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HSG.SRT.A.1Common CoreMathGeometryGrades 9-12

HSG.SRT.A.1: Verifying the Properties of Dilations

In plain English: HSG.SRT.A.1 is the Common Core geometry standard that asks students to verify by experiment what a dilation with a given center and scale factor does. A line not through the center maps to a parallel line, a line through the center maps to itself, and every segment is multiplied in length by the scale factor. It is usually taught in high school Geometry.

Verify experimentally the properties of dilations given by a center and a scale factor:.

  1. a.A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
  2. b.The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
Common Core State Standards for Mathematics · Domain: Similarity, Right Triangles, and Trigonometry (SRT) · Cluster: Understand similarity in terms of similarity transformations
Also written as HSG-SRT.A.1 or G-SRT.1 · Official standard
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01

Lesson Plan

65-70 min

Overview

Students verify, by measuring and by computing, the three properties of a dilation with center O and scale factor k: the image of a line that misses the center is a parallel line, the image of a line through the center is the same line, and every image segment is k times as long as the original. The lesson treats these as experimental results first, with rulers, protractors and dynamic geometry software, and then confirms them with coordinates using slope and the distance formula.

The work sets up the similarity standards that follow. Students need to see for themselves that a dilation keeps the shape and direction of every segment while scaling its length, because the definition of similarity in HSG.SRT.A.2 rests on exactly these facts.

Learning Objectives

By the end of this lesson, students will be able to:

  • Dilate points, segments and figures with a given center and positive scale factor, on paper and in coordinates
  • Verify by measurement and by slope that a dilation maps a line not through the center to a parallel line
  • Verify that a dilation maps a line through the center onto itself
  • Verify that an image segment is longer or shorter than the original in the ratio given by the scale factor
  • Find the scale factor of a dilation from measured or computed lengths

Prior Knowledge Required

Students should already be comfortable with:

  • Describing the effect of dilations on figures using coordinates 8.G.A.3
  • Similar figures as the result of rigid motions and dilations, informally 8.G.A.4
  • Transformations as functions that take points to points HSG.CO.A.2
  • Slope of a line and the fact that parallel lines have equal slopes
  • The distance formula, or the Pythagorean theorem on a grid 8.G.B.8

Lesson Procedure

65-70 minutes of class time across 5 phases.

  1. Warm-Up10 minutes

    Give each student grid paper with the origin marked O and the point A(3, 1). Post the prompt:

    Warm-Up Prompt

    "A dilation with center O and scale factor 2 sends every point to the point on the same ray from O that is twice as far away. Plot the image of A. Then plot the image of B(1, 2). Before you measure anything: is the image of segment AB longer, shorter or the same length? Which way does it point?"

    Students should place A′ at (6, 2) and B′ at (2, 4). Collect guesses about length and direction without confirming them. Many students predict "twice as long", and some predict that the image segment will tilt differently. Tell the class that today they will test these predictions with measurement, software and coordinates, and that a prediction only counts once it survives the test.

  2. Direct Instruction20 minutes

    Definition. A dilation with center O and scale factor k > 0 maps O to itself and maps every other point P to the point P′ on ray OP with OP′ = k · OP. If k > 1 the image is farther from O, and if 0 < k < 1 it is closer. With the center at the origin, (x, y) → (kx, ky). With center C(a, b), P′ = C + k(P - C), so (x, y) → (a + k(x - a), b + k(y - b)).

    Use Diagram 1 to model the experiment: dilate a triangle, then measure. Students state the three properties in their own words only after seeing the evidence. Then work the examples, asking students to predict each result before computing it.

    • Segment, center at the origin

      Dilate A(1, 2) and B(4, 1) with center O(0, 0) and k = 2. Compare slopes and lengths.

      Equation: A′(2, 4), B′(8, 2); both slopes -1/3; AB = √10, A′B′ = 2√10, ratio 2

    • Line through the center

      Dilate points of the line y = 2x with center O and k = 3: P(1, 2) and Q(-2, -4).

      Equation: P′(3, 6) and Q′(-6, -12) satisfy y = 2x, so the line maps onto itself

    • Center not at the origin, k < 1

      Dilate D(6, 5) and E(10, 1) with center C(2, 1) and k = 1/2.

      Equation: D′(4, 3), E′(6, 1); both slopes -1; DE = 4√2, D′E′ = 2√2, ratio 1/2

    • Scale factor from measurements

      On paper, OP = 4 cm and OP′ = 10 cm. A segment PQ measures 3.2 cm.

      Equation: k = 10/4 = 2.5, so P′Q′ should measure 2.5 · 3.2 = 8 cm

    • Line not through the center

      Dilate the line y = x + 2 with center O and k = 3, using (0, 2) and (1, 3).

      Equation: Images (0, 6) and (3, 9) lie on y = x + 6: same slope, different line

    Summarize the properties with Diagram 2. A line m that misses the center maps to a line m′ with the same slope, so m′ ∥ m. A line ℓ through the center maps onto itself, because every point of ℓ moves along its own ray from O, and that ray lies on ℓ. Stress that the second fact explains a special case of the first: a line through the center is "parallel to itself" only in the sense that it does not move at all.

  3. Guided Practice15 minutes

    Pairs work on grid paper with the segment F(-3, 1), G(1, 3).

    1. Dilate FG with center O(0, 0) and k = 2, then again with k = 1/2. Record both images. (F′(-6, 2), G′(2, 6); F″(-1.5, 0.5), G″(0.5, 1.5).)
    2. Compute the slope of FG and of each image. (All equal 1/2.) Measure the angle each segment makes with a horizontal grid line to confirm with a protractor.
    3. Compute FG, F′G′ and F″G″ with the distance formula. (2√5, 4√5 and √5.) Write each image length as a multiple of FG.
    4. Pick two points on the line y = -3x, such as (1, -3) and (-1, 3), dilate them with center O and k = 2, and test whether the images are on y = -3x.

    Circulate and listen for two errors: adding k to each coordinate instead of multiplying, and multiplying the coordinates by k when the center is not the origin. Ask pairs that finish early to explain why k = 1/2 makes the segment shorter but leaves its slope the same.

  4. Independent Practice15-20 minutes

    Students work alone on this set, then compare with a partner:

    1. Dilate U(2, 3) and V(4, 2) with center Z(1, 1) and k = 3. (U′(4, 7), V′(10, 4).)
    2. Show that UV and U′V′ are parallel by comparing slopes. (Both -1/2.)
    3. Show that U′V′ = 3 · UV. (UV = √5, U′V′ = 3√5.)
    4. The line through Z and U has slope 2. Show that U′ is on that same line, and explain why this had to happen.
    5. On blank paper, draw any point O and any segment, dilate it by k = 1.5 with a ruler, and measure both segments to the nearest millimeter. Compute the ratio and explain why it may not come out exactly 1.5.

  5. Closure5 minutes

    Exit ticket: (1) Find the image of (6, -8) under the dilation with center O(0, 0) and k = 0.5. (Answer: (3, -4).) (2) A segment 7 cm long is dilated with k = 3. How long is the image? (21 cm.) (3) True or false: a dilation always moves every line to a different, parallel line. Explain. (False: a line through the center stays where it is.)

Differentiation Strategies

For Struggling Students

  • Start with the center at the origin and whole-number scale factors, and have students draw the ray from O through each point before placing the image
  • Give a three-column table (point, vector from center, image) so that students compute P - C, multiply by k, and add C back in separate steps
  • Let students check parallel lines by counting rise and run on the grid before they compute slopes

For Advanced Students

  • Ask students to prove the parallel property in general: dilate A(x₁, y₁) and B(x₂, y₂) about the origin and show that the slope of A′B′ equals the slope of AB
  • Given a segment and its dilated image, ask students to locate the center by drawing lines through corresponding points, and explain why the lines meet at one point
  • Ask what happens with k = 1, and why a dilation with k = 1 is the identity transformation

Assessment Guidance

What to Look For

Check that students place images on the ray from the center, not only on the grid, and that they subtract the center before scaling when the center is not the origin. When students verify a property, look for evidence: two equal slopes or two measured angles for parallel lines, a point substituted into an equation for a line through the center, and a computed or measured ratio for lengths. Measured ratios such as 1.48 or 1.53 for k = 1.5 are expected; ask students to name measurement error as the reason, not a failure of the property.

02

Classroom Activities

3 Activities

1

Ruler and Protractor Dilation Lab

20 minPairs

Students dilate a triangle by hand with two scale factors, then measure sides and angles to test the length and parallel properties. This is the "verify experimentally" part of the standard done with physical tools.

Setup

Each pair gets a blank sheet, a centimeter ruler, a protractor and two colored pencils. Students mark a center O near the left edge and draw a triangle ABC with sides between 3 cm and 6 cm, at least 3 cm from O.

Procedure

  • Draw rays OA, OB and OC. Measure OA, double it, and mark A′ on ray OA; repeat for B′ and C′ (k = 2)
  • Measure halfway along each ray to mark A″, B″ and C″ (k = 1/2)
  • Measure all nine sides and record them in a table with columns original, k = 2 image, k = 1/2 image, and the two ratios
  • With the protractor, measure the angle between side AB and ray OA, then the angle between A′B′ and the same ray. Equal corresponding angles along the ray show AB ∥ A′B′
  • Draw line OA extended in both directions. Explain where A′ and A″ landed and what that says about a line through the center

Discussion Questions

  • Your ratios were close to 2 and 1/2 but not exact. What caused the differences?
  • Did any side of the image cross the matching side of the original? Why not?
  • Where would the image of O itself be?

Modification for Distance Learning

Students do the same construction on a shared digital whiteboard with a grid, using the grid to measure, and post a photo of their table to the class board.

2

Drag the Center: A Software Investigation

20 minPairs

Students use dynamic geometry software to test the properties for many centers and scale factors at once, including the case where the center lies on the line.

Setup

In GeoGebra or Desmos Geometry, students create points D and E, the line DE, a point O not on the line, and a slider k from 0.2 to 3. They apply the dilation tool to line DE and to segment DE with center O and factor k, and display the slope of both lines and the lengths DE and D′E′.

Procedure

  • Move the slider to four values of k, including one below 1, and record k, both slopes, both lengths and D′E′ ÷ DE in a table with four rows
  • Drag O to three new positions not on the line and record whether the slopes still match
  • Drag O onto line DE. Record what happens to the image line, and describe where D′ and E′ go
  • Write each observed property as a sentence that begins "For every center and every scale factor I tried..."

Challenge Variation

Set k = 1 and describe the image. Then ask: is there any position of O that makes the image line perpendicular to the original? Students explain why the answer is no.

3

Coordinate Verification Stations

20 minGroups of 3-4

Groups rotate through four station cards. At each one they apply a dilation in coordinates and confirm a property with slope or distance, turning the experimental results into computed ones.

Station Cards

  • Station 1: dilate (-1, 2) and (1, 1) with center O(0, 0) and k = 3. Compare slopes and lengths. ((-3, 6), (3, 3); slopes -1/2; √5 and 3√5)
  • Station 2: dilate (4, 0) and (8, 8) with center (0, 4) and k = 1/2. ((2, 2), (4, 6); slopes 2; 4√5 and 2√5)
  • Station 3: the line through the center (-2, 1) and the point (0, 2) is dilated with k = 3. Find the image of (0, 2) and show it is on the same line. ((4, 4), slope 1/2 from the center)
  • Station 4: A(2, 1) maps to A′(5, 3) and B(4, 1) maps to B′(9, 3) under one dilation. Find the scale factor from A′B′ ÷ AB, and find the center where lines AA′ and BB′ meet. (k = 2, center (-1, -1))

Procedure

  • At each station one student computes the images, one checks the slope or line equation, one checks the lengths, and the fourth (or the first again) writes the property that was verified
  • Roles rotate at every station; groups have 5 minutes per card
  • Each group ends by writing which station verified which property: parallel lines, a line through the center, or the length ratio

03

Diagrams & Visual Aids

2 diagrams

Diagram 1: Measuring a Dilated Triangle

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 O P Q R P′ Q′ R′ Center O(0, 0), scale factor k = 2 P(2, 1) → P′(4, 2) Q(4, 1) → Q′(8, 2) R(2, 3) → R′(4, 6) Side lengths PQ = 2, P′Q′ = 4 PR = 2, P′R′ = 4 QR = 2√2, Q′R′ = 4√2 Every ratio image : original = 2 Slopes PQ and P′Q′: 0 (horizontal) QR and Q′R′: -1 PR and P′R′: vertical
Triangle PQR with P(2, 1), Q(4, 1), R(2, 3) and its image under the dilation with center O and scale factor 2, drawn to scale. Each image vertex lies on the ray from O through the original vertex, twice as far away. Every image side is twice as long as the original side and has the same slope.

Diagram 2: Lines Through and Not Through the Center

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 O A A′ B B′ C C′ D D′ m m′ ℓ = ℓ′ Center O(0, 0), scale factor k = 2 Line m: y = -x/2 + 2 (misses O) A(0, 2) → A′(0, 4) B(4, 0) → B′(8, 0) Image m′: y = -x/2 + 4 Same slope, new line: m′ ∥ m Line ℓ: y = x (through O) C(1, 1) → C′(2, 2) D(3, 3) → D′(6, 6) Images stay on ℓ: ℓ′ = ℓ
The dilation with center O and scale factor 2, drawn to scale. Line m misses O, and its image m′ is a different line with the same slope, so m′ ∥ m. Line ℓ passes through O, and the images of its points stay on ℓ, so the line maps onto itself.

04

Homework Assignment

~30 min

HSG.SRT.A.1 Homework: Properties of Dilations

Directions: Show all work. When you claim that two lines are parallel, give both slopes. When you compare lengths, give both lengths and their ratio. Draw a sketch for every coordinate problem.

Part 1: Lengths and Scale Factors (Problems 1-3)

  1. Dilate J(1, -1) and K(3, 2) with center O(0, 0) and scale factor 3. (a) Give J′ and K′. (b) Show that JK and J′K′ are parallel. (c) Find JK and J′K′ and show that J′K′ = 3 · JK.
  2. On paper, a student marks a center O and a point A with OA = 3.0 cm, then marks A′ on ray OA with OA′ = 7.5 cm. (a) What scale factor did the student use? (b) Segment AB measures 2.4 cm. How long should A′B′ be? (c) The student measures A′B′ as 5.9 cm. Is the property wrong, or is something else going on? Explain.
  3. Dilate M(5, -1) and N(2, 8) with center C(-1, 2) and scale factor 1/3. (a) Give M′ and N′. (b) Compare the slopes of MN and M′N′. (c) Find MN and M′N′ and state their ratio.

Part 2: Lines Through and Not Through the Center (Problems 4-6)

  1. The line y = -2x + 6 is dilated with center O(0, 0) and scale factor 1/2. (a) Dilate the points (0, 6) and (3, 0) and write the equation of the image line. (b) Explain how your answer shows the image is parallel to the original. (c) What is the image of the line y = -2x under the same dilation, and why?
  2. The dilation has center P(2, 3) and scale factor 4. (a) The line y = x/2 + 2 passes through P. Find the image of (4, 4) and show it is on the same line. (b) The line y = x/2 does not pass through P. Find the images of (0, 0) and (2, 1) and write the equation of the image line. (c) Which of your two answers is a new line, and how is it related to y = x/2?
  3. Rosa dilates a 5 cm segment ST with some center O and gets an image S′T′ that is 3 cm long. (a) What is the scale factor? (b) If OS = 6 cm, how far from O is S′? (c) Rosa's drawing shows S′T′ crossing ST at a point between S and T. Explain why her drawing must contain an error if O is not on line ST.

Rubric

CriterionFull Credit (2 pts)Partial Credit (1 pt)No Credit (0 pts)
Dilating PointsImages correct, including a center that is not the originMethod correct with one arithmetic errorCoordinates added to or multiplied without using the center
Parallel and Same-Line EvidenceSlopes or equations given and interpreted for both casesEvidence given but not interpretedClaim made without evidence
Length RatiosBoth lengths and the ratio correct and matched to kLengths correct, ratio missing or wrongLengths missing or incorrect
ExplanationsMeasurement error and Rosa's error explained using the propertiesExplanation partly correctNo explanation

05

Quiz: 20 Questions

Interactive, with answers

Instructions

Choose an answer to see whether it is right and why. Work the short-answer questions on grid paper before you open the answer. Reset quiz clears all answers.

Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.

0 of 20 answered · 0 correct

  1. Question 1 of 20 · Multiple Choice

    Under the dilation with center O(0, 0) and scale factor 4, what is the image of (2, -3)?

  2. Question 2 of 20 · Multiple Choice

    A dilation has center (1, 2) and scale factor 3. What is the image of (3, 5)?

  3. Question 3 of 20 · Multiple Choice

    A segment 10 cm long is dilated with scale factor 0.6. How long is the image?

  4. Question 4 of 20 · Multiple Choice

    Segment AB is 6 units long and its image A′B′ under a dilation is 21 units long. What is the scale factor?

  5. Question 5 of 20 · Multiple Choice

    The line y = 3x - 2 is dilated with center O(0, 0) and scale factor 2. What is the equation of the image line?

  6. Question 6 of 20 · Multiple Choice

    The line y = -4x is dilated with center O(0, 0) and scale factor 5. What is the image?

  7. Question 7 of 20 · Multiple Choice

    A line n does not pass through the center of a dilation with scale factor 3. Which statement about its image n′ is true?

  8. Question 8 of 20 · Multiple Choice

    In geometry software, a student dilates line DE with center O and then drags O onto line DE. What should the student observe?

  9. Question 9 of 20 · Multiple Choice

    Under a dilation with center O, point P is 5 cm from O and its image P′ is 2 cm from O. What is the scale factor?

  10. Question 10 of 20 · Multiple Choice

    Segment QR has slope -2/5, and line QR does not pass through the center. After a dilation with scale factor 3, what is the slope of Q′R′?

  11. Question 11 of 20 · Multiple Choice

    The segment with endpoints (4, 6) and (10, -2) is dilated with center O(0, 0) and scale factor 1/2. How long is the image?

  12. Question 12 of 20 · Multiple Choice

    A student dilates triangle ABC and measures the sides. Which set of data supports a scale factor of 1.5?

  13. Question 13 of 20 · Multiple Choice

    Where is the image of the center of a dilation?

  14. Question 14 of 20 · Multiple Choice

    A line passes through the center (2, -1) of a dilation and has slope 3. The dilation has scale factor 5. Which equation describes the image line?

  15. Question 15 of 20 · Short Answer

    Dilate A(-2, 4) and B(6, 0) with center O(0, 0) and scale factor 3/2. Show that A′B′ is parallel to AB and that A′B′ = (3/2) · AB.

  16. Question 16 of 20 · Short Answer

    A dilation has center (3, 3) and scale factor 2. Find the image of (5, 4). Then show that the image lies on the line through (3, 3) and (5, 4).

  17. Question 17 of 20 · Short Answer

    Find the equation of the image of the line y = -x + 8 under the dilation with center O(0, 0) and scale factor 1/4.

  18. Question 18 of 20 · Short Answer

    Describe an experiment with a ruler and a protractor that verifies that a dilation maps a line not through the center to a parallel line.

  19. Question 19 of 20 · Short Answer

    A document camera projects a 3 cm segment on a worksheet as a 45 cm segment on the screen. Treat the projection as a dilation. (a) What is the scale factor? (b) Another segment measures 42 cm on the screen. How long is it on the worksheet?

  20. Question 20 of 20 · Short Answer

    Under a dilation with center O(0, 0), the point A(4, 2) maps to A′(10, 5). Find the scale factor and the image of B(-2, 6).

0 of 20 answered · 0 correct

06

Frequently Asked Questions

10 Questions

What does HSG.SRT.A.1 mean?

HSG.SRT.A.1 means students check by experiment what a dilation with a given center and scale factor does to lines and segments. The two properties are named in parts a and b: a line not through the center maps to a parallel line (and a line through the center stays put), and every segment is multiplied in length by the scale factor.

Is HSG.SRT.A.1 taught in Geometry or Algebra 1?

It is taught in high school Geometry, usually at the start of the similarity unit. It follows the rigid-motion work of HSG.CO.A.2 through HSG.CO.B.8 and builds on grade 8, where students described the effect of dilations with coordinates (8.G.A.3).

What does "verify experimentally" mean in this standard?

It means students gather evidence by constructing and measuring, not by writing a formal proof. Rulers and protractors, tracing paper, grid paper and dynamic geometry software all count. Coordinate calculations with slope and distance are a strong way to confirm what the measurements suggest. A general proof is a reasonable extension for advanced students.

Why does a line through the center of a dilation stay the same?

Because each point moves along its own ray from the center, and for a point on a line through the center, that ray lies on the line. The point moves to a different spot, but it stays on the same line. So the line as a whole maps onto itself, even though most of its points move.

Is the image of a line under a dilation always parallel to it?

It is either parallel or the same line. If the line misses the center, the image is a different, parallel line. If the line passes through the center, the image is the line itself. In both cases the direction of the line does not change, so the slope stays the same.

What happens when the scale factor is between 0 and 1?

The image is a reduction: every point moves toward the center, and every segment gets shorter by the factor k. A scale factor of 1/2 halves every length. The parallel property still holds. A scale factor of exactly 1 leaves every point where it is.

Can the scale factor of a dilation be negative or zero?

In this standard the scale factor is positive, and many Geometry courses use only positive values. Some textbooks allow a negative scale factor, which places the image on the opposite ray from the center; that is the same as a positive dilation followed by a 180° rotation about the center. A scale factor of 0 would send every point to the center, so it is not a dilation.

What mistakes do students make with dilations?

A common error is multiplying the coordinates by k when the center is not the origin. The fix is to subtract the center, multiply by k, and add the center back. Other frequent errors are adding k instead of multiplying, and dividing the lengths the wrong way when finding k from an original and an image.

Does a dilation change angle measures?

No: a dilation keeps every angle measure. That follows from the parallel property, because each side of an angle maps to a parallel side. Students use this fact in HSG.SRT.A.2, where similar triangles have equal corresponding angles and proportional sides.

How does HSG.SRT.A.1 connect to later standards?

It is the foundation of similarity. HSG.SRT.A.2 defines similar figures as figures related by rigid motions and dilations, HSG.SRT.A.3 uses that definition to establish the AA criterion, and HSG.SRT.B.5 applies similarity to solve problems. The same scaling idea appears in scale drawings (7.G.A.1) and in right triangle trigonometry.