100 points and !

chanel says the sequence of transformations that maps trapezoid pqrs onto trapezoid klmn is: rotate it 90 degrees counterclockwise, and then dilate it by a scale factor of 1/2.

preston says the sequence of transformations that maps trapezoid pqrs onto trapezoid klmn is: reflect it across the x-axis, and then dilate it by a scale factor of 1/2.

part a
start with chanel’s sequence. how does a 90-degree counterclockwise rotation about the origin change the coordinates of a shape?

part b
complete the table to find the rule for the rotation, the coordinates of trapezoid pqrs, and the coordinates of the rotated image, trapezoid p′q′r′s′.
original coordinates new coordinates
(x, y) ( , )
p( , ) p′( , )
q( , ) q′( , )
r( , ) r′( , )
s( , ) s′( , )

part c
how does a dilation by a scale factor of 1/2 change the coordinates of a shape?

part d
complete the table to find the rule for the dilation, the coordinates of trapezoid p′q′r′s′, and the coordinates of the dilated image, trapezoid p″q″r″s″.
original coordinates new coordinates
(x, y) ( , )
p′( , ) p″( , )
q′( , ) q″( , )
r′( , ) r″( , )
s′( , ) s″( , )
part e
now, look at preston’s sequence. how does a reflection across the x-axis change the coordinates of a shape?

part f
complete the table to find the rule for the reflection, the coordinates of trapezoid pqrs, and the coordinates of the reflected image, trapezoid p′q′r′s′.
original coordinates new coordinates
(x, y) ( , )
p( , ) p′( , )
q( , ) q′( , )
r( , ) r′( , )
s( , ) s′( , )

part g
how does a dilation by a scale factor of 1/2 change the coordinates of a shape?

part h
complete the table to find the rule for the dilation, the coordinates of trapezoid p′q′r′s′, and the coordinates of the dilated image, trapezoid p″q″r″s″.
original coordinates new coordinates
(x, y) ( , )
p′( , ) p″( , )
q′( , ) q″( , )
r′( , ) r″( , )
s′( , ) s″( , )

part i
whose sequence proves trapezoid pqrs is similar to trapezoid klmn? why?

First please check the attached figure a, as you missed to add the figure. Hence, after a little research I was able to find the figure and I took that figure as a reference to answer your query, which anyways will clear your concept.

Tracking Chanel's Sequence:

Part (A) : Impact of rotating 90° counterclockwise about the origin  

This is a fact that any point let say (x, y) is rotated 90° counterclockwise about the origin by transforming the point to (-y, x).  

So, the trapezoid PQRS with coordinates P(6,-4), Q(2,-2), R(2,-6) and S(6,-6) after rotating 90° counterclockwise about the origin would be transformed or changed to P'(4,6), Q'(2,2), R'(6,2), and S'(6,6).  

Part (B) : Completing the table to determine the rule of rotating 90° counterclockwise about the origin  and filling the coordinate table

As the rule suggests that any point let say (x, y) is rotated 90° counterclockwise about the origin by transforming the point to (-y, x), meaning, the positions of (x, y) are switched and the sign of original y-coordinate i.e is changed to its opposite.

So, the coordinates PQRS and P'Q'R'S' behave like this after rotating 90° counterclockwise about the origin.

Original Coordinates                 New Coordinates

(x, y)                     ⇒                       (-y , x)

P(6,-4)                  ⇒                       P'(4, 6)

Q(2,-2)                 ⇒                       Q'(2, 2)

R(2, -6)                 ⇒                       R'(6, 2)

S(6, -6)                 ⇒                       S'(6, 6)

Part (C) : Impact of dilation on the coordinates of a shape by a scale factor of 1/2

It is an established fact that the coordinates are reduced by a factor of 1/2 if there is a dilation be a scale factor of 1/2.

Part (D) : Completing the table to determine the rule of of dilation on the coordinates of a shape by a scale factor of 1/2 and filling the coordinates

As the rule suggests that if any point let say (x, y) is dilated by a scale factor of 1/2, it would transform the point to P(1/2x, 1/2y).

Hence, following would be the coordinates after dilation by a scale factor of 1/2.

Original Coordinates                 New Coordinates

(x, y)                       ⇒                       (1/2x , 1/2y)

P'(4, 6)                   ⇒                        P"(2, 3)

Q'(2, 2)                  ⇒                        Q"(1, 1)

R'(6, 2)                   ⇒                        R"(3, 1)

S'(6, 6)                   ⇒                        S"(3, 3)

Tracking Preston's Sequence:

Part (E) : Impact of reflection across x-axis  on the coordinates of shape

This is a fact that when any point let say (x, y) is reflected across x-axis, it transforms or changes the point to (x, -y).

So, the trapezoid PQRS with coordinates P(6,-4), Q(2,-2), R(2,-6) and S(6,-6) after reflection across x-axis would be transformed or changed to P'(6, 4), Q'(2, 2), R'(2, 6), and S'(6,6).

Part (F) : Completing the table to determine the rule of reflection across x-axis  and filling the coordinate table

As the rule suggests that This is a fact that when any point let say (x, y) is reflected across x-axis, it transforms or changes the point to (x, -y), meaning, the the x coordinate of (x, y) remains the same but the sign of original y-coordinate i.e is changed to its opposite.

So, the coordinates PQRS and P'Q'R'S' behave like this after reflection across x-axis.

Original Coordinates                 New Coordinates

(x, y)                     ⇒                       (x , -y)

P(6,-4)                  ⇒                       P'(6, 4)

Q(2,-2)                 ⇒                       Q'(2, 2)

R(2, -6)                 ⇒                       R'(2, 6)

S(6, -6)                  ⇒                       S'(6, 6)

Part (G) : Impact of dilation on the coordinates of a shape by a scale factor of 1/2

It is an established fact that the coordinates are reduced by a factor of 1/2 if there is a dilation be a scale factor of 1/2.

Part (H) : Completing the table to determine the rule of of dilation on the coordinates of a shape by a scale factor of 1/2 and filling the coordinates

As the rule suggests that if any point let say (x, y) is dilated by a scale factor of 1/2, it would transform the point to P(1/2x, 1/2y).

Hence, following would be the coordinates after dilation by a scale factor of 1/2.

Original Coordinates                 New Coordinates

(x, y)                      ⇒                       (1/2x , 1/2y)  

P'(6, 4)                   ⇒                       P''(3, 2)

Q'(2, 2)                  ⇒                        Q''(1, 1)

R'(2, 6)                   ⇒                        R''(1, 3)

S'(6, 6)                   ⇒                        S''(3, 3)

Part (I) Comparing the Chanel's Sequence and Preston's Sequence answers with the final figure with the following coordinates:

K(3, 2)

L(1, 1)

M(1, 3)

N(3, 3)

If we check the attached figure a and compare the results of Chanel's Sequence and Preston's Sequence with the attached figure a. It is clear that Preston's sequence was right.

Hence, Preston is found correct as his sequence is a clear prove that trapezoid PQRS is similar to trapezoid KLMN, as shown in attached figure a. This was the result of sequence of reflection across x-axis  and dilation on the coordinates of a shape by a scale factor of 1/2. And this sequence was

Step-by-step explanation:

x = 2

step-by-step explanation:

if sr parallel to ba then the sides bc and ac are in proportion, that is

= ← substitute values

= ( cross- multiply )

27x = 6(4x + 1)

27x = 24x + 6 ( subtract 24x from both sides )

3x = 6 ( divide both sides by 3 )

x = 2

your answer is 125y5