Tüm alıştırma soruları

Soru 2181Soru →

In the standard $(x,y)$ coordinate plane, a triangle has vertices at $A(2, 4)$ , $B(6, 4)$ , and $C(2, 10)$ . The triangle undergoes a series of transformations: first, it is reflected across the line $y = x$ ; next, the resulting image is dilated by a scale factor of $\frac{1}{2}$ with the center of dilation at $(2, 2)$ ; finally, this image is translated $1$ unit to the left and $3$ units down. What are the coordinates of the final image of vertex $C$ ?

(4, -2)

(7, 5)

(5, -1)

(5, 5)

(-5, -3)

Cevabı ve açıklamayı göster

Cevap: $(5, -1)$

Cevap

(5, -1)

The correct answer is

(5, -1)

because applying the three transformations sequentially yields the correct final coordinates: first, reflecting

C(2, 10)

across the line

y = x

gives

C'(10, 2)

; second, dilating

C'(10, 2)

by a scale factor of

\frac{1}{2}

centered at

(2, 2)

gives

C''(6, 2)

; and finally, translating

C''(6, 2)

by

1

unit left and

3

units down results in

(5, -1)

.

Adım Adım Çözüm

1

Reflect the initial coordinates of vertex

C(2, 10)

across the line

y = x

.

C'(10, 2)

Reflecting a point

(x, y)

across the line

y = x

interchanges the coordinates, mapping

(x, y)

to

(y, x)

.

2

Dilate the point

C'(10, 2)

by a scale factor of

k = \frac{1}{2}

centered at

P(2, 2)

.

C''(6, 2)

The formula for a dilation centered at

(a, b)

is

(x'', y'') = (a + k(x' - a), b + k(y' - b))

. Substituting

a=2, b=2, k=\frac{1}{2}, x'=10, y'=2

gives

(2 + \frac{1}{2}(10-2), 2 + \frac{1}{2}(2-2)) = (2+4, 2+0) = (6, 2)

.

3

Translate the point

C''(6, 2)

by

1

unit to the left and

3

units down.

C'''(5, -1)

Translating

1

unit left subtracts

1

from the

x

-coordinate (

6 - 1 = 5

), and translating

3

units down subtracts

3

from the

y

-coordinate (

2 - 3 = -1

).

Anahtar Kavram

Composite transformations in the coordinate plane involving reflections, dilations with non-origin centers, and translations.

Tahmini Süre:2m 0s

Soru 2182Soru →

In the standard $(x, y)$ coordinate plane, a line with a negative slope passes through the point $(4, 3)$ and has an $x$ -intercept that is twice its $y$ -intercept. If the equation of this line is written in the form $Ax + By = C$ , where $A$ , $B$ , and $C$ are integers with no common factor greater than 1, and $A > 0$ , what is the value of $C$ ?

Cevabı ve açıklamayı göster

Cevap: 10

Cevap

The value of

C

is 10.

The correct answer is 10. By setting the intercepts as

(2b, 0)

and

(0, b)

, we find the slope is

m = -\frac{1}{2}

. Applying the point-slope formula with

(4, 3)

gives

y - 3 = -\frac{1}{2}(x - 4)

, which simplifies to

y = -\frac{1}{2}x + 5

. Converting this to standard form with

A > 0

yields

x + 2y = 10

, where

C = 10

.

Adım Adım Çözüm

1

Represent the coordinates of the intercepts using a variable.

The

y

-intercept is

(0, b)

and the

x

-intercept is

(2b, 0)

.

The problem states that the

x

-intercept is twice the

y

-intercept.

2

Calculate the slope (

m

) of the line using the intercepts.

m = \frac{b - 0}{0 - 2b} = -\frac{1}{2}

The slope of a line passing through

(x_1, y_1)

and

(x_2, y_2)

is given by

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

.

3

Find the equation of the line using the point-slope form with point

(4, 3)

.

y - 3 = -\frac{1}{2}(x - 4) \Rightarrow y = -\frac{1}{2}x + 5

The point-slope form of a linear equation is

y - y_1 = m(x - x_1)

.

4

Convert the equation to the standard form

Ax + By = C

.

x + 2y = 10

Multiplying the equation by 2 and moving the

x

term to the left side results in integer coefficients where the coefficient of

x

is positive (

A = 1 > 0

).

5

Identify the value of

C

from the standard form equation.

C = 10

Comparing

x + 2y = 10

with

Ax + By = C

shows that

A=1

,

B=2

, and

C=10

. These integers share no common factors greater than 1.

Anahtar Kavram

Converting a linear equation to standard form using given geometric features and a coordinate point.

Alternatif Yöntem

Instead of using the point-slope formula, substitute the point

(4, 3)

directly into the intercept form of a linear equation,

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

. Since

a = 2b

, this becomes

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

. Simplifying this gives

\frac{2}{b} + \frac{3}{b} = 1 \Rightarrow \frac{5}{b} = 1 \Rightarrow b = 5

. Thus,

a = 10

. The intercept form is

\frac{x}{10} + \frac{y}{5} = 1

. Multiplying by 10 to clear denominators gives

x + 2y = 10

, where

C = 10

.

Tahmini Süre:1m 30s

Soru 2183Soru →

In the standard $(x, y)$ coordinate plane, the perpendicular bisector of the line segment with endpoints $(1, 2)$ and $(5, 10)$ intersects the curve $y = x^2$ at two points. If one of these intersection points lies in the second quadrant, what is the $y$ -coordinate of this point?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The correct answer is

9

.

The perpendicular bisector of the segment connecting

(1, 2)

and

(5, 10)

passes through their midpoint

(3, 6)

and has a slope of

-\frac{1}{2}

(the negative reciprocal of

2

). The equation of this line is

x + 2y = 15

. Substituting

y = x^2

yields the quadratic equation

2x^2 + x - 15 = 0

, which factors into

(2x - 5)(x + 3) = 0

. The solution in the second quadrant corresponds to the negative

x

-coordinate,

x = -3

. Squaring this value gives a

y

-coordinate of

9

.

Adım Adım Çözüm

1

Find the midpoint of the segment with endpoints

(1, 2)

and

(5, 10)

.

The midpoint is

(3, 6)

.

The perpendicular bisector of a segment passes through its midpoint.

2

Calculate the slope of the segment and the slope of the perpendicular bisector.

The slope of the segment is

2

, and the perpendicular slope is

-\frac{1}{2}

.

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

3

Write the equation of the perpendicular bisector.

The equation of the line is

x + 2y = 15

.

Using the point-slope form with point

(3, 6)

and slope

-\frac{1}{2}

yields

y - 6 = -\frac{1}{2}(x - 3)

, which simplifies to

x + 2y = 15

.

4

Find the intersection points of the line and the curve

y = x^2

.

The

x

-coordinates of the intersection points are

2.5

and

-3

.

Substituting

y = x^2

into

x + 2y = 15

gives the quadratic equation

2x^2 + x - 15 = 0

, which factors as

(2x - 5)(x + 3) = 0

.

5

Identify the point in the second quadrant and find its

y

-coordinate.

The point is

(-3, 9)

, so the

y

-coordinate is

9

.

A point in the second quadrant must have a negative

x

-coordinate (

x = -3

) and a positive

y

-coordinate (

y = 9

).

Anahtar Kavram

Perpendicular bisectors and systems of linear-quadratic equations in coordinate geometry

Soru 2184Soru →

A delivery drone starts a flight with a battery charge of $95\%$ . The battery charge decreases at a constant rate of $1.5\%$ per minute of flight time. During the flight, the drone lands on a charging station for $15$ minutes, during which its battery charge increases at a constant rate of $2.4\%$ per minute. After this charging period, the drone resumes its flight. If the drone ends its flight with a battery charge of $65\%$ , and its total flight time (excluding the time spent charging) was $t$ minutes, what is the value of $t$ ?

Cevabı ve açıklamayı göster

Cevap: 44

Cevap

44

The correct answer is found by setting up a linear equation that models the drone's battery charge changes. The drone starts with

95\%

battery, loses

1.5\%

per minute for

t

minutes, and gains

36\%

from charging (

15 \times 2.4\%

). Setting this equal to the final charge of

65\%

gives the equation

95 - 1.5t + 36 = 65

. Simplifying gives

131 - 1.5t = 65

, which leads to

-1.5t = -66

and

t = 44

.

Adım Adım Çözüm

1

Calculate the total percentage of battery charge gained while charging.

36\%

The drone charges at a rate of

2.4\%

per minute for

15

minutes, so the total gain is

15 \times 2.4\% = 36\%

.

2

Set up a linear equation for the final battery charge.

95 - 1.5t + 36 = 65

The final battery charge (

65\%

) is the initial charge (

95\%

) minus the battery consumed during flight (

1.5\%

per minute for

t

minutes) plus the charge gained (

36\%

).

3

Combine constant terms on the left side of the equation.

131 - 1.5t = 65

Adding

95

and

36

simplifies the expression on the left side.

4

Isolate the variable term by subtracting

131

from both sides.

-1.5t = -66

Subtracting

131

from both sides leaves only the variable term on the left.

5

Divide both sides by

-1.5

to solve for

t

.

t = 44

Dividing

-66

by

-1.5

isolates

t

to find the total flight time.

Anahtar Kavram

Solving linear equations in a real-world context by setting up an algebraic equation.

Tahmini Süre:1m 30s

Soru 2185Soru →

What is the complete solution set for the inequality $|7 - 3n| < 11$ ?

n < -\frac{4}{3}

or

n > 6

-\frac{4}{3} < n < 6

-6 < n < \frac{4}{3}

n > -\frac{4}{3}

n < -\frac{4}{3}

Cevabı ve açıklamayı göster

Cevap: $-\frac{4}{3} < n < 6$

Cevap

-\frac{4}{3} < n < 6

To solve

|7 - 3n| < 11

, rewrite it as the compound inequality

-11 < 7 - 3n < 11

. Subtracting

7

from all parts yields

-18 < -3n < 4

. Finally, dividing by

-3

and reversing the inequality signs gives

6 > n > -\frac{4}{3}

, which simplifies to the interval

-\frac{4}{3} < n < 6

. This matches the correct option.

Adım Adım Çözüm

1

Express the absolute value inequality as a compound inequality.

-11 < 7 - 3n < 11

An absolute value inequality of the form

|u| < c

is equivalent to the compound inequality

-c < u < c

.

2

Subtract

7

from all three parts of the inequality.

-18 < -3n < 4

To isolate the term containing

n

, we subtract

7

from all parts of the inequality.

3

Divide all three parts by

-3

and reverse the inequality signs.

6 > n > -\frac{4}{3}

, which is equivalent to

-\frac{4}{3} < n < 6

Dividing an inequality by a negative number requires reversing the direction of the inequality signs to maintain a true statement.

Anahtar Kavram

Absolute Value Equations and Inequalities

Soru 2186Soru →

In $\triangle PQR$ , the lengths of sides $PQ$ and $PR$ are equal. If the measure of $\angle P$ is $70^\circ$ , what is the measure, in degrees, of $\angle Q$ ?

Cevabı ve açıklamayı göster

Cevap: 55

Cevap

The measure of

\angle Q

is

55^\circ

.

Since

PQ = PR

,

\triangle PQR

is an isosceles triangle with base

QR

, which means the base angles

\angle Q

and

\angle R

have equal measures. The sum of the interior angles of a triangle is

180^\circ

. Setting up the equation gives

70^\circ + 2(\angle Q) = 180^\circ

. Subtracting

70^\circ

from both sides yields

2(\angle Q) = 110^\circ

. Dividing by 2, we find that the measure of

\angle Q

is

55^\circ

.

Adım Adım Çözüm

1

Identify the properties of the given triangle.

\triangle PQR

is an isosceles triangle with base

QR

and base angles

\angle Q = \angle R

.

A triangle with two equal sides is isosceles, and the angles opposite those sides are equal in measure.

2

Apply the triangle angle sum theorem.

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

The sum of the measures of the interior angles of any triangle is always

180^\circ

.

3

Substitute the known values and solve for

\angle Q

.

70^\circ + 2(\angle Q) = 180^\circ \implies 2(\angle Q) = 110^\circ \implies \angle Q = 55^\circ

Substituting

\angle P = 70^\circ

and

\angle R = \angle Q

allows us to solve the linear equation for the unknown angle measure.

Anahtar Kavram

Isosceles Triangle Properties and Triangle Angle Sum Theorem

Tahmini Süre:45s

Soru 2187Soru →

For the functions $f(x) = |4x - 17|$ and $g(x) = 3 - 2x$ , what is the value of $f(g(5))$ ?

Cevabı ve açıklamayı göster

Cevap: 45

Cevap

The correct answer is 45.

Evaluating the inner function first yields

g(5) = 3 - 2(5) = -7

. Substituting this result into the outer function gives

f(-7) = |4(-7) - 17| = |-45| = 45

.

Adım Adım Çözüm

1

Evaluate the inner function

g(5)

g(5) = 3 - 2(5) = -7

In a composite function of the form

f(g(x))

, the inner function

g(x)

must be evaluated first at the given input.

2

Evaluate the outer function

f(x)

at the output of the inner function

f(-7) = |4(-7) - 17| = |-28 - 17| = |-45| = 45

The output of the inner function,

-7

, becomes the input for the outer function

f(x)

.

Anahtar Kavram

Function Evaluation and Composition

Soru 2188Soru →

In the standard $(x,y)$ coordinate plane, the point $A(-4, 3)$ is reflected across the $x$ -axis and then translated $5$ units to the right to map onto point $A'$ . What are the coordinates of $A'$ ?

(1, -3)

(-9, -3)

(1, 3)

(9, 3)

(-9, 3)

Cevabı ve açıklamayı göster

Cevap: $(1, -3)$

Cevap

(1, -3)

Reflecting the point

A(-4, 3)

across the

x

-axis negates its

y

-coordinate, resulting in

(-4, -3)

. Then, translating this point

5

units to the right increases its

x

-coordinate by

5

, which yields

(-4 + 5, -3) = (1, -3)

.

Adım Adım Çözüm

1

Reflect the point

A(-4, 3)

across the

x

-axis.

The rule for reflection across the

x

-axis is

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

. Applying this rule to

A(-4, 3)

yields the intermediate point

(-4, -3)

.

A reflection across the

x

-axis negates the

y

-coordinate of the point while keeping the

x

-coordinate the same.

2

Translate the reflected point

(-4, -3)

by

5

units to the right.

The rule for translating a point

h

units to the right is

(x, y) \rightarrow (x + h, y)

. Adding

5

to the

x

-coordinate yields

(-4 + 5, -3) = (1, -3)

.

Translating a point to the right increases its

x

-coordinate by the translation distance.

Anahtar Kavram

Applying a sequence of coordinate transformations (reflection followed by translation) to a point.

Tahmini Süre:45s

Soru 2189Soru →

A company's net profit is calculated by subtracting its monthly operating costs from its monthly revenue. In a certain month, the company's revenue is $260,000$ dollars minus twice its monthly operating costs. If the company wants its monthly net profit to be at least $110,000$ dollars, which of the following inequalities represents all possible values of its monthly operating costs, $C$ , in dollars?

C \ge 50,000

C \le 50,000

C \le 75,000

C \ge 75,000

C \le 150,000

Cevabı ve açıklamayı göster

Cevap: $C \le 50,000$

Cevap

The inequality

C \le 50,000

The correct inequality is obtained by setting up the net profit as the revenue (

260,000 - 2C

) minus the operating costs (

C

), which simplifies to

260,000 - 3C

. Setting this profit to be at least

110,000

gives the inequality

260,000 - 3C \ge 110,000

. Subtracting

260,000

from both sides results in

-3C \ge -150,000

. Finally, dividing both sides by

-3

and reversing the inequality sign because of the division by a negative number yields

C \le 50,000

.

Adım Adım Çözüm

1

Define the expression for net profit by subtracting the operating costs,

C

, from the revenue, which is

260,000 - 2C

.

Net profit =

(260,000 - 2C) - C = 260,000 - 3C

Net profit is defined as revenue minus operating costs.

2

Set up the inequality where the net profit is at least

110,000

.

260,000 - 3C \ge 110,000

The phrase 'at least' translates to the 'greater than or equal to' inequality sign (

\ge

).

3

Subtract

260,000

from both sides of the inequality.

-3C \ge -150,000

Isolate the term containing the variable

C

.

4

Divide both sides by

-3

and reverse the inequality sign.

C \le 50,000

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

Anahtar Kavram

Translating and Solving Algebraic Word Problems

Tahmini Süre:1m 30s

Soru 2190Soru →

For the functions $f(x) = (x - 2)^2 - 5$ and $g(x) = |x + 1| - 3$ , what is the value of $f(g(-2))$ ?

9

-5

11

31

-22

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

11

To find the value of

f(g(-2))

, first evaluate the inner function

g(x)

at

x = -2

. Substituting

-2

into

g(x) = |x + 1| - 3

gives

g(-2) = |-2 + 1| - 3 = 1 - 3 = -2

. Next, substitute this output as the input for the outer function

f(x)

. Substituting

-2

into

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

yields

f(-2) = (-2 - 2)^2 - 5 = (-4)^2 - 5 = 16 - 5 = 11

.

Adım Adım Çözüm

1

Evaluate the inner function at the given input value

g(-2) = |-2 + 1| - 3 = |-1| - 3 = 1 - 3 = -2

In function composition

f(g(x))

, the inner function must be evaluated first to determine the input for the outer function.

2

Evaluate the outer function using the output of the inner function

f(-2) = (-2 - 2)^2 - 5 = (-4)^2 - 5 = 16 - 5 = 11

Substitute the result from the inner function evaluation into the outer function to find the final value.

Anahtar Kavram

Function composition and evaluation

Tahmini Süre:1m 0s

Soru 2191Soru →

A system administrator monitors the temperatures of two servers, Server A and Server B, over a period of $t$ hours after midnight. The temperature of Server A, in degrees Celsius, is modeled by a linear function of time. At $t = 2$ hours, the temperature is $30^\circ\text{C}$ , and at $t = a$ hours (where $a > 2$ ), the temperature is $(30 + a^2 - 2a)^\circ\text{C}$ . The temperature of Server B, in degrees Celsius, is also modeled by a linear function of time. At $t = 1$ hour, Server B's temperature is $25^\circ\text{C}$ , and at $t = a$ hours, its temperature is $(a^2 + 30)^\circ\text{C}$ . If the temperature of Server B increases at a constant rate that is $1.5$ times the constant rate of temperature increase of Server A, what is the value of $a$ ?

2

3

5

11

13

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The value of

a

is 5.

To find the rate of change (slope) for each server, we apply the slope formula

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

. For Server A, the slope is

m_A = \frac{(30 + a^2 - 2a) - 30}{a - 2} = a

. For Server B, the slope is

m_B = \frac{(a^2 + 30) - 25}{a - 1} = \frac{a^2 + 5}{a - 1}

. We are given that

m_B = 1.5 m_A

, which yields the equation

\frac{a^2 + 5}{a - 1} = 1.5a

. Multiplying both sides by

a - 1

gives

a^2 + 5 = 1.5a(a - 1) = 1.5a^2 - 1.5a

. Grouping all terms on one side gives

0.5a^2 - 1.5a - 5 = 0

. Multiplying the equation by 2 results in

a^2 - 3a - 10 = 0

, which factors as

(a - 5)(a + 2) = 0

. Since

a > 2

, the only valid solution is the positive root, which is 5.

Adım Adım Çözüm

1

Calculate the constant rate of temperature increase (slope) for Server A,

m_A

, using the coordinates

(2, 30)

and

(a, 30 + a^2 - 2a)

.

m_A = \frac{(30 + a^2 - 2a) - 30}{a - 2} = \frac{a^2 - 2a}{a - 2} = \frac{a(a-2)}{a-2} = a

(since

a > 2

)

The constant rate of change of a linear temperature model is represented by the slope of the line passing through its data points.

2

Calculate the constant rate of temperature increase (slope) for Server B,

m_B

, using the coordinates

(1, 25)

and

(a, a^2 + 30)

.

m_B = \frac{(a^2 + 30) - 25}{a - 1} = \frac{a^2 + 5}{a - 1}

The constant rate of change of Server B is represented by the slope of the line passing through its data points.

3

Set up the equation using the given relationship that Server B's rate of increase is

1.5

times Server A's rate (

m_B = 1.5 m_A

).

1.5a = \frac{a^2 + 5}{a - 1}

Equating the rate of change of Server B to 1.5 times the rate of change of Server A allows us to write an equation in terms of the variable

a

.

4

Solve the equation for

a

by cross-multiplying and simplifying the resulting quadratic equation.

1.5a(a - 1) = a^2 + 5 \implies 1.5a^2 - 1.5a = a^2 + 5 \implies 0.5a^2 - 1.5a - 5 = 0 \implies a^2 - 3a - 10 = 0 \implies (a-5)(a+2) = 0

Cross-multiplying eliminates the fraction, and grouping all terms on one side forms a standard quadratic equation that can be factored.

5

Identify the valid value of

a

that satisfies the initial domain constraint.

a = 5

(since

a = -2

is rejected because the problem states

a > 2

)

The solution must satisfy the given condition that

a > 2

for the time intervals to be valid.

Anahtar Kavram

Calculating and equating rates of change (slopes) of linear functions using the slope formula

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

and solving the resulting quadratic equation.

Soru 2192Soru →

On a coordinate map, two tracking stations are located at the points $(2, -3)$ and $(-6, 12)$ . What is the straight-line distance, in map units, between the two stations?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

The straight-line distance between the two stations is 17 map units.

Applying the coordinate distance formula to the coordinates

(2, -3)

and

(-6, 12)

yields a distance of

\sqrt{(-6 - 2)^2 + (12 - (-3))^2} = \sqrt{(-8)^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17

.

Adım Adım Çözüm

1

Identify coordinates of the two stations.

(x_1, y_1) = (2, -3)

and

(x_2, y_2) = (-6, 12)

Defining the coordinate variables is necessary to apply the formula correctly.

2

Apply the distance formula.

d = \sqrt{(-6 - 2)^2 + (12 - (-3))^2}

The distance formula calculates the straight-line distance between two coordinates.

3

Simplify the arithmetic terms.

d = \sqrt{(-8)^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289}

Simplify the differences and squares under the radical.

4

Calculate the final square root.

d = 17

The square root of 289 is 17.

Anahtar Kavram

Distance Formula

Soru 2193Soru →

A circle is graphed in the standard $(x, y)$ coordinate plane such that it is tangent to the $x$ -axis at $(4, 0)$ and tangent to the $y$ -axis at $(0, 4)$ . A line segment has one of its endpoints at the center of this circle and its midpoint at the point $(1, -1)$ . What is the total length of this line segment?

2\sqrt{10}

\sqrt{34}

2\sqrt{34}

16

2\sqrt{2}

Cevabı ve açıklamayı göster

Cevap: $2\sqrt{34}$

Cevap

The total length of the line segment is

2\sqrt{34}

.

The center of the circle is determined to be

C(4, 4)

based on the points of tangency at

(4, 0)

and

(0, 4)

. The distance between the center

C(4, 4)

and the midpoint

M(1, -1)

is calculated using the distance formula to be

\sqrt{34}

. Since the midpoint divides the segment into two equal halves, the total length of the segment is twice this distance, which is

2\sqrt{34}

.

Adım Adım Çözüm

1

Determine the coordinates of the center of the circle.

The center of the circle is

C(4, 4)

.

Since the circle is tangent to the

x

-axis at

(4, 0)

, its center must lie on the vertical line

x = 4

. Since it is tangent to the

y

-axis at

(0, 4)

, its center must lie on the horizontal line

y = 4

. Their intersection gives the center of the circle at

C(4, 4)

.

2

Calculate the distance from the center of the circle to the midpoint of the segment.

The distance is

\sqrt{34}

.

Using the distance formula between the center

C(4, 4)

and the midpoint

M(1, -1)

:

d(C, M) = \sqrt{(1 - 4)^2 + (-1 - 4)^2} = \sqrt{(-3)^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}

.

3

Find the total length of the line segment.

The total length is

2\sqrt{34}

.

Since the midpoint divides the segment into two equal parts, the total length is twice the distance from one endpoint to the midpoint:

\text{Length} = 2 \times d(C, M) = 2\sqrt{34}

.

Anahtar Kavram

Using circle tangencies to find the center, and applying the distance and midpoint formulas to determine segment properties.

Tahmini Süre:1m 30s

Soru 2194Soru →

In the standard $(x, y)$ coordinate plane, if the slope of a line passing through the points $(k, k^2)$ and $(k+1, (k+1)^2)$ is an integer, then $k$ must be an integer. Is this statement true or false?

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

False

The statement is false because the simplified slope of the line is

2k + 1

. The slope

2k + 1

is an integer for any half-integer value of

k

(such as

k = 0.5

, which yields a slope of

2

), showing that

k

does not have to be an integer.

Adım Adım Çözüm

1

Use the slope formula

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

with the given coordinates

(k, k^2)

and

(k+1, (k+1)^2)

.

The slope is

m = \frac{(k+1)^2 - k^2}{(k+1) - k}

.

To express the slope of the line in terms of the variable

k

.

2

Simplify the numerator and denominator of the slope expression.

m = \frac{k^2 + 2k + 1 - k^2}{1} = 2k + 1

.

To obtain a simplified algebraic expression for the slope.

3

Set the slope equal to an arbitrary integer

n

and solve for

k

.

2k + 1 = n \implies k = \frac{n-1}{2}

.

To find the set of values for

k

that yield an integer slope.

4

Evaluate if

k

must be an integer by choosing an integer value for

n

.

If

n = 2

, then

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

, which is not an integer.

To test the truth value of the statement by finding a counterexample.

Anahtar Kavram

Determining the slope of a line from two coordinate points and analyzing the algebraic properties of the slope expression.

Soru 2195Soru →

A hyperbola is defined by the equation $9x^2 - 4y^2 - 36x - 8y - 4 = 0$ in the standard $(x, y)$ coordinate plane. What is the slope of the asymptote of this hyperbola that has a positive slope?

Cevabı ve açıklamayı göster

Cevap: 1.5

Cevap

The positive slope of the asymptotes is 1.5.

By converting the general form of the hyperbola equation into standard form, we determine that it is a horizontal hyperbola with

a = 2

and

b = 3

. The slopes of the asymptotes for a horizontal hyperbola are

\pm \frac{b}{a}

, making the positive slope equal to

\frac{3}{2} = 1.5

.

Adım Adım Çözüm

1

Group x-terms and y-terms, and move the constant to the other side.

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

This prepares the equation for completing the square.

2

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

9(x - 2)^2 - 4(y + 1)^2 = 36

Completing the square yields

9[(x-2)^2 - 4] - 4[(y+1)^2 - 1] = 4 \implies 9(x-2)^2 - 36 - 4(y+1)^2 + 4 = 4

.

3

Divide both sides by 36 to format the equation in standard hyperbola form.

\frac{(x-2)^2}{4} - \frac{(y+1)^2}{9} = 1

The standard form of a horizontal hyperbola centered at

(h, k)

is

\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1

.

4

Identify the values of a and b from the denominators.

a = 2

and

b = 3

Since

a^2 = 4

and

b^2 = 9

, taking the square roots gives

a = 2

and

b = 3

.

5

Determine the positive slope of the asymptotes using the formula for a horizontal hyperbola.

m = \frac{b}{a} = 1.5

The asymptotes for a horizontal hyperbola are given by

y - k = \pm \frac{b}{a}(x - h)

, so the positive slope is

\frac{b}{a} = \frac{3}{2} = 1.5

.

Anahtar Kavram

Rewriting a hyperbola equation from general form to standard form to find asymptote equations.

Tahmini Süre:1m 30s

Soru 2196Soru →

In quadrilateral $ABCD$ , diagonal $AC$ divides the figure into two triangles, $\triangle ABC$ and $\triangle ACD$ . The lengths of three of the sides are $AB = 6\text{ cm}$ , $BC = 8\text{ cm}$ , and $CD = 12\text{ cm}$ . If the length of side $AD$ is an integer $k\text{ cm}$ , what is the sum of the minimum and maximum possible values of $k$ ?

20

24

25

26

28

Cevabı ve açıklamayı göster

Cevap: 26

Cevap

26

To find the minimum and maximum values of the integer side

k

, we first determine the range of the shared diagonal

AC

. In the bottom triangle, the Triangle Inequality Theorem requires

2 < AC < 14

. In the top triangle, the theorem requires

|12 - AC| < k < 12 + AC

. To minimize

k

, we look at the lower bound

|12 - AC|

. Since

AC

can be

12

, the lower bound can be

0

, meaning

k > 0

, so the minimum integer value is

1

. To maximize

k

, we look at the upper bound

12 + AC

. Since

AC < 14

, the upper bound is

k < 26

, so the maximum integer value is

25

. The sum of these values is

1 + 25 = 26

.

Adım Adım Çözüm

1

Find the possible range of lengths for the diagonal

AC

using

\triangle ABC

.

2\text{ cm} < AC < 14\text{ cm}

According to the Triangle Inequality Theorem, the length of side

AC

must be greater than

|BC - AB| = 8 - 6 = 2\text{ cm}

and less than

BC + AB = 8 + 6 = 14\text{ cm}

.

2

Set up the inequality for side

AD = k

in

\triangle ACD

.

|12 - AC| < k < 12 + AC

By the Triangle Inequality Theorem applied to

\triangle ACD

, the length

k

must be greater than the absolute difference

|CD - AC|

and less than the sum

CD + AC

.

3

Find the minimum possible integer value for

k

.

k = 1

Since

AC

can be any real number between

2

and

14

, we can choose

AC = 12

, which makes the lower bound

|12 - 12| = 0

. Thus, we have

k > 0

. The smallest integer greater than

0

is

1

. We verify that

k = 1

is possible by choosing

AC = 12

, which satisfies the inequalities for both triangles.

4

Find the maximum possible integer value for

k

.

k = 25

Since

AC < 14

, the upper bound is

k < 12 + AC < 12 + 14 = 26

. Thus, we have

k < 26

. The largest integer less than

26

is

25

. We verify that

k = 25

is possible by choosing

AC = 13.5

, which satisfies the inequalities for both triangles.

5

Calculate the sum of the minimum and maximum possible integer values of

k

.

1 + 25 = 26

To find the sum of the minimum and maximum values of

k

as requested by the question.

Anahtar Kavram

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side.

Soru 2197Soru →

In the standard $(x, y)$ coordinate plane, a line segment $AB$ has endpoints $A(2, 5)$ and $B(6, -1)$ . Segment $AB$ is reflected across the line $y = -x$ to form segment $A'B'$ . Then, segment $A'B'$ is dilated by a scale factor of $3$ with a center of dilation at $C(2, -1)$ to form segment $A''B''$ . What is the sum of the $x$ - and $y$ -coordinates of the midpoint of segment $A''B''$ ?

16

-8

-16

-18

-20

Cevabı ve açıklamayı göster

Cevap: -20

Cevap

-20

To find the coordinates of the midpoint of the transformed segment

A''B''

, we can track the midpoint of the original segment

AB

through each transformation. The midpoint of

AB

is

M(4, 2)

. Reflecting

M

across the line

y = -x

swaps and negates its coordinates, resulting in

M'(-2, -4)

. Dilating

M'

by a scale factor of

3

centered at

C(2, -1)

means finding the point

M''

such that the vector

\vec{CM''}

is

3

times the vector

\vec{CM'}

. This is computed as

M'' = (2 + 3(-2 - 2), -1 + 3(-4 - (-1))) = (-10, -10)

. The sum of these coordinates is

-10 + (-10) = -20

.

Adım Adım Çözüm

1

Find the midpoint of the original segment

AB

.

The midpoint

M

of segment

AB

is

M = \left(\frac{2 + 6}{2}, \frac{5 + (-1)}{2}\right) = (4, 2)

.

Since rigid transformations (reflections) and similarity transformations (dilations) preserve midpoints, we can find the final midpoint by applying the sequence of transformations directly to the initial midpoint.

2

Reflect the midpoint

M(4, 2)

across the line

y = -x

to find the midpoint

M'

of segment

A'B'

.

Applying the reflection rule

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

to

M(4, 2)

yields

M'(-2, -4)

.

Reflecting a point across the line

y = -x

swaps its coordinates and negates both values.

3

Dilate the midpoint

M'(-2, -4)

by a scale factor of

3

with the center of dilation at

C(2, -1)

to find the midpoint

M''

of segment

A''B''

.

Using the dilation formula

(x'', y'') = (x_c + k(x' - x_c), y_c + k(y' - y_c))

with

C(2, -1)

and

k = 3

:

x'' = 2 + 3(-2 - 2) = -10

, and

y'' = -1 + 3(-4 - (-1)) = -10

. This gives the midpoint

M''(-10, -10)

.

A dilation centered at

(x_c, y_c)

scales the horizontal and vertical distances from the center to the point by the scale factor

k

.

4

Calculate the sum of the

x

- and

y

-coordinates of

M

.

The sum of the coordinates is

-10 + (-10) = -20

.

The question asks for the sum of the coordinates of the final midpoint.

Anahtar Kavram

Applying composite transformations in the coordinate plane to geometric figures and midpoints.

Soru 2198Soru →

In the standard $(x, y)$ coordinate plane, line segment $AB$ has endpoints $A(-2, 4)$ and $B(4, 2)$ . The segment is rotated $90^\circ$ counterclockwise about the origin, then reflected across the line $y = -x$ , and finally translated $3$ units to the right and $2$ units down. What are the coordinates of the midpoint of the final image of the segment?

(-4, 5)

(-3, 6)

(4, -5)

(2, 1)

(0, -1)

Cevabı ve açıklamayı göster

Cevap: $(2, 1)$

Cevap

(2, 1)

The correct midpoint is

(2, 1)

. First, the midpoint of the original segment

AB

is calculated using the midpoint formula, yielding

(1, 3)

. Rotating this point

90^\circ

counterclockwise about the origin maps

(x, y)

to

(-y, x)

, which gives

(-3, 1)

. Reflecting

(-3, 1)

across the line

y = -x

maps

(x, y)

to

(-y, -x)

, resulting in

(-1, 3)

. Finally, translating this point

3

units to the right and

2

units down yields

(-1 + 3, 3 - 2) = (2, 1)

.

Adım Adım Çözüm

1

Find the midpoint of the original line segment

AB

.

M(1, 3)

The midpoint formula is

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

. Substituting the coordinates of

A(-2, 4)

and

B(4, 2)

yields

\left(\frac{-2 + 4}{2}, \frac{4 + 2}{2}\right) = (1, 3)

.

2

Apply a

90^\circ

counterclockwise rotation about the origin to the midpoint.

M'(-3, 1)

A

90^\circ

counterclockwise rotation about the origin maps

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

. Applying this rule to

M(1, 3)

gives

(-3, 1)

.

3

Reflect the rotated point across the line

y = -x

.

M''(-1, 3)

A reflection across the line

y = -x

maps

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

. Applying this rule to

M'(-3, 1)

gives

(-1, 3)

.

4

Translate the reflected point

3

units right and

2

units down.

M'''(2, 1)

A translation of

3

units right and

2

units down maps

(x, y) \rightarrow (x + 3, y - 2)

. Translating

M''(-1, 3)

yields

(-1 + 3, 3 - 2) = (2, 1)

.

Anahtar Kavram

Composite transformations of geometric figures in the coordinate plane

Alternatif Yöntem

Instead of transforming the midpoint, you can transform the individual endpoints

A(-2, 4)

and

B(4, 2)

through all three steps. Point

A(-2, 4)

becomes

A'(-4, -2) \rightarrow A''(2, 4) \rightarrow A'''(5, 2)

. Point

B(4, 2)

becomes

B'(-2, 4) \rightarrow B''(-4, 2) \rightarrow B'''(-1, 0)

. The midpoint of

A'''

and

B'''

is then calculated to be

\left(\frac{5 + (-1)}{2}, \frac{2 + 0}{2}\right) = (2, 1)

.

Tahmini Süre:1m 30s

Soru 2199Soru →

In the standard $(x,y)$ coordinate plane, line $q$ is perpendicular to the line with equation $y = \frac{3}{4}x - 2$ . If line $q$ passes through the point $(2, 1)$ , what is the $y$ -intercept of line $q$ ?

-\frac{5}{3}

\frac{9}{4}

\frac{11}{3}

\frac{2}{3}

-\frac{1}{2}

Cevabı ve açıklamayı göster

Cevap: $\frac{11}{3}$

Cevap

The

y

-intercept of line

q

is

\frac{11}{3}

.

The slope of the given line

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

is

\frac{3}{4}

. The slope of a perpendicular line is the negative reciprocal of the original slope, which is

-\frac{4}{3}

. Using the point-slope form of a linear equation with the point

(2, 1)

gives

y - 1 = -\frac{4}{3}(x - 2)

. To find the

y

-intercept, set

x = 0

, which gives

y - 1 = -\frac{4}{3}(-2) = \frac{8}{3}

. Adding

1

to both sides yields

y = \frac{8}{3} + 1 = \frac{11}{3}

. Therefore, the correct

y

-intercept is

\frac{11}{3}

.

Adım Adım Çözüm

1

Determine the slope of the perpendicular line

q

.

The slope of the given line is

\frac{3}{4}

, so the slope of line

q

(the negative reciprocal) is

-\frac{4}{3}

.

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

2

Write the equation of line

q

using the point-slope form with the point

(2, 1)

.

y - 1 = -\frac{4}{3}(x - 2)

The point-slope form

y - y_1 = m(x - x_1)

is used to find the equation of a line given its slope and a point on the line.

3

Find the

y

-intercept by setting

x = 0

and solving for

y

.

y - 1 = -\frac{4}{3}(0 - 2) \Rightarrow y - 1 = \frac{8}{3} \Rightarrow y = \frac{8}{3} + 1 = \frac{11}{3}

The

y

-intercept of a line is the value of

y

when

x = 0

.

Anahtar Kavram

The slope of a line perpendicular to a line with slope

m

is

-\frac{1}{m}

. The equation of a line can be written in point-slope form as

y - y_1 = m(x - x_1)

.

Soru 2200Soru →

If $x$ is a real number greater than 2 such that $\log_2(x^2 - 4) - \log_2(x - 2) = 3$ , what is the value of $x$ ?

1

4

6

7

16

Cevabı ve açıklamayı göster

Cevap: 6

Cevap

The value of

x

is 6.

Applying the quotient property of logarithms simplifies the equation to

\log_2\left(\frac{x^2 - 4}{x - 2}\right) = 3

. Factoring the numerator as

(x-2)(x+2)

and canceling the common factor

(x-2)

yields

\log_2(x+2) = 3

. Rewriting this in exponential form gives

x + 2 = 2^3 = 8

, which yields

x = 6

.

Adım Adım Çözüm

1

Combine the logarithmic terms on the left side of the equation using the quotient property of logarithms:

\log_b A - \log_b B = \log_b\left(\frac{A}{B}\right)

.

\log_2\left(\frac{x^2 - 4}{x - 2}\right) = 3

Consolidating the two logarithmic terms simplifies the equation into a single logarithmic term.

2

Factor the numerator as a difference of squares and simplify the rational expression.

\log_2(x + 2) = 3

(since

x > 2

,

x - 2 \neq 0

)

Factoring

x^2 - 4

as

(x - 2)(x + 2)

allows the cancellation of the common factor

(x - 2)

.

3

Convert the logarithmic equation into its equivalent exponential form:

\log_b Y = c \Rightarrow Y = b^c

.

x + 2 = 2^3

This removes the logarithm and sets up a linear equation to solve.

4

Evaluate the exponent and solve for

x

.

x + 2 = 8 \Rightarrow x = 6

Evaluating

2^3

as 8 and subtracting 2 from both sides isolates

x

.

Anahtar Kavram

Solving logarithmic equations using properties of logarithms and algebraic factoring

Tahmini Süre:1m 30s

Cevap

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Sık Yapılan Hatalar

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Sık Yapılan Hatalar

Alternatif Yöntem

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Alternatif Yöntem

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Sık Yapılan Hatalar

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