A coordinate plane's x-axis ranges from 0 to 13 and its y-axis ranges from negative 5 to 0, both by 1-unit increments. Points upper A, upper B, upper A prime, and upper B prime are plotted forming 2 line segments, 1 solid and 1 dotted.

Question

Determine whether the dilation passes through the center of dilation.

(1 point)
Responses

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

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 cap A prime cap b prime does not pass through the center of dilation because it is a reduction of line segment cap A cap b by a scale factor of Start Fraction 1 over 4 End Fraction .

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

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

Answer 1

To determine whether the dilation passes through the center of dilation, we first need to understand the nature of dilations in a coordinate plane. A dilation involves transforming a figure by a scale factor (which can either enlarge or reduce it) around a fixed point called the center of dilation.

From the options provided, we can analyze each response to measure its validity:

1. First response options:

   • A′B′ does not pass through the center of dilation because it is taken to a line parallel to AB.
   • Modifying A′B′ does not pass through the center of dilation because it is taken to a line parallel to line segment AB.

   This response indicates that segment A′B′ is parallel to segment AB and does not intersect the center of dilation, suggesting it is simply an orientation change rather than a dilation through the center.

2. Second response options:

   • 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 A′B′ does not pass through the center of dilation because it is a reduction of line segment AB by a scale factor of \(\frac{1}{4}\).

   Here, this response suggests there is a scale involved, but without the specifics of which segment is being reduced or the correct scale factor (given as both 14 and \(\frac{1}{4}\)), this option lacks clarity.

3. Third response options:

   • A′B′ passes through the center of dilation because it is an enlargement of AB by a scale factor of 4.
   • Line segment A′B′ passes through the center of dilation because it is an enlargement of line segment AB by a scale factor of 4.

   This response indicates that there is a scale factor involved, which would imply a specific transformation based on a center of dilation.

4. Fourth response options:

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

   This is somewhat contradictory since if A′B′ is a parallel transformation, it would not normally pass through the same center of dilation as the original segment unless specifically defined in that way.

Based on this analysis, the most logical conclusion is that if A′B′ is derived from AB by a parallel transformation without intersecting a specific point of dilation, it indeed doesn't pass through it.

Therefore, the best choice among the options is:

Response 1: A′B′ does not pass through the center of dilation because it is taken to a line parallel to AB.

This considers the properties of dilations and how they relate to points and lines in geometric transformations.

Answer 2

A coordinate plane's x-axis ranges from negative 1 to 10 and its y-axis ranges from negative 8 to 2, both by 1-unit increments. Point upper C, point upper D, and a third unlabeled point are plotted and connected by lines.
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 (2,−1) and D′ is at (1,−2).
upper C prime is at open paren 2 comma negative 1 close paren and upper D prime is at left parenthesis 1 comma negative 2 right parenthesis .

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 (9,−18) and D′ is at (18,−9).

Answer 3

To determine which statement is part of the proof that the dilation of line segment \( CD \) with a scale factor of 13 and a center of dilation at the origin will result in a line segment that is 13 times the length of \( CD \), we need to review the properties of dilation.

When we dilate points \( C \) and \( D \) from the origin (0, 0) by a scale factor of 13, we will compute the new positions of \( C' \) and \( D' \):

If point \( C \) has coordinates \( (x_C, y_C) \), then after dilation, the coordinates of \( C' \) will be: \[ C' = (13 \cdot x_C, 13 \cdot y_C) \] Similarly, for point \( D \) with coordinates \( (x_D, y_D) \), the coordinates of \( D' \) will be: \[ D' = (13 \cdot x_D, 13 \cdot y_D) \]

Now, let's analyze the statements given in the options:

1. C′ is at (2,−1) and D′ is at (1,−2).
   This provides specific coordinates for points \( C' \) and \( D' \). However, without further information regarding the original coordinates of points \( C \) and \( D \), we cannot confirm if the dilation is correctly represented.

2. The length of C′D′ is approximately 1.41 units.
   If this is the length of the new line segment after dilation, it does not strongly support the idea that it is 13 times the length of \( CD \).

3. The length of C′D′ is approximately 4.24.
   If this represents the length of the dilated segment \( C'D' \), it would indicate that the original segment \( CD \) was approximately \( \frac{4.24}{13} \) units long before dilation. This is a potential candidate for the proof since it explicitly connects the dilation process with a specific length.

4. C′ is at (9,−18) and D′ is at (18,−9).
   Here, \( (9, -18) \) and \( (18, -9) \) are valid coordinates produced by dilation by a scale factor of 13 under certain original coordinates. If these are derived from valid original coordinates, this statement could also demonstrate the dilation process.

Among these options, the most relevant statement for verifying that the length of \( C'D' \) is indeed 13 times that of \( CD \) is:

The length of C′D′ is approximately 4.24.

This statement best reflects the conclusion that, due to the dilation process, the dilated segment is proportional in length to the original segment by the defined scale factor.

Answer 4

Given point A(2,2) and point T(6,4), find the coordinates of point B so that T is two-fifths of the way from A to B.(1 point)
Responses

B(10, 6)
upper B left parenthesis 10 comma 5 right parenthesis

B(12,7)
upper B left parenthesis 12 comma 7 right parenthesis

B(4, 3)
upper B left parenthesis 4 comma 2 right parenthesis

B(22,12

Answer 5

To find the coordinates of point B so that point T is two-fifths of the way from point A to B, we can apply the section formula.

Let's denote:

• Point A as \( A(2, 2) \)
• Point T as \( T(6, 4) \)
• Point B as \( B(x, y) \)

Since T is \(\frac{2}{5}\) of the way from A to B, the remaining distance will be \(\frac{3}{5}\).

Using the section formula in a weighted manner, T divides the line segment AB in the ratio \( m:n = 2:3 \).

The coordinates of point T, according to the section formula, are given by: \[ T = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

Substituting the points:

• \( A(x_1, y_1) = (2, 2) \)
• \( B(x_2, y_2) = (x, y) \)
• \( m = 2 \)
• \( n = 3 \)

Thus, for the x-coordinate of T: \[ 6 = \frac{2x + 3 \cdot 2}{2 + 3} \] \[ 6 = \frac{2x + 6}{5} \] Multiplying both sides by 5: \[ 30 = 2x + 6 \] Subtracting 6 from both sides: \[ 24 = 2x \] Dividing by 2: \[ x = 12 \]

For the y-coordinate of T: \[ 4 = \frac{2y + 3 \cdot 2}{2 + 3} \] \[ 4 = \frac{2y + 6}{5} \] Multiplying both sides by 5: \[ 20 = 2y + 6 \] Subtracting 6 from both sides: \[ 14 = 2y \] Dividing by 2: \[ y = 7 \]

Thus, the coordinates of point B are \( B(12, 7) \).

The answer is: B(12, 7).