Question

Use the image to answer the question.

A coordinate plane's axes range from 0 to 5, both by 1-unit increments. Solid triangle upper A upper B upper C and dotted triangle upper A prime upper B prime upper C prime are plotted.

Image Long DescriptionTriangle upper A prime upper B prime upper C prime is enclosed within triangle upper A upper B upper C. Triangle upper A upper B upper C has vertices located as follows: upper A at the origin, upper B at left parenthesis 2 comma 4 right parenthesis, and upper C at left parenthesis 4 comma 2 right parenthesis. Triangle upper A prime upper B prime upper C prime has vertices located as follows: upper A prime at the origin, upper B prime at left parenthesis 1 comma 2 right parenthesis, and upper C prime at left parenthesis 2 comma 1 right parenthesis.

Do the dilations of AB¯¯¯¯¯¯¯¯
, BC¯¯¯¯¯¯¯¯
, and AC¯¯¯¯¯¯¯¯
pass through the center of dilation if the center of dilation is the origin?

(1 point)
Responses

A′B′¯¯¯¯¯¯¯¯¯¯
, B′C′¯¯¯¯¯¯¯¯¯¯¯
, and A′C′¯¯¯¯¯¯¯¯¯¯¯
all pass through the center of dilation because ΔABC
is an enlargement of ΔA′B′C′
by a scale factor of 2 and the center of dilation is at (1.5, 1.5)
.
line segment cap A prime cap b prime , line segment cap b prime cap c prime , and line segment cap A prime cap c prime all pass through the center of dilation because triangle upper A upper B upper C is an enlargement of triangle upper A prime upper B prime upper C prime by a scale factor of 2 and the center of dilation is at open paren 1 point 5 comma space 1 point 5 close paren .

A′B′¯¯¯¯¯¯¯¯¯¯
and A′C′¯¯¯¯¯¯¯¯¯¯¯
do not pass through the center of dilation because they are on the same line as AB¯¯¯¯¯¯¯¯
and AC¯¯¯¯¯¯¯¯
in the preimage, and B′C′¯¯¯¯¯¯¯¯¯¯¯
does pass through the center of dilation because it is taken to a parallel line.
line segment cap A prime cap b prime and line segment cap A prime cap c prime do not pass through the center of dilation because they are on the same line as line segment cap A cap b and line segment cap A cap c in the preimage, and line segment cap b prime cap c prime does pass through the center of dilation because it is taken to a parallel line.

A′B′¯¯¯¯¯¯¯¯¯¯
, B′C′¯¯¯¯¯¯¯¯¯¯¯
, and A′C′¯¯¯¯¯¯¯¯¯¯¯
do not pass through the center of dilation because ΔABC
is an enlargement of ΔA′B′C′
by a scale factor of 2 and the center of dilation is at (1.5, 1.5)
.
line segment cap A prime cap b prime , line segment cap b prime cap c prime , and line segment cap A prime cap c prime do not pass through the center of dilation because triangle upper A upper B upper C is an enlargement of triangle upper A prime upper B prime upper C prime by a scale factor of 2 and the center of dilation is at open paren 1 point 5 comma space 1 point 5 close paren .

A′B′¯¯¯¯¯¯¯¯¯¯
and A′C′¯¯¯¯¯¯¯¯¯¯¯
pass through the center of dilation because they are on the same line as AB¯¯¯¯¯¯¯¯
and AC¯¯¯¯¯¯¯¯
in the preimage, and B′C′¯¯¯¯¯¯¯¯¯¯¯
does not pass through the center of dilation because it is taken to a parallel line.
line segment cap A prime cap b prime and line segment cap A prime cap c prime pass through the center of dilation because they are on the same line as line segment cap A cap b and line segment cap A cap c in the preimage, and line segment cap b prime cap c prime does not pass through the center of dilation because it is taken to a parallel line.
Remaining Attempts : 3

Answer 1

A′B′ and A′C′ pass through the center (the origin); B′C′ does not.

Reason: A and A′ are at the origin, so the segments from the origin to B′ and C′ (A′B′ and A′C′) lie on rays through the center. B′C′ (line y = −x + 3) does not pass through (0,0).