Coordinate Geometry Alıştırma Soruları — ACT — Sayfa 8

Soru 141Soru →

A circle drawn in the standard $(x, y)$ coordinate plane has its center at the point $(3, -2)$ and passes through the point $(7, 1)$ . What is the radius of this circle?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The radius of the circle is 5.

The radius of a circle is defined as the distance from its center to any point on the boundary. Using the distance formula between the center

(3, -2)

and the point

(7, 1)

yields

r = \sqrt{(7 - 3)^2 + (1 - (-2))^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

.

Adım Adım Çözüm

1

Identify the formula to find the distance between two points, which represents the radius of the circle.

The distance formula is

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

.

The radius of a circle is the distance from its center to any point on its circumference.

2

Substitute the coordinates of the center

(3, -2)

and the point

(7, 1)

into the formula.

r = \sqrt{(7 - 3)^2 + (1 - (-2))^2}

We substitute

x_1 = 3, y_1 = -2

and

x_2 = 7, y_2 = 1

into the distance formula.

3

Simplify the expression to determine the radius.

r = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5

Calculate the squared differences, add them, and take the square root to find the final value.

Anahtar Kavram

Calculating the radius of a circle given its center and a point on the circle using the distance formula.

Soru 142Soru →

In the standard $(x, y)$ coordinate plane, the perpendicular bisector of the line segment with endpoints $D(-5, 8)$ and $E(3, 4)$ is represented by the equation $y = mx + b$ . What is the value of $m + b$ ?

2

\frac{7}{2}

5

7

10

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

The sum of the slope and the y-intercept of the perpendicular bisector is

10

.

The slope of the segment

DE

is calculated as

m_{DE} = \frac{4 - 8}{3 - (-5)} = -\frac{1}{2}

. The perpendicular bisector has a slope

m

that is the negative reciprocal of this, which is

2

. The line must pass through the midpoint of

DE

, which is located at

\left(\frac{-5 + 3}{2}, \frac{8 + 4}{2}\right) = (-1, 6)

. Substituting

m = 2

and the point

(-1, 6)

into the slope-intercept form

y = mx + b

yields

6 = 2(-1) + b

, which simplifies to

b = 8

. Summing the slope and the

y

-intercept gives

m + b = 2 + 8 = 10

.

Adım Adım Çözüm

1

Find the midpoint of the line segment

DE

.

Midpoint

M = \left(\frac{-5 + 3}{2}, \frac{8 + 4}{2}\right) = (-1, 6)

By definition, the perpendicular bisector must pass through the midpoint of the segment it bisects.

2

Calculate the slope of the line segment

DE

.

m_{DE} = \frac{4 - 8}{3 - (-5)} = \frac{-4}{8} = -\frac{1}{2}

The slope of the segment is needed to find the slope of any line perpendicular to it.

3

Determine the slope

m

of the perpendicular bisector.

m = -\frac{1}{m_{DE}} = -\frac{1}{-1/2} = 2

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

4

Find the

y

-intercept

b

of the perpendicular bisector.

Using the slope-intercept form

y = mx + b

with the slope

m = 2

and the midpoint

M(-1, 6)

:

6 = 2(-1) + b \implies 6 = -2 + b \implies b = 8

Substituting a known point on the line allows us to solve for the vertical intercept parameter.

5

Calculate the sum

m + b

.

m + b = 2 + 8 = 10

This is the value requested by the question.

Anahtar Kavram

The perpendicular bisector of a line segment passes through its midpoint at a right angle, meaning its slope is the negative reciprocal of the segment's slope.

Tahmini Süre:2m 0s

Soru 143Soru →

In the standard $(x,y)$ coordinate plane, a triangle has vertices at $A(1, 2)$ , $B(9, 2)$ , and $C(5, 8)$ . A horizontal line defined by the equation $y = k$ divides the area of the triangle into two regions of equal area. What is the value of $k$ ?

5

8 - 3\sqrt{2}

2 + 3\sqrt{2}

8 - 2\sqrt{3}

4

Cevabı ve açıklamayı göster

Cevap: $8 - 3\sqrt{2}$

Cevap

8 - 3\sqrt{2}

The correct answer is

8 - 3\sqrt{2}

. Because the horizontal line

y=k

is parallel to the base of the triangle, it creates a smaller top triangle similar to the original one. The area of the original triangle is

24

and the area of the smaller triangle is

12

, yielding an area ratio of

\frac{1}{2}

. The height of the original triangle is

6

, and the height of the smaller triangle is

8-k

. Since the ratio of the areas of similar triangles is the square of the ratio of their heights, we write

\frac{1}{2} = \left(\frac{8-k}{6}\right)^2

. Solving this equation yields

k = 8 - 3\sqrt{2}

.

Adım Adım Çözüm

1

Calculate the area of the entire triangle

ABC

.

The base

\overline{AB}

is horizontal along

y = 2

with a length of

9 - 1 = 8

. The height is the vertical distance from

y = 2

to the vertex

C(5, 8)

, which is

8 - 2 = 6

. The area of

\triangle ABC

is

\frac{1}{2} \times 8 \times 6 = 24

.

Finding the total area is necessary to determine the target area of the divided regions.

2

Find the area of the smaller triangle formed above the line

y = k

.

The line

y = k

divides the triangle into a smaller top triangle and a bottom trapezoid. Since both regions have equal areas, the area of the smaller top triangle is

\frac{24}{2} = 12

.

The line divides the original area of

24

into two equal parts of

12

each.

3

Set up the area ratio equation using the properties of similar triangles.

The smaller top triangle is similar to

\triangle ABC

because its base is parallel to

\overline{AB}

. The ratio of their areas is the square of the ratio of their heights:

\frac{12}{24} = \left(\frac{8 - k}{6}\right)^2 \implies \frac{1}{2} = \left(\frac{8 - k}{6}\right)^2

.

For similar figures, the area ratio is equal to the square of the scale factor.

4

Solve the ratio equation for

k

.

Taking the square root of both sides gives

\frac{1}{\sqrt{2}} = \frac{8 - k}{6}

, which simplifies to

\frac{\sqrt{2}}{2} = \frac{8 - k}{6}

. Multiplying both sides by

6

gives

3\sqrt{2} = 8 - k

, which yields

k = 8 - 3\sqrt{2}

.

This isolates

k

to find the exact vertical coordinate of the dividing line.

Anahtar Kavram

Using properties of similar figures to determine areas and coordinates on the coordinate plane.

Soru 144Soru →

A drone starts at position $P$ on a grid. The drone's path is programmed with a sequence of two movements: first, it is translated $6$ units to the right and $5$ units up, and then its position is reflected across the $y$ -axis. If the drone's final position is $(-4, 8)$ , what were the coordinates of its starting position $P$ ?

(10, 3)

(-2, 13)

(-2, 3)

(10, 13)

(-10, -13)

Cevabı ve açıklamayı göster

Cevap: $(-2, 3)$

Cevap

The correct answer is

(-2, 3)

because working backward from the final position

(-4, 8)

by undoing the reflection across the

y

-axis gives

(4, 8)

, and then undoing the translation by subtracting

6

from the

x

-coordinate and

5

from the

y

-coordinate yields

(-2, 3)

.

The correct answer is

(-2, 3)

because we can find the starting position by working backward from the final position. The last transformation applied was a reflection across the

y

-axis. Undoing this reflection maps the final coordinates

(-4, 8)

to

(4, 8)

because reflecting across the

y

-axis changes the sign of the

x

-coordinate. The first transformation was a translation of

6

units to the right and

5

units up. To undo this translation, we subtract

6

from the

x

-coordinate and subtract

5

from the

y

-coordinate of

(4, 8)

, giving

(4 - 6, 8 - 5) = (-2, 3)

.

Adım Adım Çözüm

1

Identify the final transformation and undo it. The final transformation is a reflection across the

y

-axis, which maps

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

.

The coordinates before reflection are

(4, 8)

.

Undoing a reflection across the

y

-axis on

(-4, 8)

means changing the sign of the

x

-coordinate:

-(-4) = 4

, while keeping the

y

-coordinate the same.

2

Identify the first transformation and undo it. The first transformation is a translation of

6

units right and

5

units up, which maps

(x, y) \rightarrow (x+6, y+5)

.

The starting coordinates

P

are

(-2, 3)

.

To undo the translation, subtract

6

from the

x

-coordinate and

5

from the

y

-coordinate of the intermediate point

(4, 8)

:

4 - 6 = -2

and

8 - 5 = 3

.

Anahtar Kavram

Transformations in the Coordinate Plane

Alternatif Yöntem

Instead of working backward, you can test the options by applying the transformations forward. For

(-2, 3)

, translating 6 units right and 5 units up gives

(-2+6, 3+5) = (4, 8)

. Reflecting across the

y

-axis then gives

(-4, 8)

, which matches the final position.

Tahmini Süre:1m 15s

Soru 145Soru →

In the standard $(x, y)$ coordinate plane, a line $L$ passes through the points $(a, a^2)$ and $(b, b^2)$ , where $a$ and $b$ are distinct real numbers. The slope of line $L$ is $8$ . If the midpoint of the line segment connecting these two points lies on the line $y = 5x - 1$ , what is the $y$ -coordinate of this midpoint?

Cevabı ve açıklamayı göster

Cevap: 19

Cevap

The

y

-coordinate of the midpoint is

19

.

Applying the slope formula to the points

(a, a^2)

and

(b, b^2)

yields \frac{b^2-a^2}{b-a} = a+b = 8. The

x

-coordinate of the midpoint is \frac{a+b}{2} = 4. Substituting this value into the equation

y = 5x - 1

gives

y = 5(4) - 1 = 19

.

Adım Adım Çözüm

1

Express the slope of line

L

in terms of

a

and

b

and set it equal to the given slope.

a + b = 8

The slope formula is

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

. For the points

(a, a^2)

and

(b, b^2)

, the slope is \frac{b^2 - a^2}{b - a}. Factoring the numerator gives \frac{(b-a)(b+a)}{b-a} = a + b. Since the slope is given as

8

, we establish

a + b = 8

.

2

Find the

x

-coordinate of the midpoint of the segment connecting the two points.

x_M = 4

The midpoint formula for the

x

-coordinate is

x_M = \frac{x_1 + x_2}{2}

. For our points,

x_M = \frac{a+b}{2}

. Substituting the value

a+b = 8

gives

x_M = \frac{8}{2} = 4

.

3

Determine the

y

-coordinate of the midpoint using the line equation.

y_M = 19

The midpoint lies on the line

y = 5x - 1

. Substituting

x_M = 4

into this equation gives

y_M = 5(4) - 1 = 19

.

Anahtar Kavram

Slope of a Line

Alternatif Yöntem

Let

a

and

b

be the real roots of the quadratic equation

t^2 - 8t + 13 = 0

. By Vieta's formulas,

a+b = 8

and

ab = 13

. The

x

-coordinate of the midpoint is \frac{a+b}{2} = 4, and the

y

-coordinate is \frac{a^2+b^2}{2} = \frac{(a+b)^2 - 2ab}{2} = \frac{64 - 26}{2} = 19. Since the point

(4, 19)

satisfies

y = 5x - 1

, this confirms the existence of valid real coordinates

(a, a^2)

and

(b, b^2)

that produce the midpoint on the line.

Tahmini Süre:2m 0s

Soru 146Soru →

A municipal water reservoir is being drained at a constant rate. After $4$ hours of draining, the reservoir contains $18,000$ gallons of water. After $7$ hours of draining, it contains $13,500$ gallons of water. How many hours after the draining process begins will the reservoir be completely empty?

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Cevap: 16

Cevap

The reservoir will be completely empty

16

hours after draining begins.

The correct answer is

16

hours. The constant draining rate is calculated as the change in volume divided by the change in time:

\frac{13,500 - 18,000}{7 - 4} = -1,500

gallons per hour. Since there are

18,000

gallons left at the

4

-hour mark, it will take an additional

\frac{18,000}{1,500} = 12

hours to completely empty the reservoir. The total time from the start is

4 + 12 = 16

hours.

Adım Adım Çözüm

1

Calculate the draining rate (the slope of the linear function).

The rate is

-1,500

gallons per hour.

The rate of change is the change in volume divided by the change in time:

\frac{13,500 - 18,000}{7 - 4} = \frac{-4,500}{3} = -1,500

gallons per hour.

2

Determine the remaining time needed to empty the reservoir after the

7

-hour mark.

It will take an additional

9

hours.

At

7

hours, the reservoir contains

13,500

gallons. Draining at a rate of

1,500

gallons per hour, the remaining time is

\frac{13,500}{1,500} = 9

hours.

3

Calculate the total elapsed time since the draining process began.

16

hours

Adding the initial

7

hours of draining to the additional

9

hours needed gives

7 + 9 = 16

hours.

Anahtar Kavram

Linear word problems and finding intercepts

Alternatif Yöntem

We can model the volume of water

W

as a linear function of time

t

using the slope-intercept form

W(t) = mt + b

. Substituting the rate

m = -1,500

and the point

(4, 18,000)

gives

18,000 = -1,500(4) + b

, which yields the

y

-intercept (initial volume)

b = 24,000

gallons. The linear model is

W(t) = -1,500t + 24,000

. Setting

W(t) = 0

to find when the reservoir is empty gives

0 = -1,500t + 24,000

, which simplifies to

t = \frac{24,000}{1,500} = 16

hours.

Tahmini Süre:1m 30s

Soru 147Soru →

In the standard $(x, y)$ coordinate plane, point $P$ is rotated $90^\circ$ clockwise about the origin, and then translated $3$ units to the left and $4$ units up to form the image point $P'(1, 2)$ . What are the coordinates of the pre-image point $P$ ?

(-1, 3)

(-6, -2)

(2, 4)

(-2, -4)

(1, -3)

Cevabı ve açıklamayı göster

Cevap: $(2, 4)$

Cevap

The coordinates of the pre-image point

P

are

(2, 4)

.

To find the coordinates of the pre-image point, we must undo the transformations in reverse order. First, we undo the translation of

3

units left and

4

units up by translating the image point

(1, 2)

by

3

units right and

4

units down, yielding the intermediate point

(4, -2)

. Next, we undo the

90^\circ

clockwise rotation by rotating this intermediate point

90^\circ

counterclockwise about the origin. Using the rotation rule

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

on

(4, -2)

gives the pre-image point

(2, 4)

.

Adım Adım Çözüm

1

Identify the sequence of forward transformations and set up the path to work backward from the final image point

P'(1, 2)

to the pre-image point

P

.

The forward sequence is:

P \xrightarrow{\text{Rotate } 90^\circ \text{ clockwise}} P_{\text{mid}} \xrightarrow{\text{Translate left 3, up 4}} P'(1, 2)

. To find

P

, we must apply the inverse transformations in reverse order:

P'(1, 2) \xrightarrow{\text{Translate right 3, down 4}} P_{\text{mid}} \xrightarrow{\text{Rotate } 90^\circ \text{ counterclockwise}} P

.

Working backward with inverse operations is the standard mathematical procedure to retrieve a pre-image from its transformed image.

2

Apply the inverse translation to the image point

P'(1, 2)

to find the coordinates of the intermediate point

P_{\text{mid}}

.

To undo a translation of

3

units left and

4

units up, translate

3

units right and

4

units down:

x = 1 + 3 = 4

, and

y = 2 - 4 = -2

. Thus,

P_{\text{mid}} = (4, -2)

.

Applying the opposite direction and magnitude to the coordinates reverses the translation effect.

3

Apply the inverse rotation to the intermediate point

P_{\text{mid}}(4, -2)

to find the coordinates of the pre-image point

P

.

The inverse of a

90^\circ

clockwise rotation is a

90^\circ

counterclockwise rotation about the origin. The rule for a

90^\circ

counterclockwise rotation is

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

. Applying this to

(4, -2)

gives

P(-(-2), 4) = P(2, 4)

.

Rotating

90^\circ

counterclockwise reverses the initial

90^\circ

clockwise rotation, returning the point to its original pre-image coordinates.

Anahtar Kavram

Determining the coordinates of a pre-image by reversing a composite transformation in the coordinate plane.

Soru 148Soru →

A right triangle is plotted in the standard $(x, y)$ coordinate plane. The vertices of the triangle are located at $(-3, -2)$ , $(5, -2)$ , and $(-3, 4)$ . What is the length of the hypotenuse of the triangle?

\frac{4}{3}

2\sqrt{7}

\frac{3}{5}

10

14

Cevabı ve açıklamayı göster

Cevap: $10$

Cevap

10

The distance between the vertices

(-3, -2)

and

(5, -2)

along the horizontal line

y = -2

is

5 - (-3) = 8

units. The distance between the vertices

(-3, -2)

and

(-3, 4)

along the vertical line

x = -3

is

4 - (-2) = 6

units. Because horizontal and vertical segments meet at a right angle, they form the legs of a right triangle. Applying the Pythagorean theorem, the length of the hypotenuse is the square root of the sum of the squares of the legs:

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

.

Adım Adım Çözüm

1

Determine the lengths of the two perpendicular legs of the right triangle on the coordinate plane.

The horizontal leg has a length of

5 - (-3) = 8

units, and the vertical leg has a length of

4 - (-2) = 6

units.

Because the segment between

(-3, -2)

and

(5, -2)

is horizontal (constant

y = -2

) and the segment between

(-3, -2)

and

(-3, 4)

is vertical (constant

x = -3

).

2

Apply the Pythagorean theorem to calculate the length of the hypotenuse.

The hypotenuse length is

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

units.

The Pythagorean theorem states that for any right triangle with legs

a

and

b

and hypotenuse

c

,

a^2 + b^2 = c^2

.

Anahtar Kavram

Finding the lengths of segments and applying the Pythagorean theorem on the coordinate plane.

Soru 149Soru →

A circle in the standard $(x,y)$ coordinate plane is defined by the equation $(x + 3)^2 + (y - 4)^2 = 25$ . What are the coordinates of the center and the length of the radius of this circle?

Center:

(3, -4)

; Radius:

5

Center:

(-3, 4)

; Radius:

25

Center:

(-3, 4)

; Radius:

5

Center:

(3, -4)

; Radius:

25

Center:

(-3, -4)

; Radius:

5

Cevabı ve açıklamayı göster

Cevap: Center: $(-3, 4)$ ; Radius: $5$

Cevap

Center:

(-3, 4)

; Radius:

5

The standard form of a circle's equation is

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

, where the center is

(h, k)

and the radius is

r

. Matching

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

to this standard form, we find

h = -3

,

k = 4

, and

r^2 = 25

(which gives

r = 5

). Therefore, the center is

(-3, 4)

and the radius is

5

.

Adım Adım Çözüm

1

Identify the standard form of a circle's equation.

The standard equation of a circle with center

(h, k)

and radius

r

is

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

.

This formula provides the template to match the given equation and extract the parameters.

2

Rewrite the given equation

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

to match the signs and exponents of the standard form.

The equation can be written as

(x - (-3))^2 + (y - 4)^2 = 5^2

.

Rewriting the terms helps to identify the exact values of

h

,

k

, and

r

without sign confusion.

3

Extract the center

(h, k)

and radius

r

from the rewritten equation.

Comparing the terms shows

h = -3

,

k = 4

, and

r = 5

, giving a center of

(-3, 4)

and a radius of

5

.

These extracted values are the final answer.

Anahtar Kavram

Equations and Graphs of Circles

Soru 150Soru →

A circle in the standard $(x, y)$ coordinate plane is defined by the equation $(x - 9)^2 + (y + 4)^2 = 81$ . What is the radius of this circle?

Cevabı ve açıklamayı göster

Cevap: 9

Cevap

The radius of the circle is 9.

The standard equation of a circle in the coordinate plane is

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

, where

r

is the radius. For the equation

(x - 9)^2 + (y + 4)^2 = 81

, the right-hand side represents

r^2

, so

r^2 = 81

. Taking the positive square root of both sides gives

r = \sqrt{81} = 9

.

Adım Adım Çözüm

1

Identify the standard form of the circle equation.

The standard equation of a circle is

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

, where

(h, k)

is the center and

r

is the radius.

This allows us to relate the terms of the given equation to the components of the circle.

2

Match the given equation to the standard form.

By comparing

(x - 9)^2 + (y + 4)^2 = 81

to

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

, we find that

r^2 = 81

.

The constant on the right side of the standard equation represents the square of the radius.

3

Solve for the radius

r

.

r = \sqrt{81} = 9

.

Taking the square root of

r^2

gives the actual radius of the circle.

Anahtar Kavram

Identifying the radius from the standard equation of a circle

Soru 151Soru →

In the standard $(x, y)$ coordinate plane, the midpoint of a line segment with endpoints $(1, -2)$ and $(7, 10)$ is the center of a circle. If the point $(8, 7)$ lies on the circle, what is the radius of the circle?

7

25

3\sqrt{5}

5

\sqrt{26}

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The radius of the circle is 5.

The midpoint of the line segment with endpoints

(1, -2)

and

(7, 10)

is calculated as

(\frac{1+7}{2}, \frac{-2+10}{2}) = (4, 4)

, which is the center of the circle. The distance from the center

(4, 4)

to the point

(8, 7)

on the circle is the radius, which is

\sqrt{(8-4)^2 + (7-4)^2} = \sqrt{16+9} = 5

.

Adım Adım Çözüm

1

Find the midpoint of the line segment with endpoints

(1, -2)

and

(7, 10)

to determine the center of the circle.

The center of the circle is

(4, 4)

.

The midpoint of a segment with endpoints

(x_1, y_1)

and

(x_2, y_2)

is given by

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

. Calculating this gives

(\frac{1+7}{2}, \frac{-2+10}{2}) = (4, 4)

.

2

Calculate the distance between the center of the circle

(4, 4)

and the point

(8, 7)

on the circle to find the radius.

The radius of the circle is 5.

The distance formula is

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

. Substituting the coordinates gives

\sqrt{(8-4)^2 + (7-4)^2} = \sqrt{4^2 + 3^2} = \sqrt{16+9} = \sqrt{25} = 5

.

Anahtar Kavram

Finding the midpoint of a line segment to determine a circle's center and using the distance formula to calculate its radius.

Soru 152Soru →

In the standard $(x,y)$ coordinate plane, a square has two adjacent vertices at $A(1, 4)$ and $B(4, 8)$ . If the square lies entirely in the first quadrant, what is the $x$ -coordinate of the vertex diagonally opposite to $A$ ?

Cevabı ve açıklamayı göster

Cevap: 8

Cevap

The

x

-coordinate of the vertex diagonally opposite to

A

is

8

.

The vector from

A(1,4)

to

B(4,8)

is

\vec{AB} = (3,4)

with length

5

. Since the adjacent sides of a square must be perpendicular and equal in length, the vector translation must be either

(4, -3)

or

(-4, 3)

. Applying the translation

(4, -3)

to

B

and

A

gives

C(8, 5)

and

D(5, 1)

, both of which lie strictly in the first quadrant. The diagonally opposite vertex to

A

is

C

, and its

x

-coordinate is

8

.

Adım Adım Çözüm

1

Calculate the vector representing side

AB

and its length.

\vec{AB} = (3, 4)

, which has a length of

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

.

This determines the length and orientation of the square's base.

2

Determine the perpendicular vectors of equal length to find the directions of the adjacent sides.

The perpendicular vectors of length

5

are

(4, -3)

and

(-4, 3)

.

The adjacent sides of a square are perpendicular and equal in length to side

AB

.

3

Find the coordinates of the other two vertices for both possible positions of the square.

Position 1 vertices:

C = B + (4, -3) = (8, 5)

and

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

. Position 2 vertices:

C' = B + (-4, 3) = (0, 11)

and

D' = A + (-4, 3) = (-3, 7)

.

Adding the perpendicular vectors to the coordinates of the base vertices yields the remaining vertices.

4

Determine which square lies entirely in the first quadrant and identify the target coordinate.

The square

ABCD

with vertices

C(8, 5)

and

D(5, 1)

lies entirely in the first quadrant because all

x

- and

y

-coordinates are positive. The vertex diagonally opposite to

A

is

C(8, 5)

, which has an

x

-coordinate of

8

.

The alternative square

ABC'D'

has a vertex at

D'(-3, 7)

, which lies in the second quadrant.

Anahtar Kavram

Using vector methods to determine the coordinates of a geometric figure on the coordinate plane under specific quadrant constraints.

Tahmini Süre:2m 0s

Soru 153Soru →

In the standard $(x, y)$ coordinate plane, a line $L$ passes through the point $(2, -3)$ and does not pass through the origin. The distance from the origin to the line's $x$ -intercept is twice the distance from the origin to the line's $y$ -intercept. Which of the following could be the slope of line $L$ ?

-2

-\frac{1}{2}

\frac{2}{3}

1

2

Cevabı ve açıklamayı göster

Cevap: $-\frac{1}{2}$

Cevap

The slope of the line could be

-\frac{1}{2}

.

The correct answer is

-\frac{1}{2}

. By setting the

x

-intercept of the line to

(a, 0)

and the

y

-intercept to

(0, b)

, the distance condition gives

|a| = 2|b|

. Substituting the point

(2, -3)

into the intercept equation of a line,

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

, yields

\frac{2}{a} - \frac{3}{b} = 1

. Solving the two cases (

a = 2b

and

a = -2b

) gives the intercept pairs

(-4, -2)

and

(8, -4)

. Calculating the slope

m = -\frac{b}{a}

for both pairs results in

-\frac{1}{2}

and

\frac{1}{2}

. Since

-\frac{1}{2}

is one of these possible values, it is the correct choice.

Adım Adım Çözüm

1

Represent the line

L

using its intercepts. Let the

x

-intercept of

L

be

(a, 0)

and the

y

-intercept be

(0, b)

. Since the line does not pass through the origin, we have

a \neq 0

and

b \neq 0

.

The equation of the line can be written in intercept form as:

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

Using intercept form allows us to directly relate the coordinates of the intercepts to the given point and the distance condition.

2

Translate the distance condition and the given point into equations. The distance from the origin to the

x

-intercept is

|a|

, and the distance to the

y

-intercept is

|b|

. The problem states that the distance to the

x

-intercept is twice the distance to the

y

-intercept, so

|a| = 2|b|

. Additionally, the line passes through

(2, -3)

, so we substitute

x = 2

and

y = -3

into the intercept equation.

|a| = 2|b|

and

\frac{2}{a} - \frac{3}{b} = 1

This sets up a system of equations to solve for the unknown intercepts

a

and

b

.

3

Solve the system of equations by analyzing the two cases for the absolute value:

a = 2b

and

a = -2b

.

Case 1: If

a = 2b

, then

\frac{2}{2b} - \frac{3}{b} = 1 \implies \frac{1}{b} - \frac{3}{b} = 1 \implies -\frac{2}{b} = 1 \implies b = -2

, which gives

a = -4

. Case 2: If

a = -2b

, then

\frac{2}{-2b} - \frac{3}{b} = 1 \implies -\frac{1}{b} - \frac{3}{b} = 1 \implies -\frac{4}{b} = 1 \implies b = -4

, which gives

a = 8

.

Resolving the absolute value yields the exact coordinates of the intercepts for both valid scenarios.

4

Calculate the slope

m

for both cases using the formula

m = -\frac{b}{a}

.

For Case 1,

m = -\frac{-2}{-4} = -\frac{1}{2}

. For Case 2,

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

.

The slope of a line with intercepts

(a, 0)

and

(0, b)

is given by

-\frac{b}{a}

.

Anahtar Kavram

Slope of a Line

Tahmini Süre:2m 0s

Soru 154Soru →

In the standard $(x, y)$ coordinate plane, one diagonal of a square lies along the line with equation $y = 3x - 4$ . If one of the vertices of the square that does not lie on this diagonal is located at the point $(2, 8)$ , which of the following equations represents the line containing the other diagonal of the square?

3x - y = -2

3x + y = 14

x + 3y = 26

x - 3y = -22

x + 3y = 40

Cevabı ve açıklamayı göster

Cevap: $x + 3y = 26$

Cevap

The equation of the line containing the other diagonal is

x + 3y = 26

.

The correct equation is

x + 3y = 26

. Since the diagonals of a square are perpendicular, their slopes must be negative reciprocals of each other. The given diagonal has a slope of

3

, meaning the other diagonal has a slope of

-\frac{1}{3}

. Since the vertex

(2, 8)

does not satisfy the equation of the first diagonal, it must lie on the second diagonal. Substituting this point into the slope-intercept or point-slope equation yields the standard form equation

x + 3y = 26

.

Adım Adım Çözüm

1

Identify the slope of the given diagonal line.

The slope of the line

y = 3x - 4

is

3

.

The equation is in slope-intercept form

y = mx + b

, where

m

is the slope.

2

Determine the relationship between the two diagonals of a square.

The diagonals of a square are perpendicular to each other.

By geometric definition, the diagonals of any square intersect at right angles.

3

Calculate the slope of the perpendicular diagonal.

The slope of the perpendicular diagonal is

-\frac{1}{3}

.

Perpendicular lines have slopes that are negative reciprocals of each other, so the perpendicular slope is

-\frac{1}{3}

.

4

Identify the point that the perpendicular diagonal passes through.

The perpendicular diagonal passes through the vertex

(2, 8)

.

The vertex

(2, 8)

does not lie on the first diagonal because

8 \neq 3(2) - 4

. Since a square only has two diagonals, any vertex not on the first diagonal must lie on the second diagonal.

5

Find the equation of the perpendicular diagonal using point-slope form.

The equation is

x + 3y = 26

.

Using the point-slope form

y - y_1 = m(x - x_1)

with

m = -\frac{1}{3}

and point

(2, 8)

yields

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

, which simplifies to

3(y - 8) = -(x - 2) \Rightarrow 3y - 24 = -x + 2 \Rightarrow x + 3y = 26

.

Anahtar Kavram

Diagonals of a square are perpendicular, and perpendicular lines have slopes that are negative reciprocals.

Soru 155Soru →

A line segment in the standard $(x, y)$ coordinate plane has endpoints at $A(-1, 10)$ and $B(7, -2)$ . If point $M$ is the midpoint of this segment, what is the distance from $M$ to the origin $(0, 0)$ ?

Cevabı ve açıklamayı göster

Cevap: 5

Cevap

The distance from the midpoint

M

to the origin is 5.

To find the distance from the midpoint

M

to the origin

(0,0)

, we first determine the coordinates of

M

by taking the average of the coordinates of

A(-1, 10)

and

B(7, -2)

. This results in

M\left(\frac{-1+7}{2}, \frac{10+(-2)}{2}\right) = (3, 4)

. Next, we apply the distance formula between

M(3,4)

and the origin

(0,0)

to get

\sqrt{(3-0)^2 + (4-0)^2} = \sqrt{9+16} = \sqrt{25} = 5

.

Adım Adım Çözüm

1

Calculate the coordinates of the midpoint

M

of the segment

AB

.

The midpoint is

M(3, 4)

.

The midpoint formula is

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

. Substituting the coordinates of

A(-1, 10)

and

B(7, -2)

gives the x-coordinate as

\frac{-1 + 7}{2} = 3

and the y-coordinate as

\frac{10 + (-2)}{2} = 4

.

2

Calculate the distance from the midpoint

M(3, 4)

to the origin

(0, 0)

.

The distance is 5.

The distance formula is

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

. Substituting the coordinates of

M(3, 4)

and the origin

(0, 0)

gives

d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

.

Anahtar Kavram

Using the midpoint formula to find the center point of a line segment, and then using the distance formula to find the length between that point and another specified coordinate.

Soru 156Soru →

In the standard $(x, y)$ coordinate plane, a line passes through the points $(\frac{1}{2}, \frac{2}{3})$ and $(\frac{3}{4}, \frac{5}{6})$ . What is the $y$ -intercept of this line?

\frac{1}{3}

\frac{1}{6}

-1

-\frac{1}{12}

-\frac{1}{7}

Cevabı ve açıklamayı göster

Cevap: $\frac{1}{3}$

Cevap

The

y

-intercept of the line is

\frac{1}{3}

.

The correct answer is

\frac{1}{3}

. By calculating the slope of the line as

m = \frac{2}{3}

, and using the point-slope formula

y - y_1 = m(x - x_1)

with either given point, we can isolate

y

to get the equation

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

, where the constant term represents the

y

-intercept.

Adım Adım Çözüm

1

Calculate the slope (

m

) of the line passing through the points

(\frac{1}{2}, \frac{2}{3})

and

(\frac{3}{4}, \frac{5}{6})

using the slope formula

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

.

m = \frac{\frac{5}{6} - \frac{2}{3}}{\frac{3}{4} - \frac{1}{2}} = \frac{\frac{1}{6}}{\frac{1}{4}} = \frac{2}{3}

To find the equation of a line, we first need to determine its slope.

2

Use the point-slope form

y - y_1 = m(x - x_1)

with the point

(\frac{1}{2}, \frac{2}{3})

and slope

m = \frac{2}{3}

to write the equation of the line.

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

This allows us to write the linear equation in a form where we can isolate the

y

-intercept.

3

Isolate

y

to convert the equation into slope-intercept form

y = mx + b

, where

b

represents the

y

-intercept.

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

The constant term in slope-intercept form represents the

y

-intercept.

Anahtar Kavram

Linear Equations and Graphing

Alternatif Yöntem

Instead of using the point-slope form, you can substitute one of the points directly into the slope-intercept form

y = mx + b

. Substituting

(\frac{1}{2}, \frac{2}{3})

and

m = \frac{2}{3}

yields

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

, which simplifies to

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

. Solving for

b

gives

b = \frac{1}{3}

.

Tahmini Süre:1m 30s

Soru 157Soru →

On a coordinate grid, a circle is drawn such that its center is at the point $(4, -2)$ . If the circle is tangent to the $y$ -axis, what is the equation of this circle?

(x + 4)^2 + (y - 2)^2 = 16

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

(x - 4)^2 + (y + 2)^2 = 16

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

(x - 2)^2 + (y + 4)^2 = 16

Cevabı ve açıklamayı göster

Cevap: $(x - 4)^2 + (y + 2)^2 = 16$

Cevap

(x - 4)^2 + (y + 2)^2 = 16

The correct equation is

(x - 4)^2 + (y + 2)^2 = 16

. A circle with center

(h, k)

and radius

r

has the standard equation

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

. Substituting

(4, -2)

for the center gives

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

. Since the circle is tangent to the

y

-axis, the radius is the horizontal distance from the center to the line

x = 0

, which is

4

units. Squaring the radius gives

r^2 = 16

.

Adım Adım Çözüm

1

Recall the standard form of the equation of a circle.

The standard equation of a circle is

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

, where

(h, k)

is the center and

r

is the radius.

This sets up the formula needed to write the circle's equation.

2

Substitute the center coordinates

(h, k) = (4, -2)

into the formula.

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

, which simplifies to

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

.

To represent the translation of the circle's center from the origin.

3

Determine the radius of the circle using the given tangency condition.

Since the circle is tangent to the

y

-axis, the radius is the horizontal distance from the center

(4, -2)

to the

y

-axis (the line

x = 0

). This distance is

|4 - 0| = 4

units, so

r = 4

.

To find the radius from the geometric properties of the circle.

4

Square the radius and write the complete equation.

r^2 = 4^2 = 16

. The completed equation is

(x - 4)^2 + (y + 2)^2 = 16

.

To complete the standard equation form.

Anahtar Kavram

Standard form of a circle's equation and determining its radius from a tangency condition

Soru 158Soru →

In the standard $(x, y)$ coordinate plane, a triangle has vertices at $A(k, 4)$ , $B(-1, 2)$ , and $C(3, -6)$ . The altitude from vertex $A$ to side $BC$ intersects the $y$ -axis at $(0, -2)$ . What is the value of $k$ ?

-12

3

4

12

24

Cevabı ve açıklamayı göster

Cevap: 12

Cevap

12

The slope of side

BC

is calculated as

m_{BC} = \frac{-6 - 2}{3 - (-1)} = -2

. Since the altitude from vertex

A

is perpendicular to side

BC

, its slope must be the negative reciprocal of

-2

, which is

\frac{1}{2}

. The equation of the line containing this altitude, with a given

y

-intercept of

(0, -2)

, is

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

. Substituting the coordinates of vertex

A(k, 4)

into the equation gives

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

. Solving for

k

yields

k = 12

.

Adım Adım Çözüm

1

Calculate the slope of side

BC

using the slope formula

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

.

m_{BC} = \frac{-6 - 2}{3 - (-1)} = \frac{-8}{4} = -2

The altitude is perpendicular to the side

BC

, so we first need the slope of

BC

.

2

Find the slope of the altitude by taking the negative reciprocal of the slope of

BC

.

m_{\text{alt}} = -\frac{1}{m_{BC}} = -\frac{1}{-2} = \frac{1}{2}

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

3

Write the equation of the line containing the altitude using the slope-intercept form

y = mx + b

with the given

y

-intercept of

(0, -2)

.

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

The line has a slope of

\frac{1}{2}

and crosses the

y

-axis at

-2

.

4

Substitute the coordinates of vertex

A(k, 4)

into the equation and solve for

k

.

4 = \frac{1}{2}k - 2 \Rightarrow 6 = \frac{1}{2}k \Rightarrow k = 12

Vertex

A

lies on the altitude line, so its coordinates must satisfy the line's equation.

Anahtar Kavram

The slope of a line perpendicular to a given line is the negative reciprocal of the given line's slope.

Alternatif Yöntem

Alternatively, we can use the vector dot product. The vector representing side

BC

is

\vec{BC} = (3 - (-1), -6 - 2) = (4, -8)

. The vector from the

y

-intercept to vertex

A

is

\vec{v} = (k - 0, 4 - (-2)) = (k, 6)

. Since the altitude is perpendicular to

BC

, the dot product of these two vectors must equal zero:

(4)(k) + (-8)(6) = 0 \Rightarrow 4k - 48 = 0 \Rightarrow 4k = 48 \Rightarrow k = 12

.

Tahmini Süre:2m 0s

Soru 159Soru →

A circle in the standard $(x, y)$ coordinate plane passes through the points $(0, 0)$ , $(8, 0)$ , and $(2, 6)$ . What are the coordinates of the center of this circle?

(3, 3)

(4, 2)

(4, 3)

(5, 3)

(2, 4)

Cevabı ve açıklamayı göster

Cevap: $(4, 2)$

Cevap

The center of the circle is

(4, 2)

The correct answer is

(4, 2)

. A circle's center

(h, k)

is equidistant from all points on the circle. By setting up the distance formula from the center

(h, k)

to the points

(0,0)

,

(8,0)

, and

(2,6)

, we obtain the equations

h^2 + k^2 = (h-8)^2 + k^2

and

h^2 + k^2 = (h-2)^2 + (k-6)^2

. Solving the first equation yields

16h = 64

, which simplifies to

h = 4

. Substituting

h = 4

into the second equation yields

16 + k^2 = 4 + (k-6)^2

. Expanding and simplifying gives

16 = 40 - 12k

, which simplifies to

12k = 24

, or

k = 2

. Therefore, the center of the circle is

(4, 2)

.

Adım Adım Çözüm

1

Set up the distance formula equations for the center of the circle,

(h, k)

, which must be equidistant from the three points on the circle:

(0, 0)

,

(8, 0)

, and

(2, 6)

.

The distances from

(h, k)

to the points are

\sqrt{h^2 + k^2}

,

\sqrt{(h-8)^2 + k^2}

, and

\sqrt{(h-2)^2 + (k-6)^2}

. Since they are radii, we equate them:

h^2 + k^2 = (h-8)^2 + k^2 = (h-2)^2 + (k-6)^2

.

The distance from the center of a circle to any point on its circumference is equal to its radius.

2

Equate the distance to

(0,0)

and the distance to

(8,0)

, then solve for

h

.

h^2 + k^2 = (h-8)^2 + k^2 \implies h^2 = h^2 - 16h + 64 \implies 16h = 64 \implies h = 4

.

Since both points lie on the x-axis, the x-coordinate of the center must lie on the perpendicular bisector of the segment connecting them, which is

x = 4

.

3

Substitute

h = 4

into the equation equating the distance to

(0,0)

and

(2,6)

to solve for

k

.

4^2 + k^2 = (4-2)^2 + (k-6)^2 \implies 16 + k^2 = 4 + k^2 - 12k + 36 \implies 16 = 40 - 12k \implies 12k = 24 \implies k = 2

.

Substituting the known coordinate allows us to solve the remaining single-variable equation for the

y

-coordinate of the center.

4

Combine the coordinates to state the final center of the circle.

The center is

(4, 2)

.

Both coordinates

(h, k)

have been uniquely determined.

Anahtar Kavram

Circumcenter of a triangle / Center of a circle passing through three points on a coordinate plane

Tahmini Süre:2m 0s

Soru 160Soru →

In the standard $(x, y)$ coordinate plane, if the three distinct points $(k, 4)$ , $(4, k)$ , and $(5, 6)$ are collinear, then there are exactly two possible values for the slope of the line passing through these points.

Doğru

Yanlış

Cevabı ve açıklamayı göster

Cevap: False

Cevap

The statement is false because the requirement that the three points must be distinct eliminates one of the algebraic solutions for

k

, resulting in only one possible slope for the line.

The statement is false because equating the slopes between the points leads to the quadratic equation

k^2 - 11k + 28 = 0

, which has the solutions

k = 4

and

k = 7

. However, substituting

k = 4

back into the coordinates results in the points

(4, 4)

and

(4, 4)

, which are not distinct. Thus, only

k = 7

is a valid solution, which yields exactly one line with a slope of

-1

.

Adım Adım Çözüm

1

Set up the collinearity condition using the slope formula.

The slope of the segment connecting

(4, k)

and

(5, 6)

is

m_1 = \frac{6 - k}{5 - 4} = 6 - k

. The slope of the segment connecting

(k, 4)

and

(5, 6)

is

m_2 = \frac{6 - 4}{5 - k} = \frac{2}{5 - k}

. Equating the two slopes gives

6 - k = \frac{2}{5 - k}

.

For three points to be collinear, the slope between any two pairs of points must be equal.

2

Solve the rational equation for

k

.

Multiplying both sides by

5 - k

yields

(6 - k)(5 - k) = 2

, which expands to

30 - 11k + k^2 = 2

. Rearranging into standard quadratic form gives

k^2 - 11k + 28 = 0

. Factoring the quadratic yields

(k - 4)(k - 7) = 0

, giving the solutions

k = 4

or

k = 7

.

Finding the roots of the quadratic equation identifies the algebraic candidates for

k

.

3

Verify if the points are distinct for each candidate value of

k

.

If

k = 4

, the first two points are

(4, 4)

and

(4, 4)

, which are not distinct. If

k = 7

, the three points are

(7, 4)

,

(4, 7)

, and

(5, 6)

, which are distinct.

The problem specifies that the three points must be distinct, so any solution violating this condition must be discarded.

4

Determine the number of possible slopes.

For the only valid value

k = 7

, the slope of the line is

m = 6 - 7 = -1

. There is only one possible slope.

Comparing the number of valid slopes to the statement shows that the statement is false.

Anahtar Kavram

Collinearity of points in the coordinate plane and constraints on slope calculation.

Coordinate Geometry

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

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Sık Yapılan Hatalar

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Anahtar Kavram

Sık Yapılan Hatalar

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Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

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Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

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Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

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Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

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Cevap

Adım Adım Çözüm

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Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm