Question

Triangle upper X upper Y upper Z has vertices located as follows: upper X at left parenthesis negative 3 comma negative 3 right parenthesis, upper Y at left parenthesis negative 3 comma negative 6 right parenthesis, and upper Z at left parenthesis negative 6 comma negative 3 right parenthesis. Triangle upper X prime upper Y prime upper Z prime has vertices located as follows: upper X prime at left parenthesis negative 1 comma negative 1 right parenthesis, upper Y prime at left parenthesis negative 1 comma negative 2 right parenthesis, and upper Z prime at left parenthesis negative 2 comma negative 1 right parenthesis.


Image Long DescriptionTriangle upper X upper Y upper Z has vertices located as follows: upper X at left parenthesis negative 3 comma negative 3 right parenthesis, upper Y at left parenthesis negative 3 comma negative 6 right parenthesis, and upper Z at left parenthesis negative 6 comma negative 3 right parenthesis. Triangle upper X prime upper Y prime upper Z prime has vertices located as follows: upper X prime at left parenthesis negative 1 comma negative 1 right parenthesis, upper Y prime at left parenthesis negative 1 comma negative 2 right parenthesis, and upper Z prime at left parenthesis negative 2 comma negative 1 right parenthesis.

Which statement proves that the dilation of ΔXYZ
does not pass through the center of dilation if the center of dilation is the origin?

(1 point)
Responses

X′(−3⋅13, −3⋅13)=X′(−1, −1)
, Y′(−3⋅13, −6⋅13)=Y′(−1, −2)
, Z′(−6⋅13, −3⋅13)=Z′(−2, −1)
.
upper X prime left parenthesis negative 3 times Start Fraction 1 over 3 End Fraction comma space negative 3 times Start Fraction 1 over 3 End Fraction right parenthesis equals upper X prime left parenthesis negative 1 comma space negative 1 right parenthesis , upper Y prime left parenthesis negative 3 times Start Fraction 1 over 3 End Fraction comma space negative 6 times Start Fraction 1 over 3 End Fraction right parenthesis equals upper Y prime left parenthesis negative 1 comma space negative 2 right parenthesis , upper Z prime left parenthesis negative 6 times Start Fraction 1 over 3 End Fraction comma space negative 3 times Start Fraction 1 over 3 End Fraction right parenthesis equals upper Z prime left parenthesis negative 2 comma space negative 1 right parenthesis .

XY¯¯¯¯¯¯¯¯
has a length of 3 and X′Y′¯¯¯¯¯¯¯¯¯¯¯¯
has a length of 1, which proves that ΔX′Y′Z′
is a dilation of ΔXYZ
by a scale factor of 13
.
Modifying above upper X upper Y with bar has a length of 3 and Modifying above upper X prime upper Y prime with bar has a length of 1, which proves that triangle upper X prime upper Y prime upper Z prime is a dilation of triangle upper X upper Y upper Z by a scale factor of Start Fraction 1 over 3 End Fraction .

XY¯¯¯¯¯¯¯¯
and X′Y′¯¯¯¯¯¯¯¯¯¯¯¯
both have an undefined slope with a center of dilation at (0, 0)
. This proves that the lines are parallel and that the dilation does not pass through the center of dilation.
Modifying above upper X upper Y with bar and Modifying above upper X prime upper Y prime with bar both have an undefined slope with a center of dilation at left parenthesis 0 comma space 0 right parenthesis . This proves that the lines are parallel and that the dilation does not pass through the center of dilation.

XY¯¯¯¯¯¯¯¯
has a length of 3 and X′Y′¯¯¯¯¯¯¯¯¯¯¯¯
has a length of 1, which proves that ΔXYZ
is a dilation of ΔX′Y′Z′
by a scale factor of 3.

Answer 1

Choose the statement that says: "XY and X′Y′ both have an undefined slope with a center of dilation at (0, 0). This proves that the lines are parallel and that the dilation does not pass through the center of dilation."

Reason: XY (x = −3) and X′Y′ (x = −1) are vertical (undefined slope), are parallel, and neither contains (0,0), so the image line does not pass through the origin.