Tüm alıştırma soruları

Soru 2481Soru →

A kite $WXYZ$ has diagonals $WY$ and $XZ$ that intersect at point $P$ . If the length of $WP$ is $4$ centimeters, the length of $PY$ is $9$ centimeters, and the length of $XP$ is $3$ centimeters, what is the measure, in degrees, of angle $WPX$ ?

30

45

60

90

180

Cevabı ve açıklamayı göster

Cevap: 90

Cevap

90 degrees

The diagonals of a kite are always perpendicular to each other. Therefore, the angle formed at their intersection, angle

WPX

, is a right angle, which measures exactly

90

degrees. The given segment lengths are extra information.

Adım Adım Çözüm

1

Identify the fundamental property of the diagonals of a kite.

The diagonals of any kite are perpendicular to each other.

By geometric definition, the diagonals of a kite intersect at a right angle.

2

Determine the measure of the angle formed by the intersection of the diagonals.

Angle

WPX

is a right angle, which measures exactly

90

degrees.

Since the diagonals are perpendicular, their intersection forms four

90

-degree angles regardless of the lengths of the individual diagonal segments.

Anahtar Kavram

The diagonals of a kite are perpendicular (

90^\circ

).

Soru 2482Soru →

In the standard $(x, y)$ coordinate plane, a point $P$ is reflected across the $y$ -axis and then translated 4 units down and 3 units right. The coordinates of the resulting image point, $P'$ , are $(1, -2)$ . What are the coordinates of the original point $P$ ?

(-2, -2)

(-2, 2)

(2, -6)

(2, -2)

(2, 2)

Cevabı ve açıklamayı göster

Cevap: $(2, 2)$

Cevap

The coordinates of the original point

P

are

(2, 2)

.

To find the coordinates of the original point

P(x, y)

, we apply the inverse transformations in reverse order to the final image point

P'(1, -2)

. First, we undo the translation of 4 units down and 3 units right by translating

P'

4 units up and 3 units left. This results in the intermediate coordinates

(1 - 3, -2 + 4) = (-2, 2)

. Next, we undo the reflection across the

y

-axis by reflecting

(-2, 2)

across the

y

-axis again (since a reflection is its own inverse). Changing the sign of the

x

-coordinate gives

(2, 2)

. Alternatively, setting up the equations

-x + 3 = 1

and

y - 4 = -2

and solving them yields

x = 2

and

y = 2

, which corresponds to the point

(2, 2)

.

Adım Adım Çözüm

1

Set up equations to express the composite transformations of the point

P(x, y)

to

P'(1, -2)

.

Reflecting

P(x, y)

across the

y

-axis changes the sign of the

x

-coordinate, yielding

(-x, y)

. Translating this point 4 units down and 3 units right yields

(-x + 3, y - 4)

.

Establishing the mathematical relationship for each transformation is necessary to work backward to find the coordinates of the pre-image.

2

Equate the coordinates of the transformed point to the coordinates of the final image point

P'(1, -2)

.

-x + 3 = 1

and

y - 4 = -2

This sets up two independent linear equations that can be solved for the original coordinates

x

and

y

.

3

Solve the equations for

x

and

y

.

From

-x + 3 = 1

, we get

-x = -2

, which means

x = 2

. From

y - 4 = -2

, we get

y = 2

. Therefore, the coordinates of

P

are

(2, 2)

.

Solving these equations gives the exact coordinates of the original pre-image point.

Anahtar Kavram

Working backward with composite transformations in the coordinate plane

Tahmini Süre:1m 15s

Soru 2483Soru →

Elena Rostova stood on the damp deck of the research vessel, staring at the freshly extracted core sample from a white oak timber salvaged from the wreck of the Gryphon. The ship had foundered in the autumn of 1682, but the story written in its rings stretched much further back. Just last year, in 2025, Elena had published her thesis on the Little Ice Age, utilizing data she had gathered during her 2022 expedition to the Swedish archives. There, she had unearthed the ship’s original cargo manifest, which detailed the lumber's harvest in a Baltic forest during the exceptionally cold spring of 1679. Now, looking at the narrow rings under her magnifier, she traced the signature of a severe drought in 1665—a year before the Great Fire of London, which she recalled studying during her college years. This oak had weathered that dry summer as a young sapling, decades before it was felled by shipwrights to form the Gryphon's keel.

According to the passage, which of the following events occurred first chronologically?

The oak tree weathering a severe drought

The harvesting of the lumber used for the ship

The foundering of the Gryphon in the Baltic Sea

Elena's research expedition to the Swedish archives

Cevabı ve açıklamayı göster

Cevap: The oak tree weathering a severe drought

Cevap

The oak tree weathering a severe drought

The oak tree weathering a severe drought is correct because the passage explicitly states this event occurred in 1665. This date precedes all other events mentioned in the text: the lumber harvest (1679), the sinking of the ship (1682), and Elena's research expedition (2022).

Adım Adım Çözüm

1

Identify the historical and modern dates linked to each event mentioned in the passage.

The drought occurred in 1665; the lumber harvest occurred in 1679; the Gryphon sank in 1682; and Elena's archival research occurred in 2022.

Extracting the specific dates for each event is the first step to establishing a correct chronological sequence.

2

Compare the dates to determine which occurred earliest in time.

The year 1665 is the earliest year among all the events described.

Finding the smallest numerical year value identifies the event that occurred first in the timeline.

Anahtar Kavram

Reconstructing a chronological timeline from a non-linear narrative by identifying and ordering events based on explicit dates and relative temporal clues.

Tahmini Süre:1m 30s

Soru 2484Soru →

A triangle has a vertex at $P(-4, 3)$ in the standard $(x, y)$ coordinate plane. If the triangle is reflected across the line $y = -x$ and then translated $3$ units to the right and $4$ units down, what are the coordinates of the image of vertex $P$ after both transformations?

(0, 0)

(6, -8)

(-6, 8)

(7, -1)

(-1, -7)

Cevabı ve açıklamayı göster

Cevap: $(0, 0)$

Cevap

(0, 0)

The coordinate

(0, 0)

is correct because reflecting the point

P(-4, 3)

across the line

y = -x

swaps and negates the coordinates, transforming it to

(-3, 4)

. Then, translating this point

3

units to the right and

4

units down is calculated as

(-3 + 3, 4 - 4)

, which simplifies to

(0, 0)

.

Adım Adım Çözüm

1

Apply the reflection across the line

y = -x

to the coordinate

P(-4, 3)

.

The rule for reflection across the line

y = -x

is

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

. Applying this rule to

P(-4, 3)

yields

(-3, 4)

.

To find the coordinates of the vertex after the first transformation in the composite sequence.

2

Apply the translation of

3

units right and

4

units down to the intermediate point

(-3, 4)

.

Translating

3

units to the right adds

3

to the

x

-coordinate, and translating

4

units down subtracts

4

from the

y

-coordinate:

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

.

To find the final position after the second transformation in the composite sequence.

Anahtar Kavram

Applying composite transformations in the coordinate plane, specifically a reflection across the line

y = -x

followed by a translation.

Soru 2485Soru →

For what value of the constant $c$ does the quadratic equation $x^2 - 3x + c = 0$ have two complex solutions with imaginary parts equal to $\pm 2i$ ?

-1.75

1.75

2.25

3.25

6.25

Cevabı ve açıklamayı göster

Cevap: $6.25$

Cevap

The value of the constant

c

is

6.25

.

The correct answer is

6.25

. Applying the quadratic formula to

x^2 - 3x + c = 0

gives solutions of the form

1.5 \pm \frac{\sqrt{9 - 4c}}{2}

. Since these solutions are complex with imaginary parts equal to

\pm 2i

, the term under the radical must be negative, and the imaginary component is

\frac{\sqrt{9 - 4c}}{2} = 2i

. Multiplying both sides by

2

gives

\sqrt{9 - 4c} = 4i

. Squaring both sides results in

9 - 4c = 16i^2

. Substituting

i^2 = -1

gives

9 - 4c = -16

. Solving for

c

yields

-4c = -25

, which simplifies to

c = 6.25

.

Adım Adım Çözüm

1

Apply the quadratic formula to the equation

x^2 - 3x + c = 0

.

The solutions are given by

x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(c)}}{2(1)} = 1.5 \pm \frac{\sqrt{9 - 4c}}{2}

.

This expresses the solutions in terms of the constant

c

so that the imaginary part can be identified.

2

Set the imaginary term of the solutions equal to the given imaginary parts

\pm 2i

.

\frac{\sqrt{9 - 4c}}{2} = 2i \implies \sqrt{9 - 4c} = 4i

.

The question specifies that the imaginary parts of the two complex solutions are

\pm 2i

.

3

Square both sides of the equation to solve for

c

.

9 - 4c = (4i)^2 = 16i^2

. Since

i^2 = -1

, this becomes

9 - 4c = -16

.

Squaring eliminates the radical and allows for standard algebraic isolation of the variable

c

.

4

Solve the linear equation for

c

.

-4c = -25 \implies c = 6.25

.

Subtracting

9

from both sides and then dividing by

-4

isolates the constant

c

.

Anahtar Kavram

Solving quadratic equations with complex roots using the quadratic formula and the properties of the imaginary unit.

Soru 2486Soru →

A sequence of numbers $t_1, t_2, t_3, \dots$ is defined by $t_1 = 3$ and $t_{n+1} = 3t_n - 2^n$ for all integers $n \geq 1$ . What is the value of the fourth term, $t_4$ ?

5

9

43

62

119

Cevabı ve açıklamayı göster

Cevap: 43

Cevap

The value of the fourth term is 43.

To find

t_4

, we use the recursive formula

t_{n+1} = 3t_n - 2^n

. Substituting

n=1

gives

t_2 = 3(3) - 2 = 7

. Substituting

n=2

gives

t_3 = 3(7) - 4 = 17

. Substituting

n=3

gives

t_4 = 3(17) - 8 = 43

.

Adım Adım Çözüm

1

Calculate the second term,

t_2

, by substituting

n = 1

and

t_1 = 3

into the formula

t_{n+1} = 3t_n - 2^n

.

t_2 = 3t_1 - 2^1 = 3(3) - 2 = 7

To progress to the fourth term, we must first find each preceding term in the sequence.

2

Calculate the third term,

t_3

, by substituting

n = 2

and

t_2 = 7

into the formula.

t_3 = 3t_2 - 2^2 = 3(7) - 4 = 17

Using the value of the second term allows us to find the third term.

3

Calculate the fourth term,

t_4

, by substituting

n = 3

and

t_3 = 17

into the formula.

t_4 = 3t_3 - 2^3 = 3(17) - 8 = 43

This completes the recursive process to find the target term.

Anahtar Kavram

Evaluating terms of a sequence defined by a recursive formula.

Tahmini Süre:1m 30s

Soru 2487Soru →

A laboratory chamber's initial temperature is $\frac{7}{2}$ degrees Celsius, and it decreases at a constant rate of $\frac{5}{4}$ degrees Celsius per hour. The temperature of the chamber must reach at most $-\frac{1}{2}$ degrees Celsius to complete an experiment. Which of the following inequalities represents the number of hours, $h$ , the experiment must run to reach this temperature?

h \leq \frac{16}{5}

h \geq \frac{8}{5}

h \geq \frac{16}{5}

h \leq -\frac{2}{9}

h \geq \frac{1}{2}

Cevabı ve açıklamayı göster

Cevap: $h \geq \frac{16}{5}$

Cevap

h \geq \frac{16}{5}

The correct answer is the inequality stating that

h

must be greater than or equal to sixteen-fifths. This is found by setting up the linear inequality

\frac{7}{2} - \frac{5}{4}h \leq -\frac{1}{2}

, subtracting

\frac{7}{2}

from both sides to get

-\frac{5}{4}h \leq -4

, and then multiplying by

-\frac{4}{5}

while reversing the inequality sign.

Adım Adım Çözüm

1

Set up the inequality representing the temperature constraint.

\frac{7}{2} - \frac{5}{4}h \leq -\frac{1}{2}

The initial temperature is

\frac{7}{2}

, the rate of decrease is

\frac{5}{4}

per hour

h

, and the final temperature must be at most

-\frac{1}{2}

.

2

Subtract

\frac{7}{2}

from both sides of the inequality.

-\frac{5}{4}h \leq -4

This isolates the variable term on the left side of the inequality. The subtraction is

-\frac{1}{2} - \frac{7}{2} = -\frac{8}{2} = -4

.

3

Multiply both sides by

-\frac{4}{5}

and reverse the inequality sign.

h \geq \frac{16}{5}

Multiplying or dividing by a negative number requires reversing the direction of the inequality sign. The calculation is

-4 \times \left(-\frac{4}{5}\right) = \frac{16}{5}

.

Anahtar Kavram

Solving linear inequalities involving negative coefficients and applying the inequality sign-flip rule.

Tahmini Süre:1m 30s

Soru 2488Soru →

In the standard $(x, y)$ coordinate plane, line $T$ is perpendicular to the line with the equation $y = 3x - 5$ . If line $T$ passes through the point $(6, 2)$ , what is the $y$ -intercept of line $T$ ?

-16

-10

4

20

0

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct answer is 4

The slope of the line given by the equation

y = 3x - 5

is

3

. Since line

T

is perpendicular to this line, its slope must be the negative reciprocal of

3

, which is

-\frac{1}{3}

. Using the point-slope form with the point

(6, 2)

, the equation of line

T

is

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

. To find the

y

-intercept of line

T

, we set

x = 0

and solve for

y

:

y - 2 = -\frac{1}{3}(0 - 6) \implies y - 2 = 2 \implies y = 4

.

Adım Adım Çözüm

1

Identify the slope of the given line from its equation.

The slope of the line

y = 3x - 5

is

3

.

The equation is in slope-intercept form (

y = mx + b

), where

m

is the slope.

2

Determine the slope of the perpendicular line

T

.

The slope of line

T

is

-\frac{1}{3}

.

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

3

Use the point-slope form to write the equation of line

T

with the point

(6, 2)

.

The equation of line

T

is

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

.

The point-slope form is

y - y_1 = m(x - x_1)

where

(x_1, y_1)

is a point on the line and

m

is the slope.

4

Find the

y

-intercept of line

T

by setting

x = 0

.

y = 4

.

The

y

-intercept is the value of

y

where the line crosses the

y

-axis (when

x = 0

).

Anahtar Kavram

Finding the equation and y-intercept of a perpendicular line using negative reciprocal slopes

Soru 2489Soru →

In the standard $(x, y)$ coordinate plane, the midpoint of the line segment with endpoints $A(1, 3)$ and $B\left(7, \frac{x}{2}\right)$ is $M$ . If the distance from the origin $(0, 0)$ to $M$ is $\sqrt{\frac{89}{4}}$ units, and $x$ is a positive number, what is the value of $x$ ?

2

4

8

14

16

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

The correct value of

x

is

4

.

The midpoint

M

of

A(1, 3)

and

B\left(7, \frac{x}{2}\right)

has coordinates

\left(\frac{1+7}{2}, \frac{3 + \frac{x}{2}}{2}\right) = \left(4, \frac{x+6}{4}\right)

. The distance from

(0, 0)

to

M

is

\sqrt{4^2 + \left(\frac{x+6}{4}\right)^2} = \sqrt{\frac{89}{4}}

. Squaring both sides yields

16 + \frac{(x+6)^2}{16} = \frac{89}{4}

. Multiplying through by 16 gives

256 + (x+6)^2 = 356

, which simplifies to

(x+6)^2 = 100

. Taking the square root of both sides gives

x+6 = 10

(since

x

is positive), which simplifies to

x = 4

.

Adım Adım Çözüm

1

Find the coordinates of the midpoint

M(x_M, y_M)

of the segment

AB

.

M = \left(4, \frac{x+6}{4}\right)

By the midpoint formula,

x_M = \frac{1 + 7}{2} = 4

and

y_M = \frac{3 + \frac{x}{2}}{2} = \frac{\frac{6+x}{2}}{2} = \frac{x+6}{4}

.

2

Set up the distance equation from the origin

(0, 0)

to

M\left(4, \frac{x+6}{4}\right)

using the distance formula.

\sqrt{4^2 + \left(\frac{x+6}{4}\right)^2} = \sqrt{\frac{89}{4}}

The distance between the origin

(0, 0)

and any point

(x, y)

is given by

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

.

3

Solve the distance equation for the positive variable

x

.

x = 4

Squaring both sides gives

16 + \frac{(x+6)^2}{16} = \frac{89}{4}

. Multiplying by 16 yields

256 + (x+6)^2 = 356

, which simplifies to

(x+6)^2 = 100

. Since

x

is positive,

x+6 = 10 \implies x = 4

.

Anahtar Kavram

Solving coordinate geometry problems by combining the midpoint formula and the distance formula.

Soru 2490Soru →

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

Cevabı ve açıklamayı göster

Cevap: 26

Cevap

The maximum possible value of

n

is 26.

For a convex

26

-gon, the sum of the interior angles is

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

. We can choose

26

distinct integer angles from the range

[91^\circ, 179^\circ]

that sum to exactly

4320^\circ

because the maximum possible sum of

26

distinct integers in this range is

4329^\circ

, which is greater than

4320^\circ

. For

n = 27

, the sum of the interior angles must be

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

, but the maximum possible sum of

27

distinct integers in the range is only

4482^\circ

, which is less than

4500^\circ

. Therefore,

26

is the maximum value of

n

.

Adım Adım Çözüm

1

Determine the set of possible angle measures.

The angles must be integers in the range

[91^\circ, 179^\circ]

.

Interior angles of a convex polygon must be less than

180^\circ

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

90^\circ

, giving the range

[91, 179]

.

2

Write the sum of the interior angles of a convex

n

-gon.

Sum

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

.

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

n

-gon.

3

Find the maximum possible sum of

n

distinct angles in the range

[91, 179]

.

Maximum Sum

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

.

The maximum sum is achieved by selecting the largest

n

integers from the set:

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

.

4

Set up the inequality and simplify.

n^2 + n - 720 \le 0

.

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

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

. Multiplying by 2 and simplifying yields the quadratic inequality.

5

Solve the quadratic inequality for the largest integer

n

.

n = 26

.

Evaluating the quadratic expression for consecutive integers: for

n = 26

,

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

; for

n = 27

,

27^2 + 27 - 720 = 36 > 0

. Thus, 26 is the maximum possible integer value.

Anahtar Kavram

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

Soru 2491Soru →

If the diagonals of a convex quadrilateral divide the quadrilateral into four triangles of equal perimeter, then the quadrilateral must be a square. Is this statement true or false?

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

The statement is false because any non-square rhombus satisfies the condition of being divided into four triangles of equal perimeter, yet it is not a square.

The correct answer is false because the equal-perimeter condition only requires the quadrilateral to be a rhombus (having four equal sides), but does not require the interior angles to be 90 degrees. Any non-square rhombus is a valid counterexample.

Adım Adım Çözüm

1

Analyze the given condition of equal perimeters for the four triangles formed by the diagonals of a convex quadrilateral.

The diagonals of any rhombus are perpendicular and bisect each other, dividing the rhombus into four congruent right triangles.

Congruent triangles have identical side lengths, meaning their perimeters are equal. Thus, every rhombus satisfies this property.

2

Determine if all quadrilaterals satisfying this property must be squares.

A square is a regular quadrilateral, meaning it must have both equal side lengths and interior angles of 90 degrees.

To verify if the statement is true, we must test if a non-square rhombus can satisfy the condition.

3

Construct a counterexample using a specific non-square rhombus.

Consider a rhombus with side lengths of 5 units and diagonals of lengths 6 units and 8 units. The diagonals divide it into four right triangles with sides 3, 4, and 5 units. Each triangle has a perimeter of 12 units.

This rhombus has equal perimeters for all four triangles, but its interior angles are not 90 degrees, proving it is not a square.

Anahtar Kavram

The relationship between the diagonals and side properties of rhombuses and squares.

Soru 2492Soru →

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

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

The number of sides,

n

, of the convex polygon is 14.

The sum of the interior angles of a convex polygon with

n

sides is given by the formula

(n-2) \times 180^\circ

, and the sum of its exterior angles is always

360^\circ

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

6

times the sum of the exterior angles, yielding the equation

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

. Dividing both sides of the equation by

180

gives

n - 2 = 12

. Adding

2

to both sides results in

n = 14

.

Adım Adım Çözüm

1

State the sum of interior and exterior angles formulas.

Interior sum =

(n-2) \times 180^\circ

, Exterior sum =

360^\circ

.

To represent the geometric properties of the polygon algebraically.

2

Set up the equation based on the given ratio.

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

.

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

3

Solve the equation for

n

.

n = 14

.

Divide by 180 to get

n - 2 = 12

, then add 2 to both sides.

Anahtar Kavram

The sum of the interior angles of an

n

-sided convex polygon is

(n-2) \times 180^\circ

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

360^\circ

.

Soru 2493Soru →

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

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

The value of the constant

c

is

12

.

The correct answer is

12

. By representing the roots in the ratio of

3:8

as

3r

and

8r

, Vieta's formula for the sum of roots (

-\frac{b}{a}

) gives

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

. The actual roots are therefore

1.5

and

4

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

\frac{c}{a}

) gives

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

.

Adım Adım Çözüm

1

Represent the roots using the given ratio.

Let the two roots of the quadratic equation be

3r

and

8r

.

The ratio of the two solutions is specified as

3:8

.

2

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

r

.

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

.

For a quadratic equation

ax^2 + bx + c = 0

, the sum of the roots is given by

-\frac{b}{a}

.

3

Determine the numerical values of the two roots.

The roots are

3(0.5) = 1.5

and

8(0.5) = 4

.

Substitute the value of

r = 0.5

back into the expressions for the roots.

4

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

c

.

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

.

For a quadratic equation

ax^2 + bx + c = 0

, the product of the roots is given by

\frac{c}{a}

.

Anahtar Kavram

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

Alternatif Yöntem

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

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

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

3:8

, we set up the equation:

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

. Cross-multiplying gives

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

.

Tahmini Süre:1m 30s

Soru 2494Soru →

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

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of

b

is 2.

To solve

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

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

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

. Carefully distribute the negative sign to obtain

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

. Simplifying the left side gives

22 - 3x \ge 2x + 12

. Moving the variable terms to one side yields

10 \ge 5x

, which simplifies to

x \le 2

. Thus, the upper bound value

b

is 2.

Adım Adım Çözüm

1

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

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

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

2

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

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

Distributing the negative sign across

(3x - 2)

changes it to

-3x + 2

. Distributing 2 across

(x + 6)

yields

2x + 12

.

3

Combine like terms on the left side of the inequality.

22 - 3x \ge 2x + 12

Combining the constant terms

20

and

2

simplifies the expression to

22

.

4

Isolate the variable terms by adding

3x

and subtracting 12 from both sides.

10 \ge 5x

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

x

.

5

Divide both sides by 5.

2 \ge x

(or

x \le 2

)

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

Anahtar Kavram

Solving linear inequalities involving fractions and distributing negative coefficients.

Alternatif Yöntem

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

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

. This simplifies to

5.5 - 0.75x \ge 0.5x + 3

. Subtracting

0.5x

and

5.5

from both sides gives

-1.25x \ge -2.5

. Dividing by

-1.25

and reversing the inequality sign gives

x \le 2

.

Tahmini Süre:1m 30s

Soru 2495Soru →

An irregular convex hexagon has two interior angles measuring $90^\circ$ and $130^\circ$ , respectively. The remaining four interior angles have measures in the ratio $5:6:7:7$ . What is the degree measure of the largest interior angle in this hexagon?

100^\circ

120^\circ

130^\circ

140^\circ

150^\circ

Cevabı ve açıklamayı göster

Cevap: $140^\circ$

Cevap

The correct answer is

140^\circ

.

The sum of the interior angles of a hexagon is

720^\circ

. Subtracting the two given angles (

90^\circ

and

130^\circ

) leaves

500^\circ

for the remaining four angles. Since these angles are in the ratio

5:6:7:7

, we represent them as

5x

,

6x

,

7x

, and

7x

. Their sum is

25x = 500^\circ

, which gives

x = 20^\circ

. The largest of the remaining angles is

7x = 7(20^\circ) = 140^\circ

. Since

140^\circ

is greater than both

90^\circ

and

130^\circ

, it is the largest interior angle of the hexagon.

Adım Adım Çözüm

1

Calculate the sum of the interior angles of a hexagon.

Using the formula for the sum of the interior angles of a polygon with

n

sides,

(n - 2) \times 180^\circ

, for a hexagon (

n = 6

), the sum is

(6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ

.

To establish the total sum of all interior angles of the polygon.

2

Subtract the two known angle measures from the total sum.

720^\circ - 90^\circ - 130^\circ = 500^\circ

.

To find the sum of the remaining four interior angles.

3

Set up an algebraic equation to find the value of one ratio unit,

x

.

5x + 6x + 7x + 7x = 500^\circ \implies 25x = 500^\circ \implies x = 20^\circ

.

To determine the constant multiplier for the ratio of the remaining angles.

4

Calculate the measures of the remaining angles and determine the largest angle.

The remaining angles are

5(20^\circ) = 100^\circ

,

6(20^\circ) = 120^\circ

,

7(20^\circ) = 140^\circ

, and

7(20^\circ) = 140^\circ

. Comparing all six angles of the hexagon (

90^\circ, 100^\circ, 120^\circ, 130^\circ, 140^\circ, 140^\circ

), the largest angle is

140^\circ

.

To identify the maximum interior angle measure of the hexagon.

Anahtar Kavram

The sum of the interior angles of a convex polygon with

n

sides is

(n - 2) \times 180^\circ

. The individual angle measures in an irregular polygon can be determined using algebraic representations of their relationships or ratios.

Alternatif Yöntem

Once the value of the ratio unit

x = 20^\circ

is determined, you can quickly find the largest candidate angle by multiplying the largest ratio component (

7

) by

x

to get

7(20^\circ) = 140^\circ

, and then compare it to the given angles (

90^\circ

and

130^\circ

) to verify that it is indeed the largest.

Tahmini Süre:1m 30s

Soru 2496Soru →

The ratio of the measure of an interior angle of a regular polygon to the measure of its exterior angle is $3:1$ . How many sides does this polygon have?

4

6

8

10

12

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

The interior angle and exterior angle of a polygon at any vertex are supplementary, meaning they add up to

180^\circ

. Given the ratio of the interior angle to the exterior angle is

3:1

, we can express their measures as

3x

and

x

. Solving the equation

3x + x = 180^\circ

gives

4x = 180^\circ

, which means

x = 45^\circ

. Therefore, the measure of each exterior angle of the regular polygon is

45^\circ

. Since the sum of the exterior angles of any convex polygon is always

360^\circ

, the number of sides

n

is calculated by dividing

360^\circ

by the measure of one exterior angle:

n = \frac{360^\circ}{45^\circ} = 8

.

Adım Adım Çözüm

1

Set up an equation for the interior and exterior angles using the given ratio.

Let the measure of the exterior angle be

x

and the measure of the interior angle be

3x

.

The ratio of the interior angle to the exterior angle is

3:1

, so their measures can be represented as

3x

and

x

respectively.

2

Use the fact that the interior angle and exterior angle at any vertex of a polygon are supplementary (form a linear pair).

3x + x = 180^\circ \implies 4x = 180^\circ \implies x = 45^\circ

.

An interior angle and its adjacent exterior angle always lie on a straight line and sum to

180^\circ

.

3

Calculate the number of sides of the regular polygon using the measure of one exterior angle.

n = \frac{360^\circ}{45^\circ} = 8

.

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

360^\circ

. For a regular polygon with

n

sides, each exterior angle measures

\frac{360^\circ}{n}

.

Anahtar Kavram

The relationship between the interior and exterior angles of a regular polygon, and the formula relating the number of sides to the sum of the exterior angles.

Soru 2497Soru →

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

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The distance between the two intersection points is 8.

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

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

into the circle's equation

x^2 + y^2 = 25

, yielding the quadratic equation

5x^2 - 18x - 35 = 0

. Solving this gives

x = 5

and

x = -1.4

, with corresponding

y

-coordinates

y = 0

and

y = 4.8

. The distance between

(5, 0)

and

(-1.4, 4.8)

is

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

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

(0,0)

to the line

3x + 4y - 15 = 0

is

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

. Since the radius of the circle is

r = 5

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

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

. Thus, the total chord length is

2 \times 4 = 8

.

Adım Adım Çözüm

1

Express the linear equation in terms of one variable

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

This allows for substitution into the equation of the circle.

2

Substitute the expression into the circle's equation and simplify

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

To create a single quadratic equation in terms of

x

representing the intersection points.

3

Solve the quadratic equation for

x

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

or

x = -1.4

To find the

x

-coordinates of the intersection points.

4

Calculate the corresponding

y

-coordinates

For

x = 5

,

y = 0

, giving point

A(5, 0)

. For

x = -1.4

,

y = 4.8

, giving point

B(-1.4, 4.8)

.

To determine the exact coordinates of both intersection points.

5

Apply the distance formula to find the length of the segment

AB

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

To compute the final distance between the two intersection points.

Anahtar Kavram

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

Soru 2498Soru →

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

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

14

The correct answer is 14 because

\triangle ACM

is a right triangle with legs

AC = 8\sqrt{3}

and

CM = 2

. Applying the Pythagorean Theorem yields

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

, so

AM = \sqrt{196} = 14

.

Adım Adım Çözüm

1

Find the properties of the regular hexagon and the diagonal

AC

.

The interior angle at vertex

B

is

120^\circ

. Since

AB = BC = 8

, the triangle

\triangle ABC

is an isosceles triangle with angles

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

. Using the properties of

30^\circ

-

60^\circ

-

90^\circ

triangles, the diagonal length is

AC = 8\sqrt{3}

.

To find the length of the leg

AC

for the right triangle

\triangle ACM

.

2

Determine the angle

\angle ACD

to show

\triangle ACM

is a right triangle.

Since the interior angle

\angle BCD = 120^\circ

and

\angle BCA = 30^\circ

, the remaining angle is

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

. Thus,

\triangle ACM

is a right triangle with the right angle at vertex

C

.

To establish the right-angle relationship between the legs

AC

and

CM

.

3

Apply the Pythagorean Theorem to calculate the hypotenuse

AM

.

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

. Taking the square root gives

AM = 14

.

To find the final length of segment

AM

.

Anahtar Kavram

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

Soru 2499Soru →

A homeowner has a square patio with a side length of $3x - 2$ feet. She decides to cover a portion of the patio with a rectangular outdoor rug that has dimensions $2x - 1$ feet by $x + 4$ feet. Which of the following expressions represents the area, in square feet, of the patio that remains uncovered by the rug?

7x^2 - 19x

7x^2 - 7x + 8

7x^2 - 19x + 8

9x^2 - 21x + 8

7x^2 - 5x

Cevabı ve açıklamayı göster

Cevap: $7x^2 - 19x + 8$

Cevap

7x^2 - 19x + 8

The expression representing the uncovered area is obtained by subtracting the area of the rug from the area of the patio. First, the area of the square patio is calculated as

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

. Second, the area of the rug is calculated as

(2x - 1)(x + 4) = 2x^2 + 7x - 4

. Subtracting the rug's area from the patio's area requires distributing the negative sign:

(9x^2 - 12x + 4) - (2x^2 + 7x - 4) = 9x^2 - 12x + 4 - 2x^2 - 7x + 4

. Combining like terms yields

7x^2 - 19x + 8

.

Adım Adım Çözüm

1

Calculate the area of the square patio by squaring its side length.

A_{\text{patio}} = (3x - 2)^2 = 9x^2 - 12x + 4

The area of a square is equal to the square of its side length:

\text{Area} = s^2

.

2

Calculate the area of the rectangular rug by multiplying its length and width.

A_{\text{rug}} = (2x - 1)(x + 4) = 2x^2 + 8x - x - 4 = 2x^2 + 7x - 4

The area of a rectangle is equal to the product of its length and width:

\text{Area} = l \times w

.

3

Subtract the area of the rug from the area of the patio, distributing the negative sign to all terms of the rug's area.

A_{\text{uncovered}} = (9x^2 - 12x + 4) - (2x^2 + 7x - 4) = 9x^2 - 12x + 4 - 2x^2 - 7x + 4 = 7x^2 - 19x + 8

Subtracting a polynomial requires distributing the negative sign to every term inside the parentheses and then combining like terms.

Anahtar Kavram

Subtracting one polynomial from another requires distributing the negative sign to every term of the subtracted polynomial before combining like terms.

Tahmini Süre:1m 30s

Soru 2500Soru →

A triangle has side lengths of $x$ , $2x$ , and $15$ , where $x$ is an integer. What is the total number of possible values for $x$ ?

7

8

9

10

11

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

9

Applying the Triangle Inequality Theorem yields three inequalities:

x + 2x > 15

(which simplifies to

x > 5

),

x + 15 > 2x

(which simplifies to

x < 15

), and

2x + 15 > x

(which is always true since

x

is positive). Combining these results in

5 < x < 15

. The integers in this range are

\{6, 7, 8, 9, 10, 11, 12, 13, 14\}

. Counting them gives 9 possible values.

Adım Adım Çözüm

1

Set up the three inequalities using the Triangle Inequality Theorem for the sides

x

,

2x

, and

15

.

The inequalities are

x + 2x > 15

,

x + 15 > 2x

, and

2x + 15 > x

.

The sum of the lengths of any two sides of a triangle must be strictly greater than the length of the third side.

2

Solve each inequality for

x

.

3x > 15 \implies x > 5

, and

15 > x \implies x < 15

. The third inequality

x > -15

is always true for positive side lengths.

To find the range of valid values for

x

.

3

Combine the inequalities and count the possible integer values for

x

.

The combined range is

5 < x < 15

. The integer values are

\{6, 7, 8, 9, 10, 11, 12, 13, 14\}

, which gives

14 - 6 + 1 = 9

values.

Since

x

is specified as an integer, we must identify and count all integers strictly between 5 and 15.

Anahtar Kavram

Triangle Inequality Theorem

Alternatif Yöntem

Instead of algebraic manipulation, we can test values of

x

directly. For

x=5

, the sides are

5, 10, 15

, but

5+10=15

, which does not form a triangle. For

x=15

, the sides are

15, 30, 15

, but

15+15=30

, which also does not form a triangle. Testing integer values between

5

and

15

confirms they all satisfy the triangle inequality, resulting in 9 valid integers.

Tahmini Süre:1m 0s

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