29.10.2021

Which of the following dilations was applied to ABCD to create AB C D

. 4

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Mathematics
Step-by-step answer
P Answered by PhD

B

Step-by-step explanation:

Quadrilateral ABCD has its vertices at points A(1,7), B(4,6), C(6,1) and D(1,4).

Consider dilation centered at point A(1,7) with a scale factor \frac{2}{3}. This dilation has the rule

(x,y)\rightarrow \left(\dfrac{2}{3}x+\dfrac{1}{3},\dfrac{2}{3}y+\dfrac{7}{3}\right)

Thus,

A(1,7)\rightarrow A'(1,7)\ \left[\dfrac{2}{3}\cdot 1+\dfrac{1}{3}=1,\ \dfrac{2}{3}\cdot 7+\dfrac{7}{3}=7\right];B(4,6)\rightarrow B'\left(3,6\dfrac{1}{3}\right)\ \left[\dfrac{2}{3}\cdot 4+\dfrac{1}{3}=3,\ \dfrac{2}{3}\cdot 6+\dfrac{7}{3}=\dfrac{19}{3}=6\dfrac{1}{3}\right];C(6,1)\rightarrow C'\left(4\dfrac{1}{3},3\right)\ \left[\dfrac{2}{3}\cdot 6+\dfrac{1}{3}=\dfrac{13}{3}=4\dfrac{1}{3},\ \dfrac{2}{3}\cdot 1+\dfrac{7}{3}=3\right];D(1,4)\rightarrow D'(1,5)\ \left[\dfrac{2}{3}\cdot 1+\dfrac{1}{3}=1,\ \dfrac{2}{3}\cdot 4+\dfrac{7}{3}=5\right];

These are exactly coordinates of quadrilateral A'B'C'D'

Mathematics
Step-by-step answer
P Answered by PhD

B

Step-by-step explanation:

Quadrilateral ABCD has its vertices at points A(1,7), B(4,6), C(6,1) and D(1,4).

Consider dilation centered at point A(1,7) with a scale factor \frac{2}{3}. This dilation has the rule

(x,y)\rightarrow \left(\dfrac{2}{3}x+\dfrac{1}{3},\dfrac{2}{3}y+\dfrac{7}{3}\right)

Thus,

A(1,7)\rightarrow A'(1,7)\ \left[\dfrac{2}{3}\cdot 1+\dfrac{1}{3}=1,\ \dfrac{2}{3}\cdot 7+\dfrac{7}{3}=7\right];B(4,6)\rightarrow B'\left(3,6\dfrac{1}{3}\right)\ \left[\dfrac{2}{3}\cdot 4+\dfrac{1}{3}=3,\ \dfrac{2}{3}\cdot 6+\dfrac{7}{3}=\dfrac{19}{3}=6\dfrac{1}{3}\right];C(6,1)\rightarrow C'\left(4\dfrac{1}{3},3\right)\ \left[\dfrac{2}{3}\cdot 6+\dfrac{1}{3}=\dfrac{13}{3}=4\dfrac{1}{3},\ \dfrac{2}{3}\cdot 1+\dfrac{7}{3}=3\right];D(1,4)\rightarrow D'(1,5)\ \left[\dfrac{2}{3}\cdot 1+\dfrac{1}{3}=1,\ \dfrac{2}{3}\cdot 4+\dfrac{7}{3}=5\right];

These are exactly coordinates of quadrilateral A'B'C'D'

Mathematics
Step-by-step answer
P Answered by PhD

Scale factor = \frac{2}{3}

Step-by-step explanation:

Let x be scale factor.

Since we know that the dilation of a figure changes all the sides of figure with same scale factor.  

Let us find scale factor of dilation applied to quadrilateral ABCD to create AB'C'D'.

Let us find side length of AD and AD' using distance formula.

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

\text{Length of AD}=\sqrt{(1-1)^{2}+(4-7)^{2}}

\text{Length of AD}=\sqrt{(0)^{2}+(-3)^{2}}

\text{Length of AD}=\sqrt{9}=3

Now let us find length of side AD'.

\text{Length of AD'}=\sqrt{(1-1)^{2}+(5-7)^{2}}

\text{Length of AD'}=\sqrt{(0)^{2}+(-2)^{2}}

\text{Length of AD'}=\sqrt{4}=2

The scale factor (x) times the length of AD will be length of AD'.

3\cdot x=2

x=\frac{2}{3}

Therefore, dilation by a scale factor of 2/3 is applied to ABCD to create AB'C'D'.

Mathematics
Step-by-step answer
P Answered by PhD

Scale factor = \frac{2}{3}

Step-by-step explanation:

Let x be scale factor.

Since we know that the dilation of a figure changes all the sides of figure with same scale factor.  

Let us find scale factor of dilation applied to quadrilateral ABCD to create AB'C'D'.

Let us find side length of AD and AD' using distance formula.

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

\text{Length of AD}=\sqrt{(1-1)^{2}+(4-7)^{2}}

\text{Length of AD}=\sqrt{(0)^{2}+(-3)^{2}}

\text{Length of AD}=\sqrt{9}=3

Now let us find length of side AD'.

\text{Length of AD'}=\sqrt{(1-1)^{2}+(5-7)^{2}}

\text{Length of AD'}=\sqrt{(0)^{2}+(-2)^{2}}

\text{Length of AD'}=\sqrt{4}=2

The scale factor (x) times the length of AD will be length of AD'.

3\cdot x=2

x=\frac{2}{3}

Therefore, dilation by a scale factor of 2/3 is applied to ABCD to create AB'C'D'.

Mathematics
Step-by-step answer
P Answered by Specialist

(x,y) = (2.5x,2.5y)

Step-by-step explanation:

Given

See attachment for ABCD and A'B'C'D'

Required

Determine the dilation rule

Using AB and A'B' as points of references;

AB = (-1,2)\ to\ (2,1)

A'B' = (-2.5,5)\ to\ (5,2.5)

The dilation factor (k) is calculated as:

k = \frac{A'B'}{AB}

This gives:

k = \frac{(-2.5,5)}{(-1,2)} \ or\ k = \frac{(5,2.5)}{(2,1)}

Factorize

k = \frac{2.5*(-1,2)}{(-1,2)} \ or\ k = \frac{2.5*(2,1)}{(2,1)}

k = 2.5 \ or\ k = 2.5

Hence, the scale factor is 2.5

The dilation rule is:

(x,y) = k(x,y)

(x,y) = 2.5(x,y)

(x,y) = (2.5x,2.5y)


2. Quadrilateral ABCD was dilated with the origin as the center of dilation to create quadrilateral
Mathematics
Step-by-step answer
P Answered by Specialist

(x,y) = (2.5x,2.5y)

Step-by-step explanation:

Given

See attachment for ABCD and A'B'C'D'

Required

Determine the dilation rule

Using AB and A'B' as points of references;

AB = (-1,2)\ to\ (2,1)

A'B' = (-2.5,5)\ to\ (5,2.5)

The dilation factor (k) is calculated as:

k = \frac{A'B'}{AB}

This gives:

k = \frac{(-2.5,5)}{(-1,2)} \ or\ k = \frac{(5,2.5)}{(2,1)}

Factorize

k = \frac{2.5*(-1,2)}{(-1,2)} \ or\ k = \frac{2.5*(2,1)}{(2,1)}

k = 2.5 \ or\ k = 2.5

Hence, the scale factor is 2.5

The dilation rule is:

(x,y) = k(x,y)

(x,y) = 2.5(x,y)

(x,y) = (2.5x,2.5y)


2. Quadrilateral ABCD was dilated with the origin as the center of dilation to create quadrilateral
Mathematics
Step-by-step answer
P Answered by PhD

9514 1404 393

  A, C, D, E

Step-by-step explanation:

Dilation does not change angles or directions. Every image point X' lies on the line containing the center of dilation and the pre-image point X. If the scale factor is other than 1, no image segment will be congruent to its pre-image segment. So, the following are true:

BC║B'C'AC ⊥ line C'D'D' and C have the same coordinatesC' lies on line OC
Mathematics
Step-by-step answer
P Answered by PhD

9514 1404 393

  A, C, D, E

Step-by-step explanation:

Dilation does not change angles or directions. Every image point X' lies on the line containing the center of dilation and the pre-image point X. If the scale factor is other than 1, no image segment will be congruent to its pre-image segment. So, the following are true:

BC║B'C'AC ⊥ line C'D'D' and C have the same coordinatesC' lies on line OC
Mathematics
Step-by-step answer
P Answered by Specialist

Option A , B and D are true.

The statement which are true:

The length of side AD is 4 units

The length of side A'D' is 8 units.

The scale factor is, \frac{1}{2}

Step-by-step explanation:

Given in figure trapezoid ABCD;

The coordinates of ABCD are:

A= (-4, 0)

B = (-2, 4)

C = (2,4)

D = (4, 0)

Since, trapezoid ABCD was dilated to create trapezoid A'B'C'D' as shown in figure;

The coordinates of A'B'C'D' are;

A' =(-2, 0)

B'=(-1, 2)

C' = (1, 2)

D' = (2, 0)

First calculate the length of AD

Using Distance formula for any two points i.e,

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

Since, A = (-4, 0) and D = (4, 0)

then;

Length of AD =  \sqrt{(4-(-4))^2+(0-0)^2}  = \sqrt{(4+4)^2} =\sqrt{64}=8 units

Therefore, the length of side AD is, 8 units.

Similarly find the length of A'D'.

Where A' = (-2, 0) and D' =(2,0)

Using distance formula:

Length of A'D' = \sqrt{(2-(-2))^2+(0-0)^2} =\sqrt{(2+2)^2}= \sqrt{4^2} = 4

Therefore, the length of side A'D' is, 4 units.

Now, find the slope of CD and C'D'

where C =(2, 4) , D = (4, 0) , C' = (1, 2) and D' =(2,0)

using slope formula for any two points is given by:

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

Slope of CD = \frac{0-4}{4-2} = \frac{-4}{2} = -2

Similarly,

Slope of C'D' = \frac{0-2}{2-1} = \frac{-4}{2} = -2

Since, Sides CD and C'D' have same slope i.e, -2

Scale factor(k) states that every coordinate of the original figure has been multiplied by the scale factor.

If k > 1, then the  image is an enlargement.if 0<k< 1, then the image is a reduction.If k = 1, then the figure and the image are congruent.

The rule for dilation with scale factor(k) is;

(x, y) \rightarrow (kx , ky)

To find the scale factor:

A = (-4, 0) and A' = (-2, 0)

(-2, 0) \rightarrow (-4k , 0)

On comparing we ghet;

-4k = -2

Divide -4 both sides we get;

k = \frac{1}{2}

∴ The Scale factor is, k = \frac{1}{2}

Since, k < 1 which implies the image is a reduction.

Therefore, the statements which are true regarding about trapezoids are;

The length of side AD is 4 units

The length of side A'D' is 8 units.

The scale factor is, \frac{1}{2}


Trapezoid abcd was dilated to create trapezoid a'b'c'd'. which statements are true about the trapezo

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