Question

A coordinate plane shows a translation of a line segment. The original line segment is AB. The coordinates of the endpoints are A at (4, negative 2) and B at (12, negative 4). The translated line segment is A prime B prime. The coordinates of the endpoints are A prime at (1, 0) and B prime at (3, negative 1). The translation appears to be 3 units to the left and 2 units up.


Image Long DescriptionThe graph displays two distinct line segments plotted on a coordinate plane. The x-axis ranges from 0 to 13, while the y-axis spans from negative 5 to 0. The first line segment, labeled Upper A Upper B, begins at point Upper A with coordinates left parenthesis 4 comma negative 2 right parenthesis and extends to point Upper B at left parenthesis 12 comma negative 4 right parenthesis. This segment slopes downward as it moves from left to right. The second line segment, labeled Upper A prime Upper B prime, starts at point Upper A prime with coordinates left parenthesis 1 comma negative 0.5 right parenthesis and ends at point Upper B prime at left parenthesis 3 comma negative 1 right parenthesis. This segment also slopes downward but is shorter and less steep compared to segment Upper A Upper B.

Determine whether the dilation passes through the center of dilation, if the center of dilation is the origin.

(1 point)
Responses

A′B′¯¯¯¯¯¯¯¯¯¯
does not pass through the center of dilation because it is a reduction of AB¯¯¯¯¯¯¯¯
by a scale factor of 14
.
line segment upper A prime upper b prime does not pass through the center of dilation because it is a reduction of line segment upper A upper b by a scale factor of 1 over 4 .

A′B′¯¯¯¯¯¯¯¯¯¯
passes through the center of dilation because it is an enlargement of AB¯¯¯¯¯¯¯¯
by a scale factor of 4.
line segment upper A prime upper b prime passes through the center of dilation because it is an enlargement of line segment upper A upper b by a scale factor of 4.

A′B′¯¯¯¯¯¯¯¯¯¯
does not pass through the center of dilation because it is taken to a line parallel to AB¯¯¯¯¯¯¯¯
.
line segment upper a prime upper b prime does not pass through the center of dilation because it is taken to a line parallel to line segment upper A upper b .

A′B′¯¯¯¯¯¯¯¯¯¯
passes through the center of dilation because it is taken to a line parallel to AB¯¯¯¯¯¯¯¯


The coordinates of points upper C and upper D are as follows: upper C at left parenthesis 3 comma negative 6 right parenthesis and upper D at left parenthesis 6 comma negative 3 right parenthesis. The third unlabeled point is located at the origin. The line connecting upper C with upper D is solid. The line connecting upper C with the origin point is dotted and extends past both points. The line connecting upper D with the origin point is a hybrid solid and dotted line that extends past both points.


Which statement is part of the proof that the dilation of CD¯¯¯¯¯¯¯¯
with a scale factor of 13
and a center of dilation at the origin will result in a line segment that is 13
the length of CD¯¯¯¯¯¯¯¯
?

(1 point)
Responses

C′
is at (9,−18)
and D′
is at (18,−9)
.
upper C prime is at open paren 9 comma negative 18 close paren and upper D prime is at open paren 18 comma negative 9 close paren .

The length of C′D′¯¯¯¯¯¯¯¯¯¯¯
is approximately 1.41 units.
The length of line segment cap c prime cap d prime is approximately 1.41 units.

The length of C′D′¯¯¯¯¯¯¯¯¯¯¯
is approximately 4.24.
The length of line segment cap c prime cap d prime is approximately 4.24.

C′
is at (2,−1)
and D′
is at (1,−2)
.
Point upper A has coordinates left parenthesis negative 3 comma 1 right parenthesis. Point upper B has coordinates left parenthesis negative 8 comma 5 right parenthesis.


Find the coordinates of point X
on segment AB¯¯¯¯¯¯¯¯
that partitions the segment into the ratio AX:BX = 1:2
.

(1 point)
Responses

X(−413,−10)
X left parenthesis negative 4 1 over 3 comma negative 10 right parenthesis

X(−313,83)
X left parenthesis negative 3 1 over 3 comma 8 over 3 right parenthesis

X(−423,213)
X left parenthesis negative 4 2 over 3 comma 2 1 over 3 right parenthesis

X(−1013,5)
X left parenthesis negative 10 1 over 3 comma 5 right parenthesis

Answer 1

Answers with brief reasons:

1) A′B′ does NOT pass through the center (the origin). Reason: the image segment is not on the same rays from the origin as A and B (so it is not a dilation about the origin); in the diagram A′B′ is the translated/parallel copy of AB. (Select the choice saying A′B′ does not pass through the center because it is taken to a line parallel to AB.)

2) The statement needed in the proof is the coordinates of the dilated points. Under a scale factor 1/3 about the origin (the intended meaning of “13” is 1/3), C′ = (1, −2) and D′ = (2, −1). (This identification of C′ and D′ is part of the proof that the image length is 1/3 of CD.)

3) X divides AB in ratio AX:XB = 1:2, so X = A + (1/3)(B − A) = (−14/3, 7/3) = (−4 2/3, 2 1/3).