Home

Tutoring

Subjects

Live Classes

Study Coach

Essay Review

On-Demand Courses

Colleges

Games

Sign up

Log in

Opening subject page...

Loading your content

Practice

  • All Subjects
  • Algebra Flashcards
  • SAT Math Practice Tests
  • Math Question of the Day
  • Live Classes
  • On-Demand Courses

Varsity Tutors

  • Find a Tutor
  • Test Prep
  • Online Classes
  • K-12 Learning
  • College Search
  • VarsityTutors.com

© 2026 Varsity Tutors. All rights reserved.

← Back to Learn by Concept

Geometry · Learn by Concept

Geometry Help: Circle Similarity

Review real example questions for Circle Similarity in Geometry.

Question 1 / 10

0 of 10 answered

Two circles are drawn: ⊙E\odot E⊙E and ⊙F\odot F⊙F. The center points EEE and FFF are marked, and the radii are different. Which statement explains why the circles are similar by referencing a dilation correctly?

Select an answer to continue

All questions

Question 1

Two circles are drawn: ⊙E\odot E⊙E and ⊙F\odot F⊙F. The center points EEE and FFF are marked, and the radii are different. Which statement explains why the circles are similar by referencing a dilation correctly?

  1. They are similar because a dilation can change the radius while keeping the shape a circle. (correct answer)
  2. They are similar because their centers are marked, so the circles must be congruent.
  3. They are similar because the ratio of their circumferences equals the ratio of their diameters.
  4. They are similar because EEE is the center of both circles in the diagram.

Explanation: The skill here is understanding circle similarity in geometry. Circles are similar because a dilation can change the radius while preserving the overall shape. The centers E and F are key for determining the appropriate transformation center or sequence. Applying a dilation scales ⊙E to match the radius of ⊙F, potentially after aligning centers. This justifies similarity as the shape remains a circle with proportional features. A common distractor is option C, which uses circumference ratios but doesn't invoke transformations. To transfer this strategy, think in terms of transformations like dilations and translations, not formulas.

Question 2

Two circles are shown: ⊙E\odot E⊙E has center EEE and radius 777, and ⊙F\odot F⊙F has center FFF and radius 141414. Which transformation maps ⊙E\odot E⊙E to a circle similar to ⊙F\odot F⊙F?

  1. Dilate ⊙E\odot E⊙E by scale factor 222 centered at EEE, then translate the image so the center lands on FFF. (correct answer)
  2. Translate ⊙E\odot E⊙E so EEE lands on FFF; translation alone makes the radii match.
  3. Reflect ⊙E\odot E⊙E across the perpendicular bisector of EF‾\overline{EF}EF to double its radius.
  4. They are not similar because their radii are different.

Explanation: The skill here is understanding circle similarity. Circles are similar because one can be mapped to another via a dilation that scales all distances by a constant factor, preserving shape. In this problem, the circles have centers E and F with radii 7 and 14, respectively. Applying a dilation centered at E with scale factor 2, followed by a translation to move the new center to F, maps ⊙E to a circle matching ⊙F. This justifies similarity because the sequence of dilation and translation forms a similarity transformation that overlays one circle onto the other. A distractor like choice D incorrectly claims they are not similar due to different radii, but similarity allows for size differences. To approach similar problems, think in terms of transformations like dilations rather than relying on formulas.

Question 3

Two concentric circles are shown with the same center OOO. The smaller circle has radius 555 and the larger circle has radius 202020. Which statement explains why the circles are similar?

  1. A dilation centered at OOO with scale factor 444 maps the smaller circle to the larger circle. (correct answer)
  2. They are congruent because they share the same center OOO.
  3. They are similar because the larger circle has circumference 444 times the smaller circle’s circumference.
  4. They are similar because the larger circle surrounds the smaller circle in the picture.

Explanation: The skill here is understanding circle similarity. Circles are similar because one can be mapped to another via a dilation that scales all distances by a constant factor, preserving shape. In this problem, the circles share the same center O with radii 5 and 20. Applying a dilation centered at O with scale factor 4 maps the smaller circle directly to the larger one. This justifies similarity because the dilation, being a similarity transformation, maps one circle exactly onto the other while preserving shape. A distractor like choice B claims congruence due to the shared center, but different radii mean they are not congruent. To approach similar problems, think in terms of transformations like dilations rather than relying on formulas.

Question 4

A coordinate geometry student claims that the circles x2+y2=4x^2 + y^2 = 4x2+y2=4 and (x−6)2+(y−8)2=36(x-6)^2 + (y-8)^2 = 36(x−6)2+(y−8)2=36 are not similar because their equations look completely different. How should this reasoning be evaluated?

  1. The reasoning is sound; different equation forms indicate fundamentally different geometric objects that cannot be similar
  2. The reasoning is flawed; equation appearance doesn't determine similarity, and these circles can be mapped via transformation (correct answer)
  3. The reasoning is partially correct; the equations indicate different orientations that prevent similarity transformations from working
  4. The reasoning is correct for coordinate geometry; similarity only applies when circles have identical algebraic representations

Explanation: The appearance of equations doesn't determine geometric similarity. The first circle (center origin, radius 2) and second circle (center (6,8)(6,8)(6,8), radius 6) are similar via translation by (6,8)(6,8)(6,8) and dilation by factor 3. All circles are similar regardless of their algebraic representation. Choice A incorrectly links equation form to geometric properties. Choice C misinterprets orientation concepts. Choice D incorrectly restricts similarity to identical representations.

Question 5

Two circles, ⊙R\odot R⊙R and ⊙U\odot U⊙U, are drawn with different radii and with centers marked at RRR and UUU. Which transformation maps ⊙R\odot R⊙R to a circle similar to ⊙U\odot U⊙U while keeping the idea of similarity (same shape) explicit?

  1. Dilate ⊙R\odot R⊙R about its center RRR by an appropriate scale factor, then translate so RRR moves to UUU. (correct answer)
  2. Translate ⊙R\odot R⊙R so RRR moves to UUU, and the radius will automatically change to match ⊙U\odot U⊙U.
  3. Reflect ⊙R\odot R⊙R across a line through UUU to increase its radius to match ⊙U\odot U⊙U.
  4. Use the circumference formula to compute both circumferences and conclude the circles are similar.

Explanation: The skill here is understanding circle similarity in geometry. Circles are similar via dilation, which scales the radius while preserving shape. The centers R and U guide the sequence of transformations needed. Applying a dilation about R to match the radius, followed by a translation to move R to U, maps ⊙R to ⊙U. This justifies similarity as the transformations ensure same shape and adjusted size. A common distractor is option B, which wrongly assumes translation alone changes the radius. To transfer this strategy, think in terms of transformations like dilations and translations, not formulas.

Question 6

In a proof that all circles are similar, a student writes: "Since all circles have the same shape, they are similar by definition." What is the primary flaw in this reasoning?

  1. The statement is circular reasoning that doesn't establish the existence of a similarity transformation between specific circles (correct answer)
  2. The statement is incorrect because circles can have different eccentricities depending on their radii and positions
  3. The statement fails to account for the fact that similarity requires preservation of both angle and ratio measures
  4. The statement is invalid because it doesn't specify which circle is being used as the reference standard

Explanation: A valid proof of similarity must demonstrate the existence of a specific similarity transformation (combination of rigid motions and dilation) that maps one circle to another. Simply stating that circles have the same shape is circular reasoning that doesn't provide the required constructive proof. Choice B is incorrect since circles don't have varying eccentricity. Choice C misses the main issue of circular reasoning. Choice D incorrectly suggests a reference standard is needed.

Question 7

Two circles have equations (x−2)2+(y+1)2=9(x-2)^2 + (y+1)^2 = 9(x−2)2+(y+1)2=9 and (x+3)2+(y−4)2=25(x+3)^2 + (y-4)^2 = 25(x+3)2+(y−4)2=25. To prove these circles are similar, which transformation sequence correctly maps the first circle to the second?

  1. Translation by (−5,5)(-5, 5)(−5,5) then dilation by factor 53\frac{5}{3}35​ centered at (−3,4)(-3, 4)(−3,4) (correct answer)
  2. Dilation by factor 53\frac{5}{3}35​ centered at (2,−1)(2, -1)(2,−1) then translation by (−5,5)(-5, 5)(−5,5)
  3. Translation by (5,−5)(5, -5)(5,−5) then dilation by factor 35\frac{3}{5}53​ centered at (−3,4)(-3, 4)(−3,4)
  4. Rotation of 90°90°90° about origin then dilation by factor 53\frac{5}{3}35​ centered at (−3,4)(-3, 4)(−3,4)

Explanation: The first circle has center (2,−1)(2, -1)(2,−1) and radius 3; the second has center (−3,4)(-3, 4)(−3,4) and radius 5. Translation vector from (2,−1)(2, -1)(2,−1) to (−3,4)(-3, 4)(−3,4) is (−5,5)(-5, 5)(−5,5). After translation, dilation by 53\frac{5}{3}35​ (centered at the new position) scales radius from 3 to 5. Choice B dilates first, changing the required translation. Choice C uses wrong translation direction and scale factor. Choice D adds unnecessary rotation.

Question 8

In establishing that all circles are similar, which property is both necessary and sufficient for the similarity transformation?

  1. The transformation must preserve the ratio of any chord length to the corresponding arc length across both circles
  2. The transformation must map the center of one circle to the center of the other while scaling distances proportionally
  3. The transformation must preserve all angular measures while allowing proportional scaling of all linear dimensions simultaneously (correct answer)
  4. The transformation must maintain the same orientation of both circles while preserving their relative positions in the plane

Explanation: Similarity transformations are characterized by preservation of angles and proportional scaling of lengths. This property is both necessary (required for similarity) and sufficient (guarantees similarity). Choice A focuses on a specific chord-arc relationship rather than the general similarity property. Choice B describes part of the process but isn't the defining characteristic. Choice D incorrectly emphasizes orientation and position rather than the fundamental similarity properties.

Question 9

Two circles intersect at exactly two points. A student concludes that because the circles intersect, they cannot be similar since similar figures must be non-intersecting. Which statement best describes this reasoning?

  1. Correct reasoning; intersecting circles have different geometric properties and therefore cannot be similar by definition
  2. Incorrect reasoning; similarity is determined by the existence of a similarity transformation, not by spatial relationship (correct answer)
  3. Partially correct; the circles are similar only if the intersection points lie on a line through both centers
  4. Correct reasoning; similar figures must maintain the same relative position when one is transformed to match the other

Explanation: Similarity between geometric figures depends solely on whether one can be mapped to the other through similarity transformations (rigid motions + dilation), not on their current spatial relationship. All circles are similar regardless of whether they intersect, are disjoint, or one contains the other. The student confuses positional relationships with similarity properties. Choices A and D incorrectly support the flawed reasoning, while Choice C adds an irrelevant condition about intersection points.

Question 10

Circle A has center (2,5)(2, 5)(2,5) and radius 4, while Circle B has center (−1,3)(-1, 3)(−1,3) and radius 6. To demonstrate that these circles are similar, what sequence of transformations would map Circle A onto Circle B?

  1. Translation by vector (−3,−2)(-3, -2)(−3,−2) followed by dilation with scale factor 32\frac{3}{2}23​ centered at (−1,3)(-1, 3)(−1,3) (correct answer)
  2. Dilation with scale factor 32\frac{3}{2}23​ centered at (2,5)(2, 5)(2,5) followed by translation by vector (−3,−2)(-3, -2)(−3,−2)
  3. Translation by vector (−3,−2)(-3, -2)(−3,−2) followed by dilation with scale factor 23\frac{2}{3}32​ centered at (−1,3)(-1, 3)(−1,3)
  4. Reflection across the line y=xy = xy=x followed by dilation with scale factor 32\frac{3}{2}23​ centered at origin

Explanation: To map Circle A to Circle B: First translate by vector (−3,−2)(-3, -2)(−3,−2) to move center from (2,5)(2, 5)(2,5) to (−1,3)(-1, 3)(−1,3). Then dilate by factor 64=32\frac{6}{4} = \frac{3}{2}46​=23​ centered at the new position (−1,3)(-1, 3)(−1,3) to scale radius from 4 to 6. Choice B dilates first, which would change the required translation vector. Choice C uses the wrong scale factor. Choice D uses an unnecessary reflection and wrong center for dilation.