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Parallelogram upper E upper F upper G upper H is graphed on a coordinate plane. The x-axis ranges from negative 8 to 8 in increments of 1. The y-axis ranges from negative 8 to 8 in increments of 1.

Suppose the parallelogram on the coordinate plane is dilated in reference to the origin using a magnitude of 3. What would be the coordinates of the new vertices of E′F′G′H′ ?

(1 point)
Responses

E′(−4,2), F′(−1,5), G′(1,1) and H′(−2,−2)
upper E prime left parenthesis negative 4 comma 2 right parenthesis , upper F prime left parenthesis negative 1 comma 5 right parenthesis , upper G prime left parenthesis 1 comma 1 right parenthesis and upper H prime left parenthesis negative 2 comma negative 2 right parenthesis

E′(−13,53), F′(23,83), G′(43,43) and H′(13,13)
upper E prime left parenthesis negative Start Fraction 1 over 3 End Fraction comma Start Fraction 5 over 3 End Fraction right parenthesis , upper F prime left parenthesis Start Fraction 2 over 3 End Fraction comma Start Fraction 8 over 3 End Fraction right parenthesis , upper G prime left parenthesis Start Fraction 4 over 3 End Fraction comma Start Fraction 4 over 3 End Fraction right parenthesis and upper H prime left parenthesis Start Fraction 1 over 3 End Fraction comma Start Fraction 1 over 3 End Fraction right parenthesis

E′(−3,15), F′(6,24), G′(12,12) and H′(3,3)
upper E prime left parenthesis negative 3 comma 15 right parenthesis , upper F prime left parenthesis 6 comma 24 right parenthesis , upper G prime left parenthesis 12 comma negative 12 right parenthesis and upper H prime left parenthesis 3 comma 3 right parenthesis

E′(2,8), F′(5,11), G′(7,7) and H′(4,4)

11 answers

To perform a dilation of a shape with respect to the origin, you multiply the coordinates of each vertex by the scale factor. Here, the scale factor (magnitude of dilation) is 3.

If the original coordinates of the vertices of the parallelogram are:

  • \( E(x_1, y_1) \)
  • \( F(x_2, y_2) \)
  • \( G(x_3, y_3) \)
  • \( H(x_4, y_4) \)

After dilation, the new coordinates will be:

  • \( E'(3x_1, 3y_1) \)
  • \( F'(3x_2, 3y_2) \)
  • \( G'(3x_3, 3y_3) \)
  • \( H'(3x_4, 3y_4) \)

Since we don't have the original coordinates (E, F, G, H) provided in your question, we cannot calculate the new coordinates directly. However, we can look at the answer choices you provided and check which one is consistent with dilation by a factor of 3 from a set of coordinates that might reasonably represent a parallelogram.

Here are the choices:

  1. E′(−4,2), F′(−1,5), G′(1,1) and H′(−2,−2)
  2. E′(−13,53), F′(23,83), G′(43,43) and H′(13,13)
  3. E′(−3,15), F′(6,24), G′(12,12) and H′(3,3)
  4. E′(2,8), F′(5,11), G′(7,7) and H′(4,4)

Looking at the proportions, the second choice seems to have extremely large values that likely wouldn't originate from an integer coordinate set within the bounds of (-8 to 8). The first set appears small, but relatively more consistent than the others given no visible pattern of dilation. The third option appears quite large and does not look appropriate for a simple dilation. The fourth option is moderate in size but not likely representative.

Given that there is no direct correspondence without known original coordinates, the task's completion hinges on the original coordinates. If you know the original coordinates, please provide them, and we can help find the new coordinates.

For the given answers with multiple choices, one might infer that without main coordinates or original references, answering accurately is complex.

Could you please confirm the initial coordinates of vertices E, F, G, H?

A coordinate plane's x-axis ranges from negative 1 to 8 and its y-axis ranges from negative 2 to 5, both by 1-unit increments. 3 points forming a triangle are plotted, labeled upper X, upper Y, and upper Z, respectively.
If you dilate triangle XYZ to form X′Y′Z′ using a magnitude of 2 and a center of dilation at (2,0) , what are the coordinates of the dilated triangle?

(1 point)
Responses

X′(6,4), Y′(10,0), Z′(11,4)
upper X prime left parenthesis 4 comma 8 right parenthesis , upper Y prime left parenthesis 12 comma 0 right parenthesis , upper Z prime left parenthesis 14 comma 4 right parenthesis

X′(2,2), Y′(4,0), Z′(4.5,1)
upper X prime left parenthesis negative 4 comma 4 right parenthesis , upper Y prime left parenthesis 0 comma 0 right parenthesis , upper Z prime left parenthesis 1 comma 2 right parenthesis

X′(2,8), Y′(10,0), Z′(12,4)
upper X prime left parenthesis negative 2 comma 8 right parenthesis , upper Y prime left parenthesis 6 comma 0 right parenthesis , upper Z prime left parenthesis 8 comma 4 right parenthesis

X′(2,6), Y′(8,0), Z′(9.5,3)

To dilate a point \( P(x, y) \) about a center of dilation \( C(a, b) \) with a magnitude of \( k \), you can use the following formula for the coordinates \( P' \) of the dilated point:

\[ P' = (a + k(x - a), b + k(y - b)) \]

Given the center of dilation \( C(2, 0) \) and a magnitude of dilation \( k = 2 \), you can apply this to each vertex \( X, Y, Z \) of triangle XYZ.

Assuming the original coordinates of points \( X, Y, Z \) are:

  • \( X(x_1, y_1) \)
  • \( Y(x_2, y_2) \)
  • \( Z(x_3, y_3) \)

We don't currently have these coordinates specified in your question. Let's analyze the provided responses and check which point sets would reasonably correlate with the diluted points considering the dilation from the center of \( (2, 0) \).

Let's analyze each option:

  1. X′(6,4), Y′(10,0), Z′(11,4)
  2. X′(4,8), Y′(12,0), Z′(14,4)
  3. X′(2,2), Y′(4,0), Z′(4.5,1)
  4. X′(2,8), Y′(10,0), Z′(12,4)
  5. X′(2,6), Y′(8,0), Z′(9.5,3)

Dilation Process:

Assuming original coordinates for \( X, Y, Z \), denote them generically for examination since we do not have specific values.

For example, let's say the following:

  • \( X(0, 2) \)
  • \( Y(2, 0) \)
  • \( Z(1, 1.5) \)

Applying the dilation:

  • Point X: \[ X' = (2 + 2(0 - 2), 0 + 2(2 - 0)) = (2 - 4, 0 + 4) = (-2, 4) \]
  • Point Y: \[ Y' = (2 + 2(2 - 2), 0 + 2(0 - 0)) = (2, 0) \]
  • Point Z: \[ Z' = (2 + 2(1 - 2), 0 + 2(1.5 - 0)) = (2 - 2, 0 + 3) = (0, 3) \]

So a possible dilation can generate results reflecting the provided points. We need the original alternatives to determine corresponding coordinates.

Matching the Responses:

  • Possible response with X′(2, 8) needs consistent values to match; none directly line up without original values.
  • X′(2, 2) could be derived from low original values, and some responses top off at high coordinates (like (10, 0) with arbitrary jumps).

Conclusion:

The original coordinates are necessary to resolve what the dilated representation looks like truly. Whenever you've confirmed the original coordinates (or approximated ones), you'll then apply the formulas provided above accordingly to yield the resultant dilated coordinates and check them against the choices provided.

If you discover or wish to clarify the coordinates of \( X, Y, Z \), I am here to help with specific calculations as needed!

What type of transformation uses a scale factor to change the size of a figure?(1 point)
Responses

a dilation
a dilation

a translation
a translation

a rotation
a rotation

a reflection

The correct answer is a dilation.

A dilation is a transformation that changes the size of a figure by a scale factor while maintaining the shape and proportionality of the figure. Thus, it enlarges or reduces the figure based on the specified scale factor.

Translations, rotations, and reflections do not change the size of the figure. Translations move the figure without changing its size or shape, rotations turn the figure around a point, and reflections flip the figure over a line.

Triangle F G H and triangle L O W are graphed on a coordinate plane. Both the axes range from negative 6 to 6 in unit increments.
Which response explains why ΔLOW is similar to ΔFGH ?

(1 point)
Responses

Because ΔFGH is a translated form of ΔLOW, it is congruent due to SAS Similarity.
Because triangle upper F upper G upper H is a translated form of triangle upper L upper O upper W , it is congruent due to SAS Similarity.

Because ΔFGH results from similarity transformations applied to ΔLOW, they are congruent due to Side-Side-Angle Similarity.
Because triangle upper F upper G upper H results from similarity transformations applied to triangle upper L upper O upper W , they are congruent due to Side-Side-Angle Similarity.

Because ΔFGH results from similarity transformations applied to ΔLOW, they are similar due to Angle-Angle Similarity.
Because triangle upper F upper G upper H results from similarity transformations applied to triangle upper L upper O upper W , they are similar due to Angle-Angle Similarity.

ΔFGH is similar to ΔLOW because it is a 90-degree rotation.

The correct response that explains why \(\triangle LOW\) is similar to \(\triangle FGH\) is:

"Because ΔFGH results from similarity transformations applied to ΔLOW, they are similar due to Angle-Angle Similarity."

Explanation:

  • Similarity transformations include translations, rotations, reflections, and dilations. These transformations can change the position or orientation of a shape but do not affect its angles or the proportionality of the sides.
  • If two triangles have corresponding angles that are equal, then the triangles are similar. This is the basis of Angle-Angle (AA) similarity.
  • The other options either confuse congruence with similarity or incorrectly state the nature of the transformations involved.

Therefore, the response mentioning "similarity transformations" and "Angle-Angle Similarity" correctly reflects the relationship between the two triangles.

Triangles upper A upper B upper C and upper C upper M upper N are connected at a single point: vertex upper C. The interior angles by vertices upper B and upper C are marked by arcs given as 27 and 73 degrees, respectively.
Given are two right triangles, △ABC and △MNC , with ∠B=27° and ∠MCN=73° . Are the two triangles similar? Which of the following is a correct statement about △ABC and △MNC ?

(1 point)
Responses

No, △ABC and △MNC are not similar because ∠B is not equal to ∠MCN.
No, triangle upper A upper B upper C and triangle upper M upper N upper C are not similar because angle upper B is not equal to angle upper M upper C upper N .

No, △ABC and △MNC are not similar because ∠B is not equal to ∠N.
No, triangle upper A upper B upper C and triangle upper M upper N upper C are not similar because angle upper B is not equal to angle upper N .

Yes, △ABC and △MNC are similar because both have a right angle.
Yes, triangle upper A upper B upper C and triangle upper M upper N upper C are similar because both have a right angle.

Yes, △ABC and △MNC are similar because ∠B is equal to ∠MCN.

To determine whether triangles \( \triangle ABC \) and \( \triangle MNC \) are similar, we can use the Angle-Angle (AA) similarity criterion for triangles.

  1. Given Angles:

    • \( \angle B = 27^\circ \)
    • \( \angle MCN = 73^\circ \)
    • Since both triangles share the vertex \( C \), \( \angle ACB \) is common to both triangles.

  2. Determining the Other Angles:

    • Since \( \triangle ABC \) is a right triangle, it has a right angle at vertex \( A \), making \( \angle A = 90^\circ \).
    • For triangle \( MNC \), we can find \( \angle N \) using the triangle sum property: \[ \angle MNC + \angle MCN + \angle NCM = 180^\circ \] Given \( \angle MCN = 73^\circ \) and that \( \angle NMC = 90^\circ \) (because it's a right triangle), \[ \angle N + 73 + 90 = 180 \] \[ \angle N + 163 = 180 \implies \angle N = 17^\circ \]

  3. Comparing Angles:

    • In \( \triangle ABC \):
      • The angles are \( \angle A = 90^\circ \), \( \angle B = 27^\circ \), and the third angle \( \angle C(ABC) = 63^\circ \) (since \( 90 + 27 + 63 = 180 \)).
    • In \( \triangle MNC \):
      • The angles are \( \angle N = 17^\circ \), \( \angle MCN = 73^\circ \), and \( \angle M = 90^\circ \).

Conclusion: Since \( \angle B (27^\circ) \) is not equal to \( \angle MCN (73^\circ) \) and the third angles are not equal.

Based on this analysis, the correct statement about \( \triangle ABC \) and \( \triangle MNC \) is:

"No, \( \triangle ABC \) and \( \triangle MNC \) are not similar because \( \angle B \) is not equal to \( \angle MCN \)."

Thus, you should select the statement that accurately reflects this.

Which of the properties of similarity transformations help to establish the SAS (Side-Angle-Side) and SSS (Side-Side-Side) Similarity Theorems?(1 point)
Responses

Translations preserve angle measures and produce congruent sides.
Translations preserve angle measures and produce congruent sides.

Dilations preserve angle measures and create congruent sides.
Dilations preserve angle measures and create congruent sides.

Rotations preserve angle measures and create proportional side lengths.
Rotations preserve angle measures and create proportional side lengths.

Dilations preserve angle measures and create proportional side lengths.

The correct response regarding the properties of similarity transformations that help to establish the SAS (Side-Angle-Side) and SSS (Side-Side-Side) Similarity Theorems is:

"Dilations preserve angle measures and create proportional side lengths."

Explanation:

  1. Dilations: When a figure is dilated:

    • The angles of the figure remain unchanged (i.e., the angle measures are preserved).
    • The sides are resized proportionally, meaning the lengths of the sides change but maintain the same ratios.

  2. SAS Similarity Theorem: States that if two triangles have one pair of corresponding sides that are proportional and the included angles are equal, then the triangles are similar.

  3. SSS Similarity Theorem: States that if the corresponding sides of two triangles are in proportion, then the triangles are similar.

Thus, dilations are the transformations that support these similarity theorems through the preservation of angle measures and proportionality of side lengths.