Answer:
<em>Preston is found correct as his sequence is a clear prove that trapezoid PQRS is similar to trapezoid KLMN, as shown in </em><em>attached figure a</em><em>.</em>
Step-by-step explanation:
<em>First please check the </em><em>attached figure a</em><em>, 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.</em>
<em>Tracking Chanel's Sequence:</em>
<em>Part (A) : Impact of rotating 90° counterclockwise about the origin</em>
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).
<em>Part (B) : Completing the table to determine the rule of rotating 90° counterclockwise about the origin and filling the coordinate table</em>
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.
<em>Original Coordinates </em> <em>New Coordinates</em>
(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)
<em>Part (C) : Impact of dilation on the coordinates of a shape by a scale factor of 1/2</em><em> </em>
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.
<em>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 </em>
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.
<em>Original Coordinates </em> <em>New Coordinates</em>
(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)
<em>Tracking Preston's Sequence:</em>
<em>Part (E) : Impact of reflection across x-axis on the coordinates of shape</em>
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).
<em>Part (F) : Completing the table to determine the rule of reflection across x-axis and filling the coordinate table</em>
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.
<em>Original Coordinates </em> <em>New Coordinates</em>
(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)
<em>Part (G) : Impact of dilation on the coordinates of a shape by a scale factor of 1/2</em><em> </em>
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.
<em>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 </em>
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.
<em>Original Coordinates </em> <em>New Coordinates</em>
(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)
<em>Part (I) Comparing the Chanel's Sequence and Preston's Sequence answers with the final figure with the following coordinates:</em>
<em>K(3, 2)</em>
<em>L(1, 1)</em>
<em>M(1, 3)</em>
<em>N(3, 3)</em>
<em>If we check the </em><em>attached figure a</em><em> and compare the results of Chanel's Sequence and Preston's Sequence with the </em><em>attached figure a</em><em>. It is clear that Preston's sequence was right.</em>
<em> </em><em>Hence, Preston is found correct as his sequence is a clear prove that trapezoid PQRS is similar to trapezoid KLMN, as shown in </em><em>attached figure a. </em><em>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 suggested by Preston.</em>
<em>Keywords: reflection, rotation, dilation, trapezoid</em>
<em>Learn more</em><em> about translation, reflection, dilation and trapezoid from brainly.com/question/7287774</em>
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