Question

Use the image to answer the question.

A coordinate plane's axes range from negative 8 to 0, both by 1-unit increments. Triangle upper X upper Y upper Z and triangle upper X prime upper Y prime upper Z prime are plotted.
▼Click for Long Desc
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

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.

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¯¯¯¯¯¯¯¯
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.
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 upper Y upper Z is a dilation of triangle upper X prime upper Y prime upper Z prime by a scale factor of 3.
Skip to navigation
























































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

Answer 1

The first statement.

Reason: XY is the vertical line x = −3 and X′Y′ is the vertical line x = −1 (both slopes undefined). Neither line contains the origin, so the side and its image are parallel lines that do not pass through the center (0,0). Hence the dilation does not pass through the origin.