Tüm alıştırma soruları

Soru 2761Soru →

In tropical forests, lianas (woody vines) compete intensely with trees for resources. To investigate this interaction, ecologist Georgette Bonney monitored forest plots in Panama where lianas were either left intact or manually removed. Over five years, trees in the liana-free plots grew significantly faster and had lower mortality rates than trees in control plots. Because tree biomass accumulation is a principal mechanism of carbon sequestration in these forests, Bonney’s team concluded that an increase in liana abundance in tropical forests will likely __________

Which choice most logically completes the text?

indicate that trees in tropical forests grow faster and survive longer when lianas are present in their plots.

cause trees to adapt by shifting their resource allocation to outcompete lianas for sunlight.

reduce the rate at which these forests absorb and store carbon dioxide from the atmosphere.

demonstrate that resource competition invariably limits the growth of all plant species throughout Panamanian ecosystems.

Cevabı ve açıklamayı göster

Cevap: reduce the rate at which these forests absorb and store carbon dioxide from the atmosphere.

Cevap

The option stating that an increase in liana abundance will likely reduce the rate at which these forests absorb and store carbon dioxide from the atmosphere.

The correct answer is supported by the passage's premises. The study shows that trees in plots without lianas grow faster and have lower mortality rates than those with lianas, meaning lianas suppress tree growth and survival. Since tree growth (biomass accumulation) is the principal mechanism of carbon sequestration in these forests, an increase in lianas will suppress tree growth, which in turn reduces the forests' rate of carbon sequestration.

Adım Adım Çözüm

1

Identify the premises regarding lianas and tree growth in the passage.

Trees in plots without lianas grow faster and have lower mortality rates than trees in plots with lianas, indicating that lianas hinder tree growth and survival.

To establish the relationship between lianas and tree performance.

2

Connect tree performance to carbon sequestration.

Tree biomass accumulation (growth) is the main mechanism for carbon sequestration (absorption and storage of carbon dioxide) in these forests.

To relate the biological effect of lianas to the ecological consequence of carbon storage.

3

Deduce the effect of increased liana abundance.

An increase in lianas will lead to slower tree growth and higher tree mortality, which will decrease biomass accumulation and thus reduce the rate of carbon sequestration.

To find the most logical conclusion that completes the passage.

Anahtar Kavram

Logical Inferences

Soru 2762Soru →

In right triangle $PQR$ , the measure of angle $Q$ is $90^\circ$ . If $\sin(P) = \frac{5}{13}$ and the length of side $QR$ is $15$ , what is the length of side $PQ$ ?

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

The length of side

PQ

is

36

.

By definition,

\sin(P) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{QR}{PR}

. Given that

\sin(P) = \frac{5}{13}

and

QR = 15

, we set up the equation

\frac{5}{13} = \frac{15}{PR}

and solve for the hypotenuse

PR

, giving

PR = 39

. Using the Pythagorean theorem,

PQ^2 + QR^2 = PR^2

, we substitute the known values:

PQ^2 + 15^2 = 39^2 \implies PQ^2 + 225 = 1521 \implies PQ^2 = 1296

. Taking the square root of both sides gives

PQ = 36

. Alternatively, recognizing that the sides of the triangle form a

5

-

12

-

13

Pythagorean triple scaled by a factor of

3

(since

QR = 5 \times 3 = 15

and

PR = 13 \times 3 = 39

), the remaining leg

PQ

must be

12 \times 3 = 36

.

Adım Adım Çözüm

1

Set up the sine ratio for angle

P

to find the length of the hypotenuse

PR

.

PR = 39

Since

\sin(P)

is the ratio of the opposite side (

QR

) to the hypotenuse (

PR

), we can solve the equation

\frac{5}{13} = \frac{15}{PR}

to find that

PR = 39

.

2

Apply the Pythagorean theorem to solve for the length of side

PQ

.

PQ = 36

In right triangle

PQR

, the relationship between the sides is

PQ^2 + QR^2 = PR^2

. Substituting

QR = 15

and

PR = 39

gives

PQ^2 + 15^2 = 39^2

, which simplifies to

PQ^2 = 1296

, so

PQ = 36

.

Anahtar Kavram

Using trigonometric ratios to find side lengths of right triangles followed by the Pythagorean theorem.

Alternatif Yöntem

Recognize that the triangle's sides must be a multiple of the common

5

-

12

-

13

Pythagorean triple. Since the opposite side is

15

(

5 \times 3

) and the hypotenuse is

39

(

13 \times 3

), the scaling factor is

3

, meaning the adjacent side

PQ

is

12 \times 3 = 36

.

Tahmini Süre:1m 30s

Soru 2763Soru →

While preparing a presentation on the coelacanth, a student took the following notes:
* The coelacanth is a rare order of fish closely related to lungfish and tetrapods.
* It was widely believed to have gone extinct approximately 66 million years ago.
* In December 1938, a live specimen was discovered off the coast of South Africa.
* The discovery was made by museum curator Marjorie Courtenay-Latimer.
* Ichthyologist J.L.B. Smith later identified the specimen as *Latimeria chalumnae*.

The student wants to emphasize the date and the location of the first modern coelacanth discovery. Which choice most effectively uses information from the given notes to accomplish this goal?

Although the coelacanth was believed to have been extinct for 66 million years, a live specimen was discovered by Marjorie Courtenay-Latimer.

In December 1938, a live coelacanth specimen was discovered off the coast of South Africa.

A live coelacanth specimen was found off the coast of South Africa by museum curator Marjorie Courtenay-Latimer.

In December 1938, a live coelacanth specimen was discovered off the coast of South America.

Cevabı ve açıklamayı göster

Cevap: In December 1938, a live coelacanth specimen was discovered off the coast of South Africa.

Cevap

The correct answer is the option stating that in December 1938, a live coelacanth specimen was discovered off the coast of South Africa.

The correct option effectively accomplishes the goal by explicitly presenting the date (December 1938) and the location (off the coast of South Africa) of the first modern coelacanth discovery.

Adım Adım Çözüm

1

Identify the key details requested by the rhetorical goal.

The goal requires emphasizing the date (December 1938) and the location (off the coast of South Africa) of the first modern coelacanth discovery.

To satisfy the rhetorical synthesis prompt, the correct sentence must explicitly contain these target details.

2

Evaluate each choice to see which one correctly includes both specified details.

The choice stating 'In December 1938, a live coelacanth specimen was discovered off the coast of South Africa' is the only one that includes both details accurately.

This step ensures the selected choice fully addresses the prompt's requirements without omitting crucial information or misrepresenting facts.

Anahtar Kavram

Rhetorical Synthesis: Emphasizing Specific Details

Soru 2764Soru →

In a circle with center $O$ , the length of minor arc $AB$ is $3\pi$ and the area of sector $AOB$ is $18\pi$ . What is the circumference of the circle?

12\pi

18\pi

24\pi

36\pi

Cevabı ve açıklamayı göster

Cevap: $24\pi$

Cevap

The circumference of the circle is

24\pi

.

The correct answer is the option containing

24\pi

. By using the relationship

A = \frac{1}{2}rs

, where

A

is the sector area,

r

is the radius, and

s

is the arc length, we substitute the given values to get

18\pi = \frac{1}{2}r(3\pi)

. Solving for the radius yields

r = 12

. Substituting this radius into the circumference formula

C = 2\pi r

gives

2\pi(12) = 24\pi

.

Adım Adım Çözüm

1

Write the formulas for arc length

s

and sector area

A

in terms of radius

r

and central angle

\theta

in radians.

s = r\theta = 3\pi

and

A = \frac{1}{2}r^2\theta = 18\pi

.

This sets up the system of equations using the given geometric properties.

2

Express the sector area formula in terms of arc length by substituting

s = r\theta

into

A = \frac{1}{2}r(r\theta)

.

A = \frac{1}{2}rs

, which becomes

18\pi = \frac{1}{2}r(3\pi)

.

This simplifies the relationship to a single equation with one variable,

r

.

3

Solve the equation

18\pi = 1.5\pi r

for the radius

r

.

r = 12

.

Finding the radius is necessary to calculate the circumference of the circle.

4

Substitute

r = 12

into the circumference formula

C = 2\pi r

.

C = 2\pi(12) = 24\pi

.

This provides the final circumference value requested by the question.

Anahtar Kavram

Relationship between arc length, sector area, and circumference in circle geometry

Soru 2765Soru →

Line segments $AC$ and $BD$ intersect at point $E$ such that segment $AB$ is parallel to segment $CD$ . If the length of $AE$ is $5$ , the length of $CE$ is $10$ , the length of $BE$ is $x - 2$ , and the length of $DE$ is $x + 4$ , what is the value of $x$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

8

Since segment

AB

is parallel to segment

CD

, the alternate interior angles

\angle EAB

and

\angle ECD

are congruent, and vertical angles

\angle AEB

and

\angle CED

are congruent. By the Angle-Angle (AA) similarity theorem, triangle

ABE

is similar to triangle

CDE

. The ratio of the lengths of corresponding sides is equal, so

\frac{AE}{CE} = \frac{BE}{DE}

. Substituting the given lengths gives

\frac{5}{10} = \frac{x - 2}{x + 4}

. Simplifying the left side to

\frac{1}{2}

and cross-multiplying gives

x + 4 = 2(x - 2)

, which expands to

x + 4 = 2x - 4

. Solving for

x

yields

x = 8

.

Adım Adım Çözüm

1

Establish the similarity of triangles

ABE

and

CDE

.

\triangle ABE \sim \triangle CDE

Since segment

AB

is parallel to segment

CD

, alternate interior angles

\angle EAB

and

\angle ECD

are congruent, and vertical angles

\angle AEB

and

\angle CED

are congruent. Thus, the triangles are similar by AA similarity.

2

Set up a proportion using the ratio of corresponding sides.

\frac{AE}{CE} = \frac{BE}{DE}

In similar triangles, the ratio of corresponding side lengths is constant.

3

Substitute the given algebraic expressions and segment lengths into the proportion.

\frac{5}{10} = \frac{x - 2}{x + 4}

The given values are

AE = 5

,

CE = 10

,

BE = x - 2

, and

DE = x + 4

.

4

Simplify the fraction and solve the linear equation for

x

.

x = 8

Simplifying

\frac{5}{10}

yields

\frac{1}{2}

. Cross-multiplying gives

1 \cdot (x + 4) = 2 \cdot (x - 2)

, which simplifies to

x + 4 = 2x - 4

. Subtracting

x

from both sides and adding

4

to both sides gives

x = 8

.

Anahtar Kavram

Triangle similarity criteria (specifically AA similarity) and using proportions of corresponding sides in similar triangles to solve for unknown variables.

Soru 2766Soru →

In triangle $ABC$ , the side lengths are $AB = AC = 5$ and $BC = 6$ . Point $D$ is the midpoint of side $BC$ , and point $E$ lies on side $AB$ such that segment $DE$ is perpendicular to side $AB$ . What is the length of segment $DE$ ?

Cevabı ve açıklamayı göster

Cevap: 2.4

Cevap

The length of segment

DE

is

2.4

(or the equivalent fraction

12/5

).

The correct answer is

2.4

(or

12/5

). In the isosceles triangle

ABC

with

AB = AC = 5

, the median

AD

to the base

BC

is also an altitude. Since

D

is the midpoint of

BC

, we have

BD = 3

. Applying the Pythagorean theorem to right triangle

ABD

gives

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

. Since

DE

is perpendicular to

AB

, triangle

AED

is a right triangle that shares angle

A

with right triangle

ADB

. Therefore, triangle

AED

is similar to triangle

ADB

. The ratio of the opposite side to the hypotenuse in both triangles must be equal:

\frac{DE}{BD} = \frac{AD}{AB}

, which gives

\frac{DE}{3} = \frac{4}{5}

, or

DE = 2.4

.

Adım Adım Çözüm

1

Determine the properties of the altitude

AD

in the isosceles triangle

ABC

.

AD

is perpendicular to

BC

, and

BD = 3

.

In an isosceles triangle, the median to the base is also the altitude to the base. Since

D

is the midpoint of

BC

,

BD = \frac{BC}{2} = \frac{6}{2} = 3

, and

\angle ADB = 90^\circ

.

2

Calculate the length of segment

AD

using the Pythagorean theorem in right triangle

ABD

.

AD = 4

Applying the Pythagorean theorem to right triangle

ABD

gives

AD^2 + BD^2 = AB^2

. Substituting the known lengths yields

AD^2 + 3^2 = 5^2

, which simplifies to

AD^2 = 25 - 9 = 16

, so

AD = 4

.

3

Find the length of segment

DE

using triangle similarity.

DE = 2.4

Since segment

DE

is perpendicular to side

AB

,

\angle AED = 90^\circ

. The right triangles

AED

and

ADB

share the angle at

A

, so they are similar by AA similarity (

\triangle AED \sim \triangle ADB

). This allows us to set up the ratio of corresponding sides:

\frac{DE}{BD} = \frac{AD}{AB}

. Substituting the values gives

\frac{DE}{3} = \frac{4}{5}

, which results in

DE = \frac{12}{5} = 2.4

.

Anahtar Kavram

Properties of isosceles triangles, the Pythagorean theorem, and right triangle similarity theorems.

Alternatif Yöntem

Alternatively, the length of

DE

can be found using the area of right triangle

ABD

. The area of triangle

ABD

is

\frac{1}{2} \times BD \times AD = \frac{1}{2} \times 3 \times 4 = 6

. The area can also be expressed using the hypotenuse

AB

as the base and

DE

as the height:

\text{Area} = \frac{1}{2} \times AB \times DE = \frac{1}{2} \times 5 \times DE

. Setting these equal gives

\frac{5}{2} DE = 6

, which yields

DE = 2.4

.

Tahmini Süre:1m 30s

Soru 2767Soru →

In triangle $ABC$ , point $D$ lies on side $AB$ and point $E$ lies on side $AC$ such that segment $DE$ is parallel to segment $BC$ . The length of segment $AD$ is $2x + 1$ , the length of segment $DB$ is $x + 1$ , the length of segment $AE$ is $10$ , and the length of segment $EC$ is $6$ . What is the length of segment $AB$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The length of segment

AB

is 8.

By the Triangle Proportionality Theorem, since segment

DE

is parallel to segment

BC

, the segments on the transversal sides are proportional:

\frac{AD}{DB} = \frac{AE}{EC}

. Substituting the given expressions and values yields

\frac{2x + 1}{x + 1} = \frac{10}{6}

. Simplifying the fraction on the right side to

\frac{5}{3}

and cross-multiplying gives

3(2x + 1) = 5(x + 1)

. Solving this equation yields

x = 2

. The length of segment

AB

is the sum of

AD

and

DB

, which is

(2x + 1) + (x + 1) = 3x + 2

. Substituting

x = 2

gives

AB = 8

.

Adım Adım Çözüm

1

Set up the proportion using the Triangle Proportionality Theorem.

\frac{AD}{DB} = \frac{AE}{EC}

Since segment

DE

is parallel to segment

BC

, it divides the sides of triangle

ABC

proportionally.

2

Substitute the given values and simplify the constant ratio.

\frac{2x + 1}{x + 1} = \frac{5}{3}

The length of segment

AE

is 10 and

EC

is 6, so

\frac{AE}{EC} = \frac{10}{6} = \frac{5}{3}

.

3

Cross-multiply and solve the linear equation for

x

.

x = 2

Cross-multiplying gives

3(2x + 1) = 5(x + 1)

, which simplifies to

6x + 3 = 5x + 5

. Subtracting

5x

and 3 from both sides yields

x = 2

.

4

Calculate the total length of segment

AB

.

AB = 8

The total length

AB

is the sum of

AD

and

DB

. Thus,

AB = (2x + 1) + (x + 1) = 3x + 2

. Substituting

x = 2

gives

3(2) + 2 = 8

.

Anahtar Kavram

Triangle Proportionality Theorem and Similar Triangles

Alternatif Yöntem

Instead of using the Triangle Proportionality Theorem directly, we can use the similarity of triangles

ADE

and

ABC

. Since

DE \parallel BC

, we have

\triangle ADE \sim \triangle ABC

by AA similarity. This gives the ratio of corresponding side lengths:

\frac{AD}{AB} = \frac{AE}{AC}

. Substituting the expressions yields

\frac{2x + 1}{3x + 2} = \frac{10}{16} = \frac{5}{8}

. Cross-multiplying gives

8(2x + 1) = 5(3x + 2) \implies 16x + 8 = 15x + 10 \implies x = 2

. Then,

AB = 3x + 2 = 3(2) + 2 = 8

.

Tahmini Süre:1m 30s

Soru 2768Soru →

In the $xy$ -plane, the circle defined by the equation $x^2 + y^2 - 10x + 12y = k$ has a radius of $9$ , where $k$ is a constant. What is the value of $k$ ?

-52

70

20

142

Cevabı ve açıklamayı göster

Cevap: 20

Cevap

20

To find the value of

k

, we first rewrite the given equation in the standard form of a circle's equation,

(x - h)^2 + (y - k_{center})^2 = r^2

. Starting with

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

, we complete the square for the

x

-terms by adding

(\frac{-10}{2})^2 = 25

to both sides, and for the

y

-terms by adding

(\frac{12}{2})^2 = 36

to both sides. This yields

(x^2 - 10x + 25) + (y^2 + 12y + 36) = k + 25 + 36

, which simplifies to

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

. In this standard form, the right-hand side represents the square of the radius, so

r^2 = k + 61

. Given that the radius is

9

, we have

r^2 = 9^2 = 81

. Setting

k + 61 = 81

and subtracting

61

from both sides gives

k = 20

.

Adım Adım Çözüm

1

Write the given equation of the circle.

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

To establish the starting equation before completing the square.

2

Complete the square for the

x

and

y

terms by adding

(\frac{-10}{2})^2 = 25

and

(\frac{12}{2})^2 = 36

to both sides.

(x^2 - 10x + 25) + (y^2 + 12y + 36) = k + 25 + 36

To express the quadratic expressions as perfect squares.

3

Rewrite the equation in standard form.

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

To match the standard equation of a circle,

(x-h)^2 + (y-k_{center})^2 = r^2

, where the right-hand side represents the square of the radius.

4

Equate the constant term on the right-hand side to

r^2

using the given radius

r = 9

, and solve for

k

.

k + 61 = 9^2 \implies k + 61 = 81 \implies k = 20

To calculate the value of the constant

k

that satisfies the radius requirement.

Anahtar Kavram

Equations of Circles in the Coordinate Plane

Soru 2769Soru →

In a circle with center $O$ and radius $5$ , points $A$ , $B$ , and $C$ lie on the circle. If the measure of the inscribed angle $\angle ABC$ is $\frac{2\pi}{5}$ radians, what is the length of minor arc $AC$ ?

\pi

2\pi

4\pi

8\pi

Cevabı ve açıklamayı göster

Cevap: $4\pi$

Cevap

4\pi

According to the Inscribed Angle Theorem, the measure of a central angle is twice the measure of an inscribed angle that subtends the same arc. Given that the inscribed angle

\angle ABC

has a measure of

\frac{2\pi}{5}

radians, the corresponding central angle

\angle AOC

has a measure of

2 \times \frac{2\pi}{5} = \frac{4\pi}{5}

radians. Using the formula for arc length,

s = r\theta

, where

r = 5

is the radius and

\theta = \frac{4\pi}{5}

is the central angle in radians, the length of minor arc

AC

is

5 \times \frac{4\pi}{5} = 4\pi

.

Adım Adım Çözüm

1

Find the measure of the central angle

\angle AOC

that subtends the same minor arc

AC

as the inscribed angle

\angle ABC

.

The measure of central angle

\angle AOC

is

2 \times \frac{2\pi}{5} = \frac{4\pi}{5}

radians.

By the Inscribed Angle Theorem, the measure of a central angle subtending an arc is twice the measure of any inscribed angle subtending the same arc.

2

Calculate the length of minor arc

AC

using the formula

s = r\theta

.

The arc length is

s = 5 \times \frac{4\pi}{5} = 4\pi

.

The formula for the arc length of a circle is

s = r\theta

, where

r

is the radius and

\theta

is the central angle measure in radians.

Anahtar Kavram

Inscribed Angle Theorem and Arc Length in Radians

Soru 2770Soru →

Points $A$ , $B$ , and $C$ lie on a circle with center $O$ . The length of the minor arc $AC$ is $\frac{4}{9}$ of the circumference of the circle. What is the measure, in degrees, of the inscribed angle $\angle ABC$ ?

Cevabı ve açıklamayı göster

Cevap: 80

Cevap

The measure of the inscribed angle is 80 degrees.

The correct answer is 80. The minor arc

AC

constitutes

\frac{4}{9}

of the circle's circumference, which corresponds to an arc measure of

\frac{4}{9} \times 360^\circ = 160^\circ

. By the Inscribed Angle Theorem, the measure of the inscribed angle

\angle ABC

is half the measure of the intercepted arc, which is

\frac{1}{2} \times 160^\circ = 80^\circ

.

Adım Adım Çözüm

1

Determine the degree measure of the minor arc

AC

.

The measure of minor arc

AC

is

160^\circ

.

Since a full circle has a circumference corresponding to

360^\circ

, minor arc

AC

has a degree measure of

\frac{4}{9} \times 360^\circ = 160^\circ

.

2

Calculate the measure of the inscribed angle

\angle ABC

.

The measure of

\angle ABC

is

80^\circ

.

According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of the arc it intercepts. Thus, the measure of

\angle ABC

is

\frac{1}{2} \times 160^\circ = 80^\circ

.

Anahtar Kavram

Inscribed Angle Theorem and Arc Measure

Soru 2771Soru →

In the figure below, $AD$ is the angle bisector of $\angle BAC$ in triangle $ABC$ . The length of segment $AB$ is $2x$ , the length of segment $AC$ is $3x-3$ , the length of segment $BD$ is $6$ , and the length of segment $CD$ is $8$ . What is the value of $x$ ?

2

3

6

9

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The value of

x

is 9.

According to the Angle Bisector Theorem, the bisector of an angle in a triangle divides the opposite side into segments that are proportional to the adjacent sides. This gives the proportion

\frac{AB}{AC} = \frac{BD}{CD}

. Substituting the given values, we get

\frac{2x}{3x-3} = \frac{6}{8}

. Simplifying the right side to

\frac{3}{4}

and cross-multiplying yields

2x(4) = 3(3x-3)

, which simplifies to

8x = 9x - 9

. Solving for

x

gives

x = 9

.

Adım Adım Çözüm

1

Apply the Angle Bisector Theorem, which states that an angle bisector in a triangle divides the opposite side into two segments that are proportional to the other two sides.

\frac{AB}{AC} = \frac{BD}{CD}

Because

AD

is the angle bisector of

\angle BAC

, it splits

BC

at

D

proportionally to the adjacent sides

AB

and

AC

.

2

Substitute the given side lengths into the proportion.

\frac{2x}{3x-3} = \frac{6}{8}

To set up the algebraic equation for

x

using the given lengths

AB = 2x

,

AC = 3x-3

,

BD = 6

, and

CD = 8

.

3

Simplify the ratio on the right side of the equation and cross-multiply to solve for

x

.

8x = 9x - 9

, which simplifies to

x = 9

.

First simplify

\frac{6}{8}

to

\frac{3}{4}

, then cross-multiply:

2x(4) = 3(3x-3)

, which gives

8x = 9x - 9

. Subtracting

8x

and adding

9

yields

x = 9

.

Anahtar Kavram

Angle Bisector Theorem in Triangles

Soru 2772Soru →

A sector of a circle with center $O$ has a central angle of $150^\circ$ and an area of $15\pi$ . What is the length of the minor arc that bounds this sector?

5\pi

\frac{5\pi}{2}

12\pi

30\pi

Cevabı ve açıklamayı göster

Cevap: $5\pi$

Cevap

5\pi

To find the arc length, we first determine the radius of the circle using the sector area. The area of a sector with a central angle of

150^\circ

is

\frac{150}{360} = \frac{5}{12}

of the total circle area. Setting up the equation

15\pi = \frac{5}{12}\pi r^2

allows us to solve for

r^2 = 36

, which gives a radius of

r = 6

. Using the radius, we find the length of the minor arc by taking the same fraction of the total circumference:

\frac{5}{12} \times 2\pi(6) = 5\pi

. This matches the correct option.

Adım Adım Çözüm

1

Set up the equation for the area of the sector to solve for the radius

r

.

15\pi = \frac{150}{360} \pi r^2

The area of a sector is given by the formula

A = \frac{\theta}{360} \pi r^2

, where

\theta

is the central angle in degrees.

2

Simplify the fraction and solve for

r^2

and

r

.

15\pi = \frac{5}{12} \pi r^2 \implies 15 = \frac{5}{12} r^2 \implies r^2 = 36 \implies r = 6

Dividing both sides by

\pi

and multiplying by

\frac{12}{5}

isolates

r^2

, and taking the square root gives the radius

r

.

3

Calculate the length of the minor arc using the radius and central angle.

Arc Length

= \frac{150}{360} \times 2\pi(6) = \frac{5}{12} \times 12\pi = 5\pi

The arc length formula is

L = \frac{\theta}{360} \times 2\pi r

, representing the fraction of the total circumference.

Anahtar Kavram

The relationship between a circle's sector area, central angle, radius, and arc length.

Alternatif Yöntem

Instead of solving for the radius first, note that the ratio of the sector area to the total area is equal to the ratio of the arc length to the total circumference. Since Sector Area

= \frac{1}{2} r L

(where

L

is the arc length and

r

is the radius), we have

15\pi = \frac{1}{2} r L

. Since we also know that the area

15\pi = \frac{5}{12} \pi r^2 \implies r = 6

, substituting this directly into the area-arc relation gives

15\pi = \frac{1}{2} (6) L \implies 15\pi = 3L \implies L = 5\pi

.

Tahmini Süre:1m 30s

Soru 2773Soru →

In the $xy$ -plane, a circle is defined by the equation $x^2 + y^2 + 6x - 8y = 0$ . Which of the following points lies on the circle?

(2, 4)

(-3, 4)

(3, -4)

(0, 5)

Cevabı ve açıklamayı göster

Cevap: $(2, 4)$

Cevap

The point

(2, 4)

lies on the circle.

Substituting the coordinates of the point

(2, 4)

into the circle's equation

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

yields

2^2 + 4^2 + 6(2) - 8(4) = 4 + 16 + 12 - 32 = 0

, which is a true statement. Therefore, this point lies on the circle.

Adım Adım Çözüm

1

Group the

x

and

y

terms and complete the square for both variables.

The expression

(x^2 + 6x) + (y^2 - 8y) = 0

becomes

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

, which simplifies to

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

.

Converting the general form equation of a circle into the standard form

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

helps identify its center and radius.

2

Determine the center and the radius from the standard form equation.

The center is

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

and the radius is

r = \sqrt{25} = 5

.

Understanding the circle's parameters allows us to analyze which points lie on its perimeter.

3

Substitute the given options into the standard form equation to verify which point lies on the circle.

For

(2, 4)

, we get

(2 + 3)^2 + (4 - 4)^2 = 5^2 + 0^2 = 25

, which is true. For the other points, the equations are not satisfied.

A point lies on a circle if its distance to the center is exactly equal to the radius, meaning its coordinates satisfy the circle's equation.

Anahtar Kavram

Equations of Circles in the Coordinate Plane

Tahmini Süre:1m 30s

Soru 2774Soru →

In the $xy$ -plane, the graph of the equation $x^2 + y^2 - 10x - 24y + 69 = 0$ is a circle. What is the distance between the center of this circle and the origin?

Cevabı ve açıklamayı göster

Cevap: 13

Cevap

The distance between the center of the circle and the origin is 13.

Completing the square on the given equation yields standard form

(x - 5)^2 + (y - 12)^2 = 100

, identifying the center of the circle as

(5, 12)

. The distance from

(5, 12)

to

(0,0)

is calculated using the distance formula:

\sqrt{5^2 + 12^2} = \sqrt{169} = 13

.

Adım Adım Çözüm

1

Group the

x

and

y

terms and complete the square for each variable.

(x - 5)^2 + (y - 12)^2 = 100

By rewriting

x^2 - 10x

as

(x - 5)^2 - 25

and

y^2 - 24y

as

(y - 12)^2 - 144

, the equation becomes

(x - 5)^2 - 25 + (y - 12)^2 - 144 + 69 = 0

. Combining the constant terms gives

(x - 5)^2 + (y - 12)^2 - 100 = 0

, which simplifies to standard form.

2

Identify the center of the circle from the standard form

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

.

The center is

(5, 12)

.

Comparing

(x - 5)^2 + (y - 12)^2 = 100

to the standard form shows that

h = 5

and

k = 12

.

3

Use the distance formula to calculate the distance between the center

(5, 12)

and the origin

(0, 0)

.

13

The distance formula is

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

. Substituting the points gives

d = \sqrt{(5 - 0)^2 + (12 - 0)^2} = \sqrt{25 + 144} = \sqrt{169} = 13

.

Anahtar Kavram

Equations of Circles in the Coordinate Plane

Soru 2775Soru →

In right triangle $XYZ$ , the measure of angle $Y$ is $90^\circ$ . If the length of side $XY$ is $12$ and the area of the triangle is $30$ , what is the value of $\cos(X)$ ?

\frac{12}{13}

\frac{5}{13}

\frac{5}{12}

\frac{24}{25}

Cevabı ve açıklamayı göster

Cevap: $\frac{12}{13}$

Cevap

\frac{12}{13}

The correct answer is the fraction expressing twelve thirteenths. The area of thirty and the leg length of twelve imply that the other leg has a length of five. Applying the Pythagorean theorem to legs of twelve and five yields a hypotenuse of thirteen. Because cosine is defined as the ratio of the adjacent leg to the hypotenuse, the cosine of angle

X

is twelve over thirteen.

Adım Adım Çözüm

1

Find the length of leg

YZ

using the area formula.

YZ = 5

The area of a right triangle is

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

. Here,

\text{Area} = 30

and base

XY = 12

, so

30 = \frac{1}{2} \times 12 \times YZ \implies 30 = 6 \times YZ \implies YZ = 5

.

2

Calculate the length of the hypotenuse

XZ

using the Pythagorean theorem.

XZ = 13

In right triangle

XYZ

, the Pythagorean theorem states that

XZ^2 = XY^2 + YZ^2

. Substituting the known leg lengths,

XZ^2 = 12^2 + 5^2 = 144 + 25 = 169

. Taking the square root gives

XZ = 13

.

3

Determine the value of

\cos(X)

using the trigonometric definition.

\cos(X) = \frac{12}{13}

By definition, the cosine of an acute angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. For angle

X

, the adjacent side is

XY

and the hypotenuse is

XZ

, so

\cos(X) = \frac{XY}{XZ} = \frac{12}{13}

.

Anahtar Kavram

Using the area formula, Pythagorean theorem, and trigonometric ratios to solve right triangles.

Soru 2776Soru →

In the $xy$ -plane, a circle is represented by the equation $x^2 + y^2 - 10x + 8y - 8 = 0$ . What is the diameter of this circle?

7

14

49

98

Cevabı ve açıklamayı göster

Cevap: 14

Cevap

14

To find the diameter of the circle, we first rewrite the equation in standard form,

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

, by completing the square. Grouping the terms gives

(x^2 - 10x) + (y^2 + 8y) = 8

. Adding

25

and

16

to both sides yields

(x - 5)^2 + (y + 4)^2 = 49

. Since

r^2 = 49

, the radius of the circle is

7

. The diameter is twice the radius, which is

14

.

Adım Adım Çözüm

1

Group the

x

-terms and

y

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

(x^2 - 10x) + (y^2 + 8y) = 8

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

2

Complete the square for both the

x

and

y

expressions by adding

( \frac{-10}{2} )^2 = 25

and

( \frac{8}{2} )^2 = 16

to both sides of the equation.

(x^2 - 10x + 25) + (y^2 + 8y + 16) = 8 + 25 + 16

Completing the square converts the equation into the standard form of a circle's equation.

3

Rewrite the left side as squared binomials and simplify the right side.

(x - 5)^2 + (y + 4)^2 = 49

This puts the equation in the standard form

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

.

4

Identify

r^2

from the standard form equation and calculate the radius

r

.

r^2 = 49 \implies r = 7

In the standard form equation of a circle, the constant on the right side is the square of the radius.

5

Calculate the diameter by doubling the radius.

Diameter

= 2r = 2(7) = 14

The diameter of a circle is twice its radius.

Anahtar Kavram

Equations of Circles in the Coordinate Plane

Soru 2777Soru →

A rectangular field has a diagonal path of length $40$ meters. The length of the field is $8$ meters greater than its width. What is the width, in meters, of the field?

Cevabı ve açıklamayı göster

Cevap: 24

Cevap

The correct answer is 24.

Letting

w

represent the width of the rectangular field in meters, the length is

w + 8

meters. The diagonal forms the hypotenuse of a right triangle with the width and the length as legs. By the Pythagorean theorem,

w^2 + (w+8)^2 = 40^2

. Expanding this yields

2w^2 + 16w + 64 = 1600

, which simplifies to

w^2 + 8w - 768 = 0

. Factoring this equation gives

(w-24)(w+32) = 0

. Since width must be positive, the width is 24.

Adım Adım Çözüm

1

Represent the dimensions of the rectangular field.

Let the width of the field be

w

meters, so the length is

w + 8

meters.

The problem states the length is 8 meters greater than the width.

2

Apply the Pythagorean theorem.

w^2 + (w + 8)^2 = 40^2

The diagonal of a rectangle forms a right triangle with the width and length as its legs.

3

Simplify the quadratic equation.

w^2 + 8w - 768 = 0

Expanding the equation gives

2w^2 + 16w - 1536 = 0

, which simplifies when divided by 2.

4

Solve for the variable

w

.

(w - 24)(w + 32) = 0

, so

w = 24

or

w = -32

.

Factoring the quadratic equation gives the possible values for the width.

5

Determine the valid physical width.

w = 24

Since the width of a field must be positive, we reject the negative solution.

Anahtar Kavram

Using the Pythagorean theorem to relate the sides of a right triangle in a geometric word problem.

Soru 2778Soru →

A drone is hovering directly above a point on the ground. A surveyor's rangefinder, positioned $1.6$ meters above the ground, measures the angle of elevation to the drone to be $60^\circ$ . The rangefinder is located at a horizontal distance of $12\sqrt{3}$ meters from the point directly beneath the drone. What is the height, in meters, of the drone above the ground?

13.6

37.6

36.0

25.6

Cevabı ve açıklamayı göster

Cevap: $37.6$

Cevap

The height of the drone above the ground is

37.6

meters.

To find the height of the drone, we first represent the situation with a right triangle. The horizontal distance from the point directly below the drone to the rangefinder is

12\sqrt{3}

meters. The angle of elevation is

60^\circ

. In this right triangle, the side opposite the

60^\circ

angle represents the vertical height of the drone above the level of the rangefinder, which we can call

h

. Using the tangent ratio, we have

\tan(60^\circ) = \frac{h}{12\sqrt{3}}

. Since

\tan(60^\circ) = \sqrt{3}

, we find

h = 12\sqrt{3} \times \sqrt{3} = 36

meters. Finally, we add the height of the rangefinder above the ground to find the total height of the drone:

36 + 1.6 = 37.6

meters.

Adım Adım Çözüm

1

Identify the right triangle components from the given scenario.

A right triangle is formed where the horizontal leg (adjacent to the

60^\circ

angle) has a length of

12\sqrt{3}

meters, and the vertical leg (opposite the

60^\circ

angle) represents the height

h

of the drone above the level of the rangefinder.

This sets up the geometric model to use trigonometric ratios.

2

Use the tangent trigonometric ratio to find the vertical height

h

above the rangefinder level.

\tan(60^\circ) = \frac{h}{12\sqrt{3}} \Rightarrow \sqrt{3} = \frac{h}{12\sqrt{3}} \Rightarrow h = 12\sqrt{3} \times \sqrt{3} = 36

meters.

The tangent function relates the opposite side to the adjacent side in a right triangle.

3

Calculate the total height of the drone above the ground by adding the height of the rangefinder.

\text{Total Height} = 36 + 1.6 = 37.6

meters.

The rangefinder is positioned

1.6

meters above the ground, so the drone's height relative to the ground is the sum of its height above the rangefinder and the rangefinder's height.

Anahtar Kavram

Solving right triangles using trigonometric ratios (specifically tangent) or special 30-60-90 right triangle properties, and accounting for the height of the observer.

Soru 2779Soru →

In the figure below, triangle $ABC$ is a right triangle with the right angle at $B$ . Point $D$ lies on side $AC$ such that segment $BD$ is perpendicular to side $AC$ . If $\cos(A) = \frac{4}{5}$ and the length of segment $AD$ is $16$ , what is the length of segment $CD$ ?

8

9

12

15

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The length of segment CD is 9.

The correct answer is the option indicating that the length of segment CD is 9. By applying the definition of cosine in right triangle ADB, the hypotenuse AB is found to be 20. The Pythagorean theorem then yields the length of the altitude BD as 12. Because triangle ADB is similar to triangle BDC, the ratio of the shorter leg to the longer leg is consistent between the triangles, giving the proportion CD/BD = BD/AD. Solving this proportion results in CD = 9.

Adım Adım Çözüm

1

Find the length of AB using the definition of cosine in right triangle ADB.

AB = 20

In right triangle ADB,

\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{AD}{AB}

. Substituting the given values:

\frac{4}{5} = \frac{16}{AB}

, which simplifies to

AB = 20

.

2

Find the length of BD using the Pythagorean theorem in right triangle ADB.

BD = 12

Since ADB is a right triangle,

AD^2 + BD^2 = AB^2

. Substituting the known lengths:

16^2 + BD^2 = 20^2 \implies 256 + BD^2 = 400 \implies BD^2 = 144 \implies BD = 12

.

3

Use similar triangles to set up a proportion and solve for CD.

CD = 9

Triangles ADB and BDC are similar. Comparing the ratio of the shorter leg to the longer leg in both triangles gives

\frac{CD}{BD} = \frac{BD}{AD}

. Substituting the known values:

\frac{CD}{12} = \frac{12}{16} \implies CD = 12 \times \frac{3}{4} = 9

.

Anahtar Kavram

Using trigonometry and similar right triangles to find unknown segment lengths.

Soru 2780Soru →

In the $xy$ -plane, a circle with center at the origin passes through the point $(0, 8)$ . A line tangent to the circle at point $T$ passes through the point $P(15, 8)$ . What is the length of segment $PT$ ?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

The length of segment

PT

is

15

.

The radius of the circle is

8

because the circle is centered at

(0,0)

and passes through

(0,8)

. The distance from the center

O(0,0)

to point

P(15,8)

is

OP = \sqrt{15^2 + 8^2} = 17

. The radius

OT

is perpendicular to the tangent segment

PT

at point

T

, forming a right triangle

\triangle OTP

with hypotenuse

OP

and legs

OT

and

PT

. Using the Pythagorean theorem,

PT = \sqrt{17^2 - 8^2} = 15

.

Adım Adım Çözüm

1

Determine the radius of the circle

Radius

r = 8

The circle is centered at the origin

(0, 0)

and passes through

(0, 8)

, so the distance from the center to this point is the radius.

2

Calculate the distance from the origin

O(0,0)

to the point

P(15, 8)

Distance

OP = 17

Using the distance formula in the coordinate plane:

OP = \sqrt{(15-0)^2 + (8-0)^2} = \sqrt{225 + 64} = 17

.

3

Apply the Pythagorean theorem to the right triangle

\triangle OTP

Length

PT = 15

Since the tangent line

PT

is perpendicular to the radius

OT

at the point of tangency

T

,

\triangle OTP

is a right triangle with hypotenuse

OP = 17

and leg

OT = 8

. Thus,

PT = \sqrt{OP^2 - OT^2} = \sqrt{17^2 - 8^2} = 15

.

Anahtar Kavram

Tangent lines to circles and the Pythagorean theorem in the coordinate plane

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Alternatif Yöntem

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Alternatif Yöntem

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Alternatif Yöntem

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Sık Yapılan Hatalar

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