)^2 = a^2 + 2ab + b^2

. Substituting

a = x^2

and

b = y^{-1}

yields

(x^2 + y^{-1})^2 = (x^2)^2 + 2(x^2)(y^{-1}) + (y^{-1})^2 = x^4 + 2x^2 y^{-1} + y^{-2}

. Dividing each term by the denominator

x^4

and applying the quotient rule for exponents

\frac{x^m}{x^n} = x^{m-n}

results in

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

.

Adım Adım Çözüm

1

Expand the binomial in the numerator.

(x^2 + y^{-1})^2 = (x^2)^2 + 2(x^2)(y^{-1}) + (y^{-1})^2 = x^4 + 2x^2 y^{-1} + y^{-2}

Apply the binomial expansion formula

(a + b)^2 = a^2 + 2ab + b^2

where

a = x^2

and

b = y^{-1}

.

2

Divide each term in the expanded numerator by the denominator.

\frac{x^4}{x^4} + \frac{2x^2 y^{-1}}{x^4} + \frac{y^{-2}}{x^4}

Distribute the division over the terms in the numerator.

3

Simplify each fraction using exponent properties.

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

Use the quotient rule

\frac{x^m}{x^n} = x^{m-n}

to simplify the variables.

Anahtar Kavram

Properties of Exponents in Algebraic Expressions

Soru 2280Soru →

A student designed an experiment to investigate how the concentration of fertilizer affects the growth of *Arabidopsis thaliana* plants. The student prepared four pots, each containing five seedlings. Pot 1 was watered with a 0% fertilizer solution (distilled water), Pot 2 with a 1% solution, Pot 3 with a 5% solution, and Pot 4 with a 10% solution. Pots 1 and 2 were placed on a sunny windowsill, while Pots 3 and 4 were placed on a shaded shelf in the same room. All pots received the same volume of liquid daily. After three weeks, the average height of the plants in each pot was measured. Which of the following identifies the primary confounding variable in this experimental design?

The different locations of the pots, which exposed the plants to varying levels of sunlight

The different concentrations of fertilizer solution applied to each pot

The use of Pot 1 as a control group receiving 0% fertilizer solution

The number of seedlings planted in each pot

Cevabı ve açıklamayı göster

Cevap: The different locations of the pots, which exposed the plants to varying levels of sunlight

Cevap

The different locations of the pots, which exposed the plants to varying levels of sunlight

The correct answer is the option identifying the different locations of the pots. In a well-designed experiment, only the independent variable (fertilizer concentration) should vary between groups. Because the pots were placed in different locations (sunny windowsill vs. shaded shelf), the plants received different amounts of sunlight. Sunlight is a critical factor for plant growth, so this difference acts as a confounding variable, preventing the student from determining whether the fertilizer or the light levels caused the differences in growth.

Adım Adım Çözüm

1

Identify the independent variable and the dependent variable in the experiment.

The independent variable is the concentration of fertilizer (0%, 1%, 5%, 10%), and the dependent variable is the average height of the plants after three weeks.

To evaluate experimental validity, we must first understand what is being manipulated and what is being measured.

2

Analyze the experimental conditions to identify any external factors that were not held constant across all treatment groups.

Pots 1 and 2 were placed on a sunny windowsill, whereas Pots 3 and 4 were placed on a shaded shelf. This introduces a second variable (sunlight exposure) that changes along with the independent variable (fertilizer concentration).

A confounding variable is an uncontrolled factor that varies systematically with the independent variable, potentially affecting the dependent variable.

3

Determine which option describes this uncontrolled variable.

The option specifying the different locations of the pots and their exposure to varying levels of sunlight correctly identifies the confounding variable.

Since both fertilizer concentration and light levels changed between the groups, any difference in plant height cannot be confidently attributed to the fertilizer alone.

Anahtar Kavram

A confounding variable is an uncontrolled factor that varies along with the independent variable, making it impossible to isolate the cause of any observed changes in the dependent variable.

Tahmini Süre:1m 0s

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