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G.11(B)·Geometry·Similarity & MeasurementSupporting

Ratios in Similar Figures

Use ratios to solve problems involving similar figures.

Students match corresponding lengths in similar figures and use a consistent scale factor or proportion to find missing dimensions in practical designs.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Geometry key. Geometry is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutorAll Geometry released practice45 minutes

Learn the concept

Match the same roles

Similar figures have congruent corresponding angles and proportional corresponding side lengths. Match length with length and width with width, even if one drawing is turned. For polygons, use angle markings or vertex order to establish correspondence.

Use a scale factor

From an original figure to a new figure, k=new lengthcorresponding original lengthk=\frac{\text{new length}}{\text{corresponding original length}}. Multiply every original side by k. A factor greater than 1 enlarges; a factor between 0 and 1 reduces.

Write a consistent proportion

For similar rectangles, L1W1=L2W2\frac{L_1}{W_1}=\frac{L_2}{W_2}, or equivalently L2L1=W2W1\frac{L_2}{L_1}=\frac{W_2}{W_1}. Keep the same comparison on both sides. Solve, attach units, and check that the dimensions share one scale factor.

Worked example

A rectangular work surface is 8 feet long and 3 feet wide. A similar work surface is 4.5 feet wide. Find its length.

  1. The widths correspond: original 3 feet and new 4.5 feet.
  2. The scale factor from the original to the new surface is 4.5 ÷ 3 = 1.5.
  3. Multiply the original length by the same factor: 8(1.5) = 12.
  4. Check: 12/8 = 4.5/3 = 1.5.

12 feet

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Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

7th Grade Math Attributes of Similarity

Five Minute Math · Aaron Daffern, Dallas ISD

Identify corresponding dimensions and use proportional side lengths in similar figures.

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TEKSForge practice · unscored

  1. 1. A rectangular sign measures 35 by 20 centimeters. A similar sign has corresponding shorter side 12 centimeters. Find its longer side.

    Show answer and reasoning

    21 centimeters

    The reduction factor is 1220=35\frac{12}{20}=\frac35. Multiply 35(35)=2135\left(\frac35\right)=21.

  2. 2. In similar triangles ABC and DEF, AB corresponds to DE and BC corresponds to EF. AB = 9, DE = 15, and BC = 12. Find EF.

    Show answer and reasoning

    20 units

    The scale factor is 159=53\frac{15}{9}=\frac53, so EF=12(53)=20EF=12\left(\frac53\right)=20.

  3. 3. Are rectangles measuring 6 by 10 and 9 by 13 similar?

    Show answer and reasoning

    No

    Although both dimensions increased by 3, their corresponding ratios differ: 96=1.5\frac96=1.5 but 1310=1.3\frac{13}{10}=1.3. Similarity requires multiplication by one common factor.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Geometry EOC · 2013 · Item 42 · 1 point

The top of a bench has a length of 5 ft and a width of 2 ft. A second bench is similar to the first bench. The top of the second bench is 3 ft wide. What is the length of the top of the second bench?

  1. 6 ft
  2. 7 ft
  3. 4.5 ft
  4. 7.5 ft
Show STAAR answer and explanation

7.5 ft

The scale factor is 3/2=1.53/2=1.5. Multiply the original length by the same factor: 5(1.5)=7.55(1.5)=7.5 ft.

6 ft: This adds the width increase of 1 ft instead of scaling lengths.

7 ft: Adding the two original dimensions does not preserve the length-to-width ratio.

4.5 ft: This scales the new width by 1.5 rather than scaling the original length.

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Learning objectives

By the end of the lesson, students can:

  • Identify corresponding dimensions in similar figures.
  • Find and apply a linear scale factor in either direction.
  • Write and solve a proportion while keeping corresponding measurements aligned.

Instructional sequence

  1. 1

    Engage

    5 min

    Compare two rectangular designs with equal length-to-width ratios. Ask why adding the same amount to both dimensions would usually change the shape.

  2. 2

    Direct instruction

    10 min

    Match corresponding sides and calculate new divided by original as the scale factor. Write an equivalent within-figure proportion.

  3. 3

    Guided practice

    12 min

    Solve enlargement and reduction problems using a correspondence table, including fractional scale factors.

  4. 4

    Design applications

    12 min

    Find missing dimensions of similar tabletops, signs, and polygonal designs. Check that every known corresponding pair has the same ratio.

  5. 5

    Closure

    6 min

    Complete the exit ticket with a scale-factor estimate and a ratio check.

Misconceptions to address

““Add the width increase to the length.””

Teacher move: Similarity preserves ratios, not differences. Multiply every corresponding length by the same scale factor.

““The scale factor is whichever side ratio I notice first.””

Teacher move: State the direction first. Going from original to new uses new divided by original; reversing direction uses the reciprocal.

““Cross multiplication fixes mismatched sides.””

Teacher move: A proportion must first compare corresponding lengths in a consistent order. Correct arithmetic cannot repair an incorrect setup.