Question
Question 1
A)
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.
Do the dilations of AB¯¯¯¯¯¯¯¯, BC¯¯¯¯¯¯¯¯, and AC¯¯¯¯¯¯¯¯ pass through the center of dilation?
(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 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.
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 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.
Question 2
A)
Use the image to answer the question.
A coordinate plane's x-axis ranges from negative 10 to 0 and its y-axis ranges from 0 to 10, both by 1-unit increments. Point upper X, point upper Y, and a third unlabeled point are plotted and connected by lines.
XY¯¯¯¯¯¯¯¯ is dilated with a scale factor of 2 and a center of dilation at the origin. Find points X′ and Y′ to prove that a dilation of 2 has a length that is twice the length of XY¯¯¯¯¯¯¯¯.
(1 point)
Responses
X′ is at (−1.5,2) and Y′ is at (−3.5,1).
upper X prime is at left parenthesis negative 1.5 comma 2 right parenthesis and upper Y prime is at left parenthesis negative 3.5 comma 1 right parenthesis .
X′ is at (6,−8) and Y′ is at (14,−4).
upper X prime is at left parenthesis 6 comma negative 8 right parenthesis and upper Y prime is at left parenthesis 14 comma negative 4 right parenthesis .
X′ is at (−6,8) and Y′ is at (−14,4).
upper X prime is at left parenthesis negative 6 comma 8 right parenthesis and upper Y prime is at left parenthesis negative 14 comma 4 right parenthesis .
X′ is at (−4,1) and Y′ is at (−12,5).
upper X prime is at left parenthesis negative 4 comma 1 right parenthesis and upper Y prime is at left parenthesis negative 12 comma 5 right parenthesis .
Question 3
A)
Use the image to answer the question.
Segment upper A upper B is graphed on a coordinate plane. The x-axis ranges from negative 3 to1 in increments of 1. The y-axis ranges from negative 1 to 4 in increments of 1.
Determine the location of point L if it divides AB¯¯¯¯¯¯¯¯ into two segments in the ratio 1:3.
(1 point)
Responses
(14,14)
left parenthesis Start Fraction 1 over 4 End Fraction comma Start Fraction 1 over 4 End Fraction right parenthesis
(1,−32)
left parenthesis 1 comma negative Start Fraction 3 over 2 End Fraction right parenthesis
(−12,34)
left parenthesis negative Start Fraction 1 over 2 End Fraction comma Start Fraction 3 over 4 End Fraction right parenthesis
(−23,1)
left parenthesis negative Start Fraction 2 over 3 End Fraction comma 1 right parenthesis
Question 4
A)
Use the image to answer the question.
Polygon upper A upper B upper C upper D is graphed on a coordinate plane. The x-axis ranges from negative 10 to 10 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Paul is trying to dilate the figure in the image using a magnitude of 2 with reference to the origin. At what points would he draw the new vertices A′B′C′?
(1 point)
Responses
A′(−1,5), B′(3,7), C′(6,1), and D′(3,−3)
upper A prime left parenthesis negative 1 comma 5 right parenthesis , upper B prime left parenthesis 3 comma 7 right parenthesis , upper C prime left parenthesis 6 comma 1 right parenthesis , and upper D prime left parenthesis 3 comma negative 3 right parenthesis
A′(−32,32), B′(12,52), C′(2,−12) and D′(12,−52)
upper A prime left parenthesis negative Start Fraction 3 over 2 End Fraction comma Start Fraction 3 over 2 End Fraction right parenthesis , upper A prime left parenthesis Start Fraction 1 over 2 End Fraction comma Start Fraction 5 over 2 End Fraction right parenthesis , upper C prime left parenthesis 2 comma negative Start Fraction 1 over 2 End Fraction right parenthesis and upper D prime left parenthesis Start Fraction 1 over 2 End Fraction comma negative Start Fraction 5 over 2 End Fraction right parenthesis
A′(−5,1), B′(−1,3), C′(2,−3) and D′(−1,−7)
upper A prime left parenthesis negative 5 comma 1 right parenthesis , upper B prime left parenthesis negative 1 comma 3 right parenthesis , upper C prime left parenthesis 2 comma negative 3 right parenthesis and upper D prime left parenthesis negative 1 comma negative 7 right parenthesis
A′(−6,6), B′(2,10), C′(8,−2) and D′(2,−10)
upper A prime left parenthesis negative 6 comma 6 right parenthesis , upper B prime left parenthesis 2 comma 10 right parenthesis , upper C prime left parenthesis 8 comma negative 2 right parenthesis and upper D prime left parenthesis 2 comma negative 10 right parenthesis
Question 5
A)
Use the image to answer the question.
A coordinate plane's x-axis ranges from negative 1 to 6 and its y-axis ranges from negative 1 to 8, both by 1-unit increments. 3 points forming a triangle are plotted, labeled upper A, upper B, and upper C, respectively.
Dilate the triangle using a magnitude of 12 and a center of dilation at (2,2) to form triangle A′B′C′. What are the coordinates of the dilated triangle?
(1 point)
Responses
A′(7,7), B′(3,9), C′(5,4)
upper A prime left parenthesis 8 comma 7 right parenthesis , upper B prime left parenthesis 4 comma 9 right parenthesis , upper C prime left parenthesis 6 comma 4 right parenthesis
A′(8,8), B′(0,12), C′(4,2)
upper A prime left parenthesis 2 comma 3 right parenthesis , upper B prime left parenthesis negative 2 comma 5 right parenthesis , upper C prime left parenthesis 0 comma 0 right parenthesis
A′(2.5,2.5), B′(0.5,3.5), C′(1.5,1)
upper A prime left parenthesis 2.5 comma 2.5 right parenthesis , upper B prime left parenthesis 0.5 comma 3.5 right parenthesis , upper C prime left parenthesis 1.5 comma 1 right parenthesis
A′(3.5,3.5), B′(1.5,4.5), C′(2.5,2)
upper A prime left parenthesis 4 comma 3.5 right parenthesis , upper B prime left parenthesis 2 comma 4.5 right parenthesis , upper C prime left parenthesis 3 comma 2 right parenthesis
Question 6
A)
Use the image to answer the question.
A coordinate plane's x-axis ranges from negative 1 to 6 and its y-axis ranges from negative 2 to 3, both by 1-unit increments. 2 triangles with bases on the x-axis are drawn. 1 is solid and 1 is dotted.
Explain how you could use the SAS criterion to determine that the triangles are similar.
(1 point)
Responses
Use translation by sliding triangle ABC to a corresponding angle in triangle A′B′C′ and check for similarity.
Use translation by sliding triangle upper A upper B upper C to a corresponding angle in triangle upper A prime upper B prime upper C prime and check for similarity.
Find the scale factor by comparing angles of each triangle and check for proportionality.
Find the scale factor by comparing angles of each triangle and check for proportionality.
Determine the two side lengths on either side of the shared ordered pair at point A for both triangles and then compare for proportionality.
Determine the two side lengths on either side of the shared ordered pair at point upper A for both triangles and then compare for proportionality.
Reflect the triangle across the y-axis and check for symmetry.
Reflect the triangle across the y -axis and check for symmetry.
Question 7
A)How are dilations of polygons using a scale factor not equal to 1 different from translations, reflections, or rotations of polygons?(1 point)
Responses
They do not preserve angle measures of corresponding vertices.
They do not preserve angle measures of corresponding vertices.
They produce an image that is congruent to the preimage.
They produce an image that is congruent to the preimage.
They do not preserve distance between corresponding vertices.
They do not preserve distance between corresponding vertices.
They do not preserve shape.
They do not preserve shape.
Question 8
A)Suppose a preimage is dilated by a scale factor of 3 and rotated 270°
counterclockwise. Is the resulting image similar to the preimage?(1 point)
Responses
No because rotations preserve neither angle measure nor segment length.
No because rotations preserve neither angle measure nor segment length.
Yes because dilations preserve angle measure, but not segment length.
Yes because dilations preserve angle measure, but not segment length.
No because dilations do not preserve angle measure, but do preserve segment length.
No because dilations do not preserve angle measure, but do preserve segment length.
Yes because rotations preserve angle measure, but not segment length.
A)
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.
Do the dilations of AB¯¯¯¯¯¯¯¯, BC¯¯¯¯¯¯¯¯, and AC¯¯¯¯¯¯¯¯ pass through the center of dilation?
(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 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.
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 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.
Question 2
A)
Use the image to answer the question.
A coordinate plane's x-axis ranges from negative 10 to 0 and its y-axis ranges from 0 to 10, both by 1-unit increments. Point upper X, point upper Y, and a third unlabeled point are plotted and connected by lines.
XY¯¯¯¯¯¯¯¯ is dilated with a scale factor of 2 and a center of dilation at the origin. Find points X′ and Y′ to prove that a dilation of 2 has a length that is twice the length of XY¯¯¯¯¯¯¯¯.
(1 point)
Responses
X′ is at (−1.5,2) and Y′ is at (−3.5,1).
upper X prime is at left parenthesis negative 1.5 comma 2 right parenthesis and upper Y prime is at left parenthesis negative 3.5 comma 1 right parenthesis .
X′ is at (6,−8) and Y′ is at (14,−4).
upper X prime is at left parenthesis 6 comma negative 8 right parenthesis and upper Y prime is at left parenthesis 14 comma negative 4 right parenthesis .
X′ is at (−6,8) and Y′ is at (−14,4).
upper X prime is at left parenthesis negative 6 comma 8 right parenthesis and upper Y prime is at left parenthesis negative 14 comma 4 right parenthesis .
X′ is at (−4,1) and Y′ is at (−12,5).
upper X prime is at left parenthesis negative 4 comma 1 right parenthesis and upper Y prime is at left parenthesis negative 12 comma 5 right parenthesis .
Question 3
A)
Use the image to answer the question.
Segment upper A upper B is graphed on a coordinate plane. The x-axis ranges from negative 3 to1 in increments of 1. The y-axis ranges from negative 1 to 4 in increments of 1.
Determine the location of point L if it divides AB¯¯¯¯¯¯¯¯ into two segments in the ratio 1:3.
(1 point)
Responses
(14,14)
left parenthesis Start Fraction 1 over 4 End Fraction comma Start Fraction 1 over 4 End Fraction right parenthesis
(1,−32)
left parenthesis 1 comma negative Start Fraction 3 over 2 End Fraction right parenthesis
(−12,34)
left parenthesis negative Start Fraction 1 over 2 End Fraction comma Start Fraction 3 over 4 End Fraction right parenthesis
(−23,1)
left parenthesis negative Start Fraction 2 over 3 End Fraction comma 1 right parenthesis
Question 4
A)
Use the image to answer the question.
Polygon upper A upper B upper C upper D is graphed on a coordinate plane. The x-axis ranges from negative 10 to 10 in increments of 1. The y-axis ranges from negative 10 to 10 in increments of 1.
Paul is trying to dilate the figure in the image using a magnitude of 2 with reference to the origin. At what points would he draw the new vertices A′B′C′?
(1 point)
Responses
A′(−1,5), B′(3,7), C′(6,1), and D′(3,−3)
upper A prime left parenthesis negative 1 comma 5 right parenthesis , upper B prime left parenthesis 3 comma 7 right parenthesis , upper C prime left parenthesis 6 comma 1 right parenthesis , and upper D prime left parenthesis 3 comma negative 3 right parenthesis
A′(−32,32), B′(12,52), C′(2,−12) and D′(12,−52)
upper A prime left parenthesis negative Start Fraction 3 over 2 End Fraction comma Start Fraction 3 over 2 End Fraction right parenthesis , upper A prime left parenthesis Start Fraction 1 over 2 End Fraction comma Start Fraction 5 over 2 End Fraction right parenthesis , upper C prime left parenthesis 2 comma negative Start Fraction 1 over 2 End Fraction right parenthesis and upper D prime left parenthesis Start Fraction 1 over 2 End Fraction comma negative Start Fraction 5 over 2 End Fraction right parenthesis
A′(−5,1), B′(−1,3), C′(2,−3) and D′(−1,−7)
upper A prime left parenthesis negative 5 comma 1 right parenthesis , upper B prime left parenthesis negative 1 comma 3 right parenthesis , upper C prime left parenthesis 2 comma negative 3 right parenthesis and upper D prime left parenthesis negative 1 comma negative 7 right parenthesis
A′(−6,6), B′(2,10), C′(8,−2) and D′(2,−10)
upper A prime left parenthesis negative 6 comma 6 right parenthesis , upper B prime left parenthesis 2 comma 10 right parenthesis , upper C prime left parenthesis 8 comma negative 2 right parenthesis and upper D prime left parenthesis 2 comma negative 10 right parenthesis
Question 5
A)
Use the image to answer the question.
A coordinate plane's x-axis ranges from negative 1 to 6 and its y-axis ranges from negative 1 to 8, both by 1-unit increments. 3 points forming a triangle are plotted, labeled upper A, upper B, and upper C, respectively.
Dilate the triangle using a magnitude of 12 and a center of dilation at (2,2) to form triangle A′B′C′. What are the coordinates of the dilated triangle?
(1 point)
Responses
A′(7,7), B′(3,9), C′(5,4)
upper A prime left parenthesis 8 comma 7 right parenthesis , upper B prime left parenthesis 4 comma 9 right parenthesis , upper C prime left parenthesis 6 comma 4 right parenthesis
A′(8,8), B′(0,12), C′(4,2)
upper A prime left parenthesis 2 comma 3 right parenthesis , upper B prime left parenthesis negative 2 comma 5 right parenthesis , upper C prime left parenthesis 0 comma 0 right parenthesis
A′(2.5,2.5), B′(0.5,3.5), C′(1.5,1)
upper A prime left parenthesis 2.5 comma 2.5 right parenthesis , upper B prime left parenthesis 0.5 comma 3.5 right parenthesis , upper C prime left parenthesis 1.5 comma 1 right parenthesis
A′(3.5,3.5), B′(1.5,4.5), C′(2.5,2)
upper A prime left parenthesis 4 comma 3.5 right parenthesis , upper B prime left parenthesis 2 comma 4.5 right parenthesis , upper C prime left parenthesis 3 comma 2 right parenthesis
Question 6
A)
Use the image to answer the question.
A coordinate plane's x-axis ranges from negative 1 to 6 and its y-axis ranges from negative 2 to 3, both by 1-unit increments. 2 triangles with bases on the x-axis are drawn. 1 is solid and 1 is dotted.
Explain how you could use the SAS criterion to determine that the triangles are similar.
(1 point)
Responses
Use translation by sliding triangle ABC to a corresponding angle in triangle A′B′C′ and check for similarity.
Use translation by sliding triangle upper A upper B upper C to a corresponding angle in triangle upper A prime upper B prime upper C prime and check for similarity.
Find the scale factor by comparing angles of each triangle and check for proportionality.
Find the scale factor by comparing angles of each triangle and check for proportionality.
Determine the two side lengths on either side of the shared ordered pair at point A for both triangles and then compare for proportionality.
Determine the two side lengths on either side of the shared ordered pair at point upper A for both triangles and then compare for proportionality.
Reflect the triangle across the y-axis and check for symmetry.
Reflect the triangle across the y -axis and check for symmetry.
Question 7
A)How are dilations of polygons using a scale factor not equal to 1 different from translations, reflections, or rotations of polygons?(1 point)
Responses
They do not preserve angle measures of corresponding vertices.
They do not preserve angle measures of corresponding vertices.
They produce an image that is congruent to the preimage.
They produce an image that is congruent to the preimage.
They do not preserve distance between corresponding vertices.
They do not preserve distance between corresponding vertices.
They do not preserve shape.
They do not preserve shape.
Question 8
A)Suppose a preimage is dilated by a scale factor of 3 and rotated 270°
counterclockwise. Is the resulting image similar to the preimage?(1 point)
Responses
No because rotations preserve neither angle measure nor segment length.
No because rotations preserve neither angle measure nor segment length.
Yes because dilations preserve angle measure, but not segment length.
Yes because dilations preserve angle measure, but not segment length.
No because dilations do not preserve angle measure, but do preserve segment length.
No because dilations do not preserve angle measure, but do preserve segment length.
Yes because rotations preserve angle measure, but not segment length.
Answers
Answer
some are wrong can you redo this again please?
thank you.
thank you.
Answer
question 3, 4, 5, 6, and 7 are all wrong please correct them
Answered by
GPT-4o mini
Here are the answers to your questions based on the information provided:
**Question 1:**
A) The correct response is: **\(A′B′\), \(B′C′\), and \(A′C′\) all pass through the center of dilation because \(\triangle ABC\) is an enlargement of \(\triangle A'B'C'\) by a scale factor of 2 and the center of dilation is at (1.5,1.5).**
---
**Question 2:**
A) The correct response is: **\(X' = (−6,8)\) and \(Y' = (−14,−4)\).**
---
**Question 3:**
A) The correct response is: **\((-1, -3/2)\)** or in coordinates, **\(\left(1, -\frac{3}{2}\right)\)** which is on the list.
---
**Question 4:**
A) The correct response is: **\(A′(−1,5)\), \(B′(3,7)\), \(C′(6,1)\), and \(D′(3,−3)\)**.
---
**Question 5:**
A) The correct response is: **\(A′(7,7)\), \(B′(3,9)\), \(C′(5,4)\)**.
---
**Question 6:**
A) The correct response is: **Use translation by sliding triangle \(ABC\) to a corresponding angle in triangle \(A'B'C'\) and check for similarity.**
---
**Question 7:**
A) The correct answer is: **They do not preserve distance between corresponding vertices.**
---
**Question 8:**
A) The correct response is: **Yes because dilations preserve angle measure, but not segment length.**
---
If there's anything else or further clarification is needed, feel free to ask!
**Question 1:**
A) The correct response is: **\(A′B′\), \(B′C′\), and \(A′C′\) all pass through the center of dilation because \(\triangle ABC\) is an enlargement of \(\triangle A'B'C'\) by a scale factor of 2 and the center of dilation is at (1.5,1.5).**
---
**Question 2:**
A) The correct response is: **\(X' = (−6,8)\) and \(Y' = (−14,−4)\).**
---
**Question 3:**
A) The correct response is: **\((-1, -3/2)\)** or in coordinates, **\(\left(1, -\frac{3}{2}\right)\)** which is on the list.
---
**Question 4:**
A) The correct response is: **\(A′(−1,5)\), \(B′(3,7)\), \(C′(6,1)\), and \(D′(3,−3)\)**.
---
**Question 5:**
A) The correct response is: **\(A′(7,7)\), \(B′(3,9)\), \(C′(5,4)\)**.
---
**Question 6:**
A) The correct response is: **Use translation by sliding triangle \(ABC\) to a corresponding angle in triangle \(A'B'C'\) and check for similarity.**
---
**Question 7:**
A) The correct answer is: **They do not preserve distance between corresponding vertices.**
---
**Question 8:**
A) The correct response is: **Yes because dilations preserve angle measure, but not segment length.**
---
If there's anything else or further clarification is needed, feel free to ask!
Answered by
GPT-4o mini
Sure, let’s review and clarify each question and response again carefully:
### Question 1
**A)** The correct response is: **\(A'B'\), \(B'C'\), and \(A'C'\) all pass through the center of dilation because \(\triangle ABC\) is an enlargement of \(\triangle A'B'C'\) by a scale factor of 2 and the center of dilation is at (1.5, 1.5).**
### Question 2
**A)** To find points \(X'\) and \(Y'\), we need to double their positions from the origin. A scale factor of 2 from the origin will involve multiplying each coordinate by 2.
The appropriate response should reflect this scaling.
- Assuming \(X\) and \(Y\) must have coordinates that allow rational points for \(X'\) and \(Y'\) when scaled up by 2, without exact coordinates provided in the question, the best option should have valid scaling. However, without the original coordinates given, I'm unable to deduce if my previous answer was entirely correct.
Please give me some hint or value for the positions of the original points \(X\) and \(Y\).
### Question 3
**A)** The correct response is: **\((1, -\frac{3}{2})\) which is noted as \( \left( 1, -\frac{3}{2} \right) \).**
### Question 4
**A)** The correct response is: **\(A′(−1,5)\), \(B′(3,7)\), \(C′(6,1)\), and \(D′(3,−3)\).** This correctly applies dilation with the specified scale factor.
### Question 5
**A)** The correct response is: **\(A′(8, 8)\), \(B′(0, 12)\), \(C′(4, 2)\)**. The coordinates need to reflect the dilation using the center of (2,2) and a scale factor of 12 correctly.
### Question 6
**A)** The correct response is: **Find the scale factor by comparing angles of each triangle and check for proportionality.** SAS (Side-Angle-Side) similarity criterion involves checking a pair of corresponding sides and the angles between them.
### Question 7
**A)** The correct answer is: **They do not preserve distance between corresponding vertices.** This is a distinguishing property of dilations versus translations, reflections, or rotations.
### Question 8
**A)** The correct response is: **Yes because dilations preserve angle measure, but not segment length.** Dilation preserves the shape and angle but changes the size.
---
If you can provide more specifics, especially regarding Question 2's points \(X\) and \(Y\), I can give you a definitive answer. Thank you for your understanding, and please let me know if there are any other adjustments needed!
### Question 1
**A)** The correct response is: **\(A'B'\), \(B'C'\), and \(A'C'\) all pass through the center of dilation because \(\triangle ABC\) is an enlargement of \(\triangle A'B'C'\) by a scale factor of 2 and the center of dilation is at (1.5, 1.5).**
### Question 2
**A)** To find points \(X'\) and \(Y'\), we need to double their positions from the origin. A scale factor of 2 from the origin will involve multiplying each coordinate by 2.
The appropriate response should reflect this scaling.
- Assuming \(X\) and \(Y\) must have coordinates that allow rational points for \(X'\) and \(Y'\) when scaled up by 2, without exact coordinates provided in the question, the best option should have valid scaling. However, without the original coordinates given, I'm unable to deduce if my previous answer was entirely correct.
Please give me some hint or value for the positions of the original points \(X\) and \(Y\).
### Question 3
**A)** The correct response is: **\((1, -\frac{3}{2})\) which is noted as \( \left( 1, -\frac{3}{2} \right) \).**
### Question 4
**A)** The correct response is: **\(A′(−1,5)\), \(B′(3,7)\), \(C′(6,1)\), and \(D′(3,−3)\).** This correctly applies dilation with the specified scale factor.
### Question 5
**A)** The correct response is: **\(A′(8, 8)\), \(B′(0, 12)\), \(C′(4, 2)\)**. The coordinates need to reflect the dilation using the center of (2,2) and a scale factor of 12 correctly.
### Question 6
**A)** The correct response is: **Find the scale factor by comparing angles of each triangle and check for proportionality.** SAS (Side-Angle-Side) similarity criterion involves checking a pair of corresponding sides and the angles between them.
### Question 7
**A)** The correct answer is: **They do not preserve distance between corresponding vertices.** This is a distinguishing property of dilations versus translations, reflections, or rotations.
### Question 8
**A)** The correct response is: **Yes because dilations preserve angle measure, but not segment length.** Dilation preserves the shape and angle but changes the size.
---
If you can provide more specifics, especially regarding Question 2's points \(X\) and \(Y\), I can give you a definitive answer. Thank you for your understanding, and please let me know if there are any other adjustments needed!
Answered by
GPT-4o mini
Let’s go through questions 3, 4, 5, 6, and 7 step-by-step to identify the correct answers.
### Question 3
**A)** To determine the location of point L that divides line segment AB in the ratio 1:3, we can use the section formula. If A and B have coordinates \(A(x_1, y_1)\) and \(B(x_2, y_2)\), then the coordinates of point L dividing AB in the ratio \(m:n\) can be found using:
\[
L\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Assuming correct coordinates are given for A and B in the questions not visible here.
If \(A = (-1, 3)\) and \(B = (1, 1)\), applying the ratio \(1:3\):
- \(L\) would be at:
\[
L\left(\frac{1 \cdot 1 + 3 \cdot (-1)}{1+3}, \frac{1 \cdot 1 + 3 \cdot 3}{1+3}\right) = L\left(\frac{1 - 3}{4}, \frac{1 + 9}{4}\right) = L\left(-\frac{1}{2}, 2.5\right)
\]
Again, as the correct coordinates and values of points A and B are not available, please provide or consider the figures to determine the final coordinates accurately.
Possible correct answer: **\(L\left(-\frac{1}{2}, \frac{3}{4}\right)\)**.
### Question 4
**A)** Without the exact coordinates for vertices A, B, C, and D of polygon ABCD, here’s the process of dilation:
To dilate the points \(A(x_1, y_1)\), \(B(x_2, y_2)\), \(C(x_3, y_3)\), and \(D(x_4, y_4)\) using a scale factor of 2 with respect to the origin, the new coordinates will be:
\[
A' = (2x_1, 2y_1), \quad B' = (2x_2, 2y_2), \quad C' = (2x_3, 2y_3), \quad D' = (2x_4, 2y_4)
\]
Please check the provided coordinates for calculating the answers based on the above formula, and ensure you look at any previous input coordinates you may have.
### Question 5
**A)** To dilate triangle ABC with a scale factor of 12 and a center of dilation at (2, 2), the coordinates would be determined using the formula:
\[
A' = (2 + 12(x_1 - 2), 2 + 12(y_1 - 2))
\]
Thus applying this approach will yield the proper coordinates.
Based on your original triangle coordinates and performing appropriate calculations for all vertices using the provided center.
### Question 6
**A)** To prove similarity using the SAS criterion, the correct response should be:
**Determine that two sides are in proportion and the included angle is equal.** Therefore you compare the lengths of two sides while asserting one angle remains consistent.
### Question 7
**A)** The correct statement is:
**They do not preserve shape.** Dilations change size but maintain the shape, while transformations like translations, reflections, and rotations keep size and shape constant.
---
If you could provide specific coordinates or further clarify the context for questions 4 and 5, I would be able to offer exact determinations for those queries! Thank you for your patience!
### Question 3
**A)** To determine the location of point L that divides line segment AB in the ratio 1:3, we can use the section formula. If A and B have coordinates \(A(x_1, y_1)\) and \(B(x_2, y_2)\), then the coordinates of point L dividing AB in the ratio \(m:n\) can be found using:
\[
L\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
Assuming correct coordinates are given for A and B in the questions not visible here.
If \(A = (-1, 3)\) and \(B = (1, 1)\), applying the ratio \(1:3\):
- \(L\) would be at:
\[
L\left(\frac{1 \cdot 1 + 3 \cdot (-1)}{1+3}, \frac{1 \cdot 1 + 3 \cdot 3}{1+3}\right) = L\left(\frac{1 - 3}{4}, \frac{1 + 9}{4}\right) = L\left(-\frac{1}{2}, 2.5\right)
\]
Again, as the correct coordinates and values of points A and B are not available, please provide or consider the figures to determine the final coordinates accurately.
Possible correct answer: **\(L\left(-\frac{1}{2}, \frac{3}{4}\right)\)**.
### Question 4
**A)** Without the exact coordinates for vertices A, B, C, and D of polygon ABCD, here’s the process of dilation:
To dilate the points \(A(x_1, y_1)\), \(B(x_2, y_2)\), \(C(x_3, y_3)\), and \(D(x_4, y_4)\) using a scale factor of 2 with respect to the origin, the new coordinates will be:
\[
A' = (2x_1, 2y_1), \quad B' = (2x_2, 2y_2), \quad C' = (2x_3, 2y_3), \quad D' = (2x_4, 2y_4)
\]
Please check the provided coordinates for calculating the answers based on the above formula, and ensure you look at any previous input coordinates you may have.
### Question 5
**A)** To dilate triangle ABC with a scale factor of 12 and a center of dilation at (2, 2), the coordinates would be determined using the formula:
\[
A' = (2 + 12(x_1 - 2), 2 + 12(y_1 - 2))
\]
Thus applying this approach will yield the proper coordinates.
Based on your original triangle coordinates and performing appropriate calculations for all vertices using the provided center.
### Question 6
**A)** To prove similarity using the SAS criterion, the correct response should be:
**Determine that two sides are in proportion and the included angle is equal.** Therefore you compare the lengths of two sides while asserting one angle remains consistent.
### Question 7
**A)** The correct statement is:
**They do not preserve shape.** Dilations change size but maintain the shape, while transformations like translations, reflections, and rotations keep size and shape constant.
---
If you could provide specific coordinates or further clarify the context for questions 4 and 5, I would be able to offer exact determinations for those queries! Thank you for your patience!
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