How does a dilation change a figure, and how do you prove two figures similar?
Perform dilations on the coordinate plane and describe their effect on lengths and angles; define similarity through a sequence of rigid motions and a dilation; and prove triangles similar using AA (and SAS, SSS similarity), then use proportions to find missing lengths.
A NY Regents Geometry answer on dilations and similarity: performing a dilation about a center, why angles are preserved while lengths scale, the AA similarity criterion, and using proportions to find missing sides.
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What this topic is asking
The Regents Geometry exam (the Similarity, Right Triangles, and Trigonometry, G-SRT, cluster) wants you to perform a dilation, know that it scales lengths by the scale factor while keeping angles unchanged, define similarity as a rigid motion followed by a dilation, and prove triangles similar (mostly by AA) then solve proportions for missing lengths. Similarity is the bridge from congruence to trigonometry.
Dilations
A dilation resizes a figure from a fixed center by a scale factor .
If the figure enlarges; if it shrinks. Crucially, a dilation multiplies lengths by but preserves angle measures, so the image has the same shape. A line through the center maps to itself; a line not through the center maps to a parallel line.
Similarity through transformations
The Regents defines similarity transformationally: two figures are similar if a sequence of rigid motions and a dilation maps one onto the other. This is why similar figures have equal corresponding angles (rigid motions and dilations preserve angles) and proportional corresponding sides (the dilation scales them by ).
For triangles, the standard proof is AA: if two angles of one triangle equal two angles of another, the triangles are similar (the third angles must match because angles sum to 180 degrees). SAS similarity (two sides in proportion with the included angle equal) and SSS similarity (all three sides in proportion) are the other criteria.
Proportions in similar figures
Once similarity is established, corresponding sides are proportional, and you solve for a missing length by cross-multiplying. A clarifying point that prevents most errors is to match corresponding parts carefully: write the ratio so that the same triangle's sides are on top and the other triangle's sides are on the bottom (or numerator/denominator consistently), and pair sides that lie opposite equal angles. A second idea worth noting is that lengths scale by , but areas scale by : doubling the sides of a figure multiplies its area by four, a relationship the Regents tests when comparing similar figures' areas.
Try this
Q1. A dilation with scale factor maps a side of length 12 to what length? [1 credit]
- Cue. Multiply by : length 6.
Q2. Two triangles have angle pairs and degrees. Are they similar? [1 credit]
- Cue. Two pairs of equal angles means AA similarity: yes.
Exam-style practice questions
Practice questions written in the style of NYSED exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
Regents (style)2 marksPart I (multiple choice). Triangle is dilated by a scale factor of 3 centered at the origin. If side , what is the length of the corresponding side ? (1) (2) (3) (4) Show worked answer →
The correct answer is (3).
A dilation multiplies every length by the scale factor. With scale factor 3, the image side is . Choice (2) wrongly adds 3 instead of multiplying; choice (4) divides, which would be a scale factor of . Angles are unchanged by a dilation, but lengths scale by the factor.
Regents (style)4 marksPart III (constructed response). In triangle , point is on and point is on with . Given , , and , find and explain the similarity used.Show worked answer →
A 4-credit question: credit for the similarity statement, the proportion, and the answer.
Because , angle angle and angle angle (corresponding angles), so triangle triangle by AA. Corresponding sides are proportional: . Here and . So , giving , so , , . A correct proportion with no AA justification, or setting up the ratio with the wrong corresponding parts, loses credit.
Related dot points
- Define the sine, cosine, and tangent ratios in a right triangle (SOHCAHTOA), use them with inverse trig functions to find missing sides and angles, apply the relationship between the sine and cosine of complementary angles, and solve angle-of-elevation and depression problems.
A NY Regents Geometry answer on right triangle trigonometry: the sine, cosine, and tangent ratios, inverse trig to find an angle, the complementary sine-cosine relationship, and angle of elevation and depression problems.
- Represent and perform reflections, rotations, and translations using rules and the coordinate plane; recognize that rigid motions preserve distance and angle; and define congruence of two figures as the existence of a sequence of rigid motions mapping one onto the other.
A NY Regents Geometry answer on rigid motions: performing reflections, rotations, and translations on the coordinate plane, why they preserve distance and angle, and how a sequence of rigid motions defines congruence.
- Apply central angle, inscribed angle, chord, tangent, and secant relationships in a circle; compute arc length and sector area; and write and use the equation of a circle in the coordinate plane.
A NY Regents Geometry answer on circles: central and inscribed angles, the chord, tangent, and secant relationships, arc length and sector area, and the standard equation of a circle on the coordinate plane.
- Use the distance, midpoint, and slope formulas to analyze figures; determine parallel and perpendicular lines from slope; partition a directed segment into a given ratio; and write equations of lines satisfying given conditions.
A NY Regents Geometry answer on coordinate geometry: the distance, midpoint, and slope formulas, parallel and perpendicular slope conditions, partitioning a segment in a ratio, and writing equations of lines.
- Use the triangle congruence criteria (SSS, SAS, ASA, AAS, HL) to prove two triangles congruent, and use CPCTC (corresponding parts of congruent triangles are congruent) to justify further equal sides or angles.
A NY Regents Geometry answer on triangle congruence: the SSS, SAS, ASA, AAS, and HL criteria, why SSA and AAA fail, and using CPCTC to conclude further equal parts once triangles are proven congruent.
Sources & how we know this
- Regents Examination in Geometry — NYSED (2024)
- New York State Next Generation Mathematics Learning Standards — NYSED (2017)