Understand Similarity Through Transformations - 8th Grade Math

Which statement correctly describes the difference between congruent figures and similar figures?
Answer: Congruent figures are similar with scale factor $k=1$ , while similar figures may have $k\ne 1$ This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides where corresponding sides have equal ratios forming scale factor k, and equal corresponding angles; it requires dilation because rigid transformations like rotation, reflection, and translation preserve size giving congruence with k=1, while dilation scales by factor k creating different sizes, such as k=2 doubling all lengths or k=1/2 halving them, with sequences typically involving dilate by k from a center then rotate, reflect, or translate as needed to position, where dilation creates the size difference and others adjust position or orientation. For example, a triangle with sides 3-4-5 is similar to one with sides 6-8-10, checking proportionality: 6/3=8/4=10/5=2 for equal ratios with scale factor k=2, and a sequence could be dilate by 2 from the origin giving a similar triangle twice as large, then translate or rotate to match position if needed. The correct difference is that congruent figures are similar with k=1, while similar figures may have k≠1, so choice C is accurate. Common errors include reversing the definitions like in D, or misstatement about angles or transformations in A and B. To distinguish, note both have equal angles, but similarity allows proportional sides with k≠1 via dilation, while congruence requires k=1 with rigid motions only. Congruence is a special case of similarity where sizes are identical.
Rectangle PQRS is similar to rectangle WXYZ. If the ratio of corresponding sides is 3:5, and rectangle PQRS can be mapped onto rectangle WXYZ through a sequence of transformations, which transformation must be included in this sequence?
Answer: A dilation with scale factor $\frac{5}{3}$ to adjust the size difference between rectangles Since the rectangles are similar with a ratio of corresponding sides of 3:5, rectangle PQRS must be dilated by scale factor 5/3 to match the size of rectangle WXYZ. This dilation is essential for similarity transformations. Choice A describes a specific translation that may or may not be needed depending on position. Choice C describes a reflection that may or may not be needed depending on orientation. Choice D describes a rotation that may or may not be needed depending on orientation. Only the dilation is guaranteed to be required.
Polygon $A$ is obtained from polygon $B$ by a dilation with scale factor $1$ . Which statement is true?
Answer: The polygons are congruent and similar. Tests understanding similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor $k \neq 1$ . Similar figures: same shape, proportional sides (corresponding sides have equal ratios forming scale factor $k$ ), equal corresponding angles. Requires dilation: rigid transformations (rotation, reflection, translation) preserve size giving congruence ( $k=1$ ), dilation scales by factor $k$ creating different sizes ( $k=2$ doubles all lengths, $k=1/2$ halves). Sequence: typically "dilate by $k$ from center, then rotate/reflect/translate as needed to position" or variations—dilation creates size difference, others adjust position/orientation. [Example: triangle with sides 3-4-5 similar to triangle with sides 6-8-10, check proportionality: $6/3=8/4=10/5=2$ (equal ratios, scale factor $k=2$ ), sequence could be "dilate by 2 from origin" giving similar triangle $2 \times$ larger, then translate/rotate to match position if needed]. The polygons are congruent since dilation by $k=1$ leaves size unchanged, and congruence is a special case of similarity. A common error is thinking dilation with $k=1$ changes size or that similarity requires $k>1$ . Checking similarity: (1) measure corresponding sides (side AB corresponds to A'B', BC to B'C', etc.), (2) calculate ratios (AB/A'B', BC/B'C', CA/C'A'), (3) verify equal (all ratios same value $k$ ? yes→similar with scale factor $k$ ), (4) check angles if uncertain (corresponding angles equal? yes→similar). Transformation sequence: identify scale factor (ratio of sides: $k=6/3=2$ ), describe dilation (dilate by 2 from origin), add rigid transformations if needed (rotate, reflect, translate to match position). Congruence is similarity with $k=1$ (special case: same size and shape). Mistakes: forgetting dilation (trying to use only rigid for different sizes—impossible), inverting scale factor (using smaller/larger instead of larger/smaller), claiming proportional when ratios differ (not checking all pairs).
Rectangle $R$ is $2$ units by $4$ units. Rectangle $S$ is $3$ units by $5$ units. Are the rectangles similar?
Answer: No, because the side-length ratios are not equal: $\tfrac{2}{3} \ne \tfrac{4}{5}$ . This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. The rectangles are not similar because side ratios are not equal: 2/3 ≠ 4/5 (or checking pairs: width 2 to 3 is 3/2, length 4 to 5 is 5/4, unequal), so choice C is correct. A common error is claiming all rectangles are similar, but only if proportions match (like squares), or thinking translations make them similar without dilation for size change. To check similarity: (1) measure corresponding sides (assume width 2 to 3, length 4 to 5), (2) calculate ratios (3/2=1.5, 5/4=1.25), (3) verify equal (no, not similar), (4) angles are all 90° but sides not proportional. Mistakes include claiming proportional when ratios differ or thinking dilations change angles, which they don't.
Triangle $ABC$ has side lengths $3,5,6$ . Triangle $DEF$ has side lengths $3,4,5$ . Are the triangles similar?
Answer: No, because the ratios of corresponding sides are not all equal This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides where corresponding sides have equal ratios forming scale factor k, and equal corresponding angles; it requires dilation because rigid transformations like rotation, reflection, and translation preserve size giving congruence with k=1, while dilation scales by factor k creating different sizes, such as k=2 doubling all lengths or k=1/2 halving them, with sequences typically involving dilate by k from a center then rotate, reflect, or translate as needed to position, where dilation creates the size difference and others adjust position or orientation. For example, a triangle with sides 3-4-5 is similar to one with sides 6-8-10, checking proportionality: 6/3=8/4=10/5=2 for equal ratios with scale factor k=2, and a sequence could be dilate by 2 from the origin giving a similar triangle twice as large, then translate or rotate to match position if needed. In this case, the triangles are not similar because ratios like 3/3=1, 5/4=1.25, 6/5=1.2 are not equal, so no consistent k, making choice C correct. Common errors include assuming all triangles are similar like in A or B, or wrongly stating dilation changes angles in D, but dilation preserves angles. To check, calculate ratios of corresponding sides assuming possible orders, verify if all equal, and if not, they are not similar. Mistakes include not checking all ratios or confusing similarity with having the same number of sides.
Triangle $JKL$ has side lengths $4$ , $6$ , and $8$ . Triangle $MNO$ has side lengths $6$ , $9$ , and $12$ . Are the triangles similar? If yes, what is the scale factor from $\triangle JKL$ to $\triangle MNO$ ?
Answer: Yes; scale factor $k=\tfrac{3}{2}$ . This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. The triangles are similar with scale factor 3/2 from JKL to MNO since 6/4=3/2, 9/6=3/2, 12/8=3/2, making choice A correct. A common error is using sum instead of ratio (like 4+6≠8) or thinking translations prevent similarity. To check similarity: (1) measure corresponding sides (assume 4 to 6, 6 to 9, 8 to 12), (2) calculate ratios (6/4=1.5, etc.), (3) verify equal (yes, k=3/2), (4) angles equal by proportionality. Mistakes include inverting k to 2/3 or claiming not similar due to translations.
Two triangles are related by a dilation with scale factor $k=1$ . Which statement best describes the relationship between the triangles?
Answer: They are congruent and therefore similar. This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. With k=1, the triangles are congruent (same size and shape), and thus also similar, making choice B correct. A common error is confusing congruence and similarity, like claiming they are similar but not congruent when k=1 means they are both. Congruence is similarity with k=1 (special case: same size and shape). Mistakes include thinking dilation with k=1 changes something or that it implies reflection only.
Triangle $MNO$ has side lengths $3$ , $5$ , and $6$ . Triangle $RST$ has side lengths $3$ , $6$ , and $10$ . Are the triangles similar?
Answer: No, because the side-length ratios are not all the same. Tests understanding similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures: same shape, proportional sides (corresponding sides have equal ratios forming scale factor k), equal corresponding angles. Requires dilation: rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves). Sequence: typically "dilate by k from center, then rotate/reflect/translate as needed to position" or variations—dilation creates size difference, others adjust position/orientation. [Example: triangle with sides 3-4-5 similar to triangle with sides 6-8-10, check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be "dilate by 2 from origin" giving similar triangle 2× larger, then translate/rotate to match position if needed]. The triangles are not similar because the side ratios are not equal: 3/3=1, 5/6≈0.83, 6/10=0.6, which differ. A common error is assuming similarity based on one matching side or misunderstanding that translations imply similarity without checking ratios. Checking similarity: (1) measure corresponding sides (side AB corresponds to A'B', BC to B'C', etc.), (2) calculate ratios (AB/A'B', BC/B'C', CA/C'A'), (3) verify equal (all ratios same value k? yes→similar with scale factor k), (4) check angles if uncertain (corresponding angles equal? yes→similar). Transformation sequence: identify scale factor (ratio of sides: k=6/3=2), describe dilation (dilate by 2 from origin), add rigid transformations if needed (rotate, reflect, translate to match position). Congruence is similarity with k=1 (special case: same size and shape). Mistakes: forgetting dilation (trying to use only rigid for different sizes—impossible), inverting scale factor (using smaller/larger instead of larger/smaller), claiming proportional when ratios differ (not checking all pairs).
Triangle $PQR$ has side lengths $PQ=6$ , $QR=9$ , and $RP=12$ . Triangle $P'Q'R'$ has side lengths $P'Q'=2$ , $Q'R'=3$ , and $R'P'=4$ . What is the scale factor that maps $\triangle PQR$ to $\triangle P'Q'R'$ ?
Answer: $k=\tfrac{1}{3}$ This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. The scale factor mapping PQR to P'Q'R' is 1/3 since 2/6=1/3, 3/9=1/3, 4/12=1/3, making choice A correct. A common error is inverting the scale factor, like using 3 instead of 1/3 by swapping which maps to which. To check similarity: (1) measure corresponding sides (PQ to P'Q', etc.), (2) calculate ratios (6/2=3, but for mapping PQR to P'Q'R' it's 2/6=1/3), (3) verify equal (all 1/3, yes—similar with k=1/3), (4) angles equal implicitly by proportionality. Mistakes include computing as difference (e.g., 6-2=4) instead of ratio or claiming not proportional when they are.
Two similar pentagons have a scale factor of 2:3. If the first pentagon undergoes a sequence of transformations to map onto the second pentagon, and the sequence includes a dilation, a rotation of 45°, and a translation, in which order must these transformations be applied?
Answer: Translation last, with dilation and rotation in either order first In similarity transformations, rigid transformations (rotations, reflections, translations) preserve shape and size, while dilations change size but preserve shape. Dilation and rotation can be applied in either order since both preserve the figure's center relative to the dilation point. However, translation typically comes last because it moves the entire figure to its final position after size and orientation adjustments are complete.
Triangle $ABC$ has side lengths $AB=3$ , $BC=4$ , and $CA=5$ . Triangle $DEF$ has side lengths $DE=6$ , $EF=8$ , and $FD=10$ . Which statement is true?
Answer: The triangles are similar; $\triangle DEF$ is a dilation of $\triangle ABC$ with scale factor $2$ . This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. Here, the triangles are similar with scale factor 2 since 6/3=2, 8/4=2, 10/5=2, so triangle DEF is a dilation of triangle ABC with k=2, making choice B correct. A common error is confusing congruence and similarity, like claiming they are congruent despite different sizes or thinking dilation changes angles, which it doesn't. To check similarity: (1) measure corresponding sides (AB to DE, BC to EF, CA to FD), (2) calculate ratios (3/6=1/2, but actually for mapping ABC to DEF it's 6/3=2), (3) verify equal (all ratios 2, yes—similar with k=2), (4) angles are equal since both are right triangles (3-4-5 is right-angled). Transformation sequence: identify scale factor (k=6/3=2), describe dilation (dilate by 2 from a center), add rigid transformations if needed; congruence is similarity with k=1, but here k=2 so similar but not congruent; mistakes include inverting scale factor or claiming not similar due to angle change.
On a coordinate plane, triangle $ABC$ has points $A(1,1)$ , $B(3,1)$ , and $C(2,3)$ . Triangle $A'B'C'$ has points $A'(-2,-2)$ , $B'(-6,-2)$ , and $C'(-4,-6)$ . Which sequence of transformations maps $\triangle ABC$ to $\triangle A'B'C'$ ?
Answer: Dilate by scale factor $2$ about the origin, then translate by $(-4,-4)$ . This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. The correct sequence is dilate by scale factor 2 about the origin (e.g., A(1,1) to (2,2), but then translate by (-4,-4) to (-2,-2); similarly for others), making choice B correct. A common error is using only rigid transformations for different sizes, which is impossible, or wrong scale factor like 1/2. To check similarity: (1) measure corresponding sides (e.g., AB distance sqrt( $(3-1)^2$ + $(1-1)^2$ )=2, A'B' sqrt( $(-6+2)^2$ + $(-2+2)^2$ )=4, ratio 4/2=2), (2) calculate ratios (all 2), (3) verify equal (yes, k=2), (4) angles equal as shapes match. Transformation sequence: identify k=2, dilate by 2 from origin, then translate by (-4,-4); mistakes include forgetting dilation or inverting k.
Triangle $XYZ$ has side lengths $7$ , $9$ , and $12$ . Triangle $X'Y'Z'$ has corresponding side lengths $14$ , $18$ , and $24$ . What is the scale factor from $\triangle XYZ$ to $\triangle X'Y'Z'$ ?
Answer: $k=2$ This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides where corresponding sides have equal ratios forming scale factor k, and equal corresponding angles; it requires dilation because rigid transformations like rotation, reflection, and translation preserve size giving congruence with k=1, while dilation scales by factor k creating different sizes, such as k=2 doubling all lengths or k=1/2 halving them, with sequences typically involving dilate by k from a center then rotate, reflect, or translate as needed to position, where dilation creates the size difference and others adjust position or orientation. For example, a triangle with sides 3-4-5 is similar to one with sides 6-8-10, checking proportionality: 6/3=8/4=10/5=2 for equal ratios with scale factor k=2, and a sequence could be dilate by 2 from the origin giving a similar triangle twice as large, then translate or rotate to match position if needed. Here, the scale factor from XYZ to X'Y'Z' is k=2 since 14/7=2, 18/9=2, and 24/12=2, confirming similarity, so choice B is correct. Common errors include computing wrong ratios like adding sides or picking unrelated numbers like 5 or 7 in C and D. To check, calculate each pair's ratio and ensure they equal k=2, verifying angles if needed by triangle properties. Congruence is similarity with k=1, and mistakes include inverting k to 1/2 or claiming non-proportional sides are similar.
Two triangles have side lengths $3,4,5$ and $6,8,10$ . Which statement is true?
Answer: They are similar with scale factor $k=2$ from the smaller triangle to the larger triangle This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides where corresponding sides have equal ratios forming scale factor k, and equal corresponding angles; it requires dilation because rigid transformations like rotation, reflection, and translation preserve size giving congruence with k=1, while dilation scales by factor k creating different sizes, such as k=2 doubling all lengths or k=1/2 halving them, with sequences typically involving dilate by k from a center then rotate, reflect, or translate as needed to position, where dilation creates the size difference and others adjust position or orientation. For example, a triangle with sides 3-4-5 is similar to one with sides 6-8-10, checking proportionality: 6/3=8/4=10/5=2 for equal ratios with scale factor k=2, and a sequence could be dilate by 2 from the origin giving a similar triangle twice as large, then translate or rotate to match position if needed. Here, the triangles with sides 3-4-5 and 6-8-10 are similar with scale factor k=2 from smaller to larger since all ratios are 2 and angles are equal as both are right triangles, so choice A is correct. Common errors include confusing similarity with congruence like in B despite different sizes, denying similarity due to size difference in C, or computing scale factor as a difference like 6-3=3 in D instead of the ratio 6/3=2. To check similarity, measure corresponding sides assuming order 3 to 6, 4 to 8, 5 to 10, calculate ratios 6/3=2, 8/4=2, 10/5=2, verify they are equal for k=2, and confirm corresponding angles are equal if needed. Congruence is similarity with k=1 as a special case of same size and shape, and mistakes include claiming proportional when ratios differ or inverting the scale factor.
Triangle XYZ undergoes a sequence of transformations to produce similar triangle X'Y'Z'. The sequence includes a reflection across the x-axis, a dilation by scale factor $\frac{3}{4}$ , and a rotation of 60° counterclockwise. If triangle XYZ has an area of 48 square units, what is the area of triangle X'Y'Z'?
Answer: The area of triangle X'Y'Z' is 27 square units because area scales by the square of the scale factor When you encounter transformation problems involving area, the key insight is understanding which transformations affect area and by how much. Different transformations have different effects on measurements. Let's work through this step-by-step. The triangle undergoes three transformations: reflection across the x-axis, dilation by scale factor $\frac{3}{4}$ , and a 60° rotation. Here's what each does to area: - Reflection: Preserves all measurements, including area - Rotation: Preserves all measurements, including area - Dilation: Changes area by the square of the scale factor Since only the dilation affects area, we calculate: Area of X'Y'Z' = Original area × (scale factor)² = 48 × $\left(\frac{3}{4}\right)^2$ = 48 × $\frac{9}{16}$ = 27 square units. Looking at the wrong answers: Choice A incorrectly applies the scale factor directly to area (48 × $\frac{3}{4}$ = 36, then somehow gets 12). Choice B makes the common error of thinking area scales linearly with the dilation factor, giving 48 × $\frac{3}{4}$ = 36. Choice C ignores the dilation entirely, focusing only on the fact that reflections and rotations preserve area. Study tip: Remember that area always scales by the square of the linear scale factor in dilations. If a shape is dilated by factor $k$ , its area changes by factor $k^2$ . Reflections and rotations never change area, but dilations always do unless the scale factor is 1.
Rectangle $R$ is $2\text{ cm}$ by $4\text{ cm}$ . Rectangle $S$ is $3\text{ cm}$ by $5\text{ cm}$ . Are the rectangles similar?
Answer: No, because the side ratios are not equal: $2/3\ne 4/5$ . Tests understanding similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures: same shape, proportional sides (corresponding sides have equal ratios forming scale factor k), equal corresponding angles. Requires dilation: rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves). Sequence: typically "dilate by k from center, then rotate/reflect/translate as needed to position" or variations—dilation creates size difference, others adjust position/orientation. [Example: triangle with sides 3-4-5 similar to triangle with sides 6-8-10, check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be "dilate by 2 from origin" giving similar triangle 2× larger, then translate/rotate to match position if needed]. The rectangles are not similar because the side ratios are not equal: 2/3 ≠ 4/5 for corresponding sides. A common error is assuming all rectangles are similar or that rigid transformations alone can make them similar without checking proportionality. Checking similarity: (1) measure corresponding sides (side AB corresponds to A'B', BC to B'C', etc.), (2) calculate ratios (AB/A'B', BC/B'C', CA/C'A'), (3) verify equal (all ratios same value k? yes→similar with scale factor k), (4) check angles if uncertain (corresponding angles equal? yes→similar). Transformation sequence: identify scale factor (ratio of sides: k=6/3=2), describe dilation (dilate by 2 from origin), add rigid transformations if needed (rotate, reflect, translate to match position). Congruence is similarity with k=1 (special case: same size and shape). Mistakes: forgetting dilation (trying to use only rigid for different sizes—impossible), inverting scale factor (using smaller/larger instead of larger/smaller), claiming proportional when ratios differ (not checking all pairs).
Triangle $XYZ$ has vertices $X(2,0)$ , $Y(4,0)$ , and $Z(2,3)$ . A dilation with scale factor $\dfrac{1}{2}$ about the origin is applied to triangle $XYZ$ to create triangle $X'Y'Z'$ . What are the coordinates of $Y'$ ?
Answer: $(2,0)$ Tests understanding similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures: same shape, proportional sides (corresponding sides have equal ratios forming scale factor k), equal corresponding angles. Requires dilation: rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves). Sequence: typically "dilate by k from center, then rotate/reflect/translate as needed to position" or variations—dilation creates size difference, others adjust position/orientation. [Example: triangle with sides 3-4-5 similar to triangle with sides 6-8-10, check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be "dilate by 2 from origin" giving similar triangle 2× larger, then translate/rotate to match position if needed]. The coordinates of Y' after dilation by 1/2 about the origin are (4/2, 0/2) = (2,0). A common error is misapplying the dilation formula, like halving only one coordinate or confusing with other transformations. Checking similarity: (1) measure corresponding sides (side AB corresponds to A'B', BC to B'C', etc.), (2) calculate ratios (AB/A'B', BC/B'C', CA/C'A'), (3) verify equal (all ratios same value k? yes→similar with scale factor k), (4) check angles if uncertain (corresponding angles equal? yes→similar). Transformation sequence: identify scale factor (ratio of sides: k=6/3=2), describe dilation (dilate by 2 from origin), add rigid transformations if needed (rotate, reflect, translate to match position). Congruence is similarity with k=1 (special case: same size and shape). Mistakes: forgetting dilation (trying to use only rigid for different sizes—impossible), inverting scale factor (using smaller/larger instead of larger/smaller), claiming proportional when ratios differ (not checking all pairs).
A triangle has vertices $A(-1,2)$ , $B(1,2)$ , and $C(0,5)$ . It is dilated by a scale factor of $3$ about the origin to form triangle $A'B'C'$ . What are the coordinates of $C'$ ?
Answer: $(0,15)$ This question tests understanding of similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures have the same shape with proportional sides (corresponding sides have equal ratios forming scale factor k) and equal corresponding angles; it requires dilation since rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), while dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves); the sequence is typically 'dilate by k from center, then rotate/reflect/translate as needed to position' or variations—dilation creates size difference, others adjust position/orientation. For example, a triangle with sides 3-4-5 is similar to a triangle with sides 6-8-10; check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be 'dilate by 2 from origin' giving similar triangle 2× larger, then translate/rotate to match position if needed. Dilation by 3 about origin maps C(0,5) to C'(0*3,5*3)=(0,15), making choice A correct. A common error is misapplying the dilation formula, like adding instead of multiplying coordinates by k. Transformation sequence: identify k=3, dilate by 3 from origin (x' = kx, y' = ky), so (0,5) to (0,15); no further transformations mentioned. Mistakes include using wrong center or confusing with translation.
Triangle $ABC$ has vertices $A(1,1)$ , $B(4,1)$ , and $C(1,3)$ . Triangle $A'B'C'$ has vertices $A'(2,2)$ , $B'(8,2)$ , and $C'(2,6)$ . Which sequence of transformations maps triangle $ABC$ to triangle $A'B'C'$ ?
Answer: Dilate by scale factor $2$ about the origin, then translate right 0 and up 0 Checking whether dilating by scale factor 2 about the origin maps each vertex of triangle $ABC$ to triangle $A'B'C'$ : $A(1,1)\to(2,2)=A'$ , $B(4,1)\to(8,2)=B'$ , $C(1,3)\to(2,6)=C'$ . Every vertex matches with no translation needed, so the correct sequence is a dilation by scale factor 2 about the origin alone. Choice B is incorrect because it adds an unnecessary rotation. Choice C is incorrect because translating before dilating changes the result: translating $A(1,1)$ to $(2,2)$ first and then dilating by 2 about the origin gives $(4,4)$ , not $A'(2,2)$ . Choice D is incorrect because reflecting across the $y$ -axis before dilating sends $A(1,1)$ to $(-2,2)$ , not $A'(2,2)$ .
A small right triangle has side lengths $3$ , $4$ , and $5$ inches. A larger right triangle has side lengths $6$ , $8$ , and $10$ inches. Which statement is true?
Answer: The triangles are similar with scale factor $2$ (larger compared to smaller). Tests understanding similarity obtained via transformations including dilation—same shape, different sizes means proportional sides with scale factor k≠1. Similar figures: same shape, proportional sides (corresponding sides have equal ratios forming scale factor k), equal corresponding angles. Requires dilation: rigid transformations (rotation, reflection, translation) preserve size giving congruence (k=1), dilation scales by factor k creating different sizes (k=2 doubles all lengths, k=1/2 halves). Sequence: typically "dilate by k from center, then rotate/reflect/translate as needed to position" or variations—dilation creates size difference, others adjust position/orientation. [Example: triangle with sides 3-4-5 similar to triangle with sides 6-8-10, check proportionality: 6/3=8/4=10/5=2 (equal ratios, scale factor k=2), sequence could be "dilate by 2 from origin" giving similar triangle 2× larger, then translate/rotate to match position if needed]. The triangles are similar with scale factor 2, as sides 6/3=2, 8/4=2, 10/5=2, and both are right triangles so angles match. A common error is confusing similarity with congruence or miscalculating scale factor as a difference like 6-3=3 instead of the ratio. Checking similarity: (1) measure corresponding sides (side AB corresponds to A'B', BC to B'C', etc.), (2) calculate ratios (AB/A'B', BC/B'C', CA/C'A'), (3) verify equal (all ratios same value k? yes→similar with scale factor k), (4) check angles if uncertain (corresponding angles equal? yes→similar). Transformation sequence: identify scale factor (ratio of sides: k=6/3=2), describe dilation (dilate by 2 from origin), add rigid transformations if needed (rotate, reflect, translate to match position). Congruence is similarity with k=1 (special case: same size and shape). Mistakes: forgetting dilation (trying to use only rigid for different sizes—impossible), inverting scale factor (using smaller/larger instead of larger/smaller), claiming proportional when ratios differ (not checking all pairs).

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