Similarity, Scale & Enlargement
Similarity Vocabulary
Draw a line from each term to its correct definition.
Identify Similar Triangles
Circle YES if the triangles could be similar, or NO if they cannot.
Triangle A: angles 40, 60, 80 deg. Triangle B: angles 40, 60, 80 deg.
Triangle A: angles 30, 60, 90 deg. Triangle B: angles 45, 45, 90 deg.
Triangle A: angles 50, 70, 60 deg. Triangle B: angles 50, 70, 60 deg.
Find the Scale Factor
Calculate the scale factor and the unknown length in each pair of similar figures.
Rectangle A: 4 cm x 6 cm. Rectangle B: 10 cm x ___. Scale factor (A to B) = ___. Missing length = ___
Triangle A: sides 3 cm, 4 cm, 5 cm. Triangle B: hypotenuse = 15 cm. Scale factor = ___. Other sides = ___, ___
Using Scale to Find Measurements
Use the given scale to answer each question.
A map has a scale of 1 : 50,000. Two towns are 6 cm apart on the map. What is the real distance in kilometres?
A model car is built to a scale of 1 : 20. The model is 22 cm long. What is the real length in metres?
A house plan has a scale of 1 : 100. A room is 4.5 cm long on the plan. What is the real length in metres?
Enlargement on a Grid
Apply the given scale factor (centre at the origin) and write the new vertices.
Triangle A has vertices at (1, 1), (3, 1) and (3, 4). Enlarge by scale factor 2. New vertices:
Rectangle B has vertices at (2, 2), (6, 2), (6, 4), (2, 4). Reduce by scale factor 1/2. New vertices:
Finding Unknown Sides Using Proportion
Use ratios of corresponding sides to find the unknown length.
Two similar triangles. Triangle 1: base = 4, height = 6. Triangle 2: base = 10. Find the height.
Two similar pentagons. Pentagon 1: perimeter = 20 cm, one side = 5 cm. Pentagon 2: corresponding side = 8 cm. Find the perimeter of Pentagon 2.
Shadow and Height Problem
Use similar triangles to solve this real-world problem.
A 1.8 m tall person casts a shadow 2.4 m long. At the same time, a tree casts a shadow 16 m long. How tall is the tree? Show your working using a proportion.
Similarity in the Real World
Explore similarity around your home.
- 1Find an object at home (e.g. a photo). Measure it and draw a scaled-down version on paper using a scale of 1 : 5.
- 2Look at a map of your local area. Use the scale bar to calculate the real distance between two places you know.
- 3Use a measuring tape, a stick and shadow measurements to estimate the height of a tall tree using similar triangles.
Scale Factor for Area and Volume
When a shape is enlarged by scale factor k, its area is multiplied by k^2 and its volume by k^3. Use this to solve each problem.
A square has side 4 cm and area 16 cm^2. It is enlarged by scale factor 3. New side = ___, New area = ___, Check: 16 x 3^2 = ___
A cube has volume 8 cm^3. It is enlarged by scale factor 2. New volume = ___
Two similar rectangles have a scale factor of 5 (length to length). If the smaller has area 6 cm^2, what is the area of the larger?
Map Reading with Scale
Use the given map scale to answer each question.
Scale: 1 : 25,000. Two towns are 8 cm apart on the map. What is the real distance in metres? In kilometres?
Scale: 1 : 200 (architect's plan). A room on the plan is 3.5 cm x 2 cm. What are the real dimensions in metres?
Two cities are 240 km apart in real life. A map shows them 6 cm apart. Write the map's scale in the form 1 : n.
Similar Triangles -- Proof Conditions
Two triangles are similar if they satisfy AA (Angle-Angle), SAS (Side-Angle-Side) or SSS (Side-Side-Side) similarity conditions. For each pair, state which condition applies (if any).
Triangle A: angles 40 deg, 70 deg, 70 deg. Triangle B: angles 40 deg, 70 deg, 70 deg. Condition:
Triangle A: sides 4, 6, 8. Triangle B: sides 6, 9, 12. Condition:
Triangle A: sides 3 and 5, included angle 60 deg. Triangle B: sides 6 and 10, included angle 60 deg. Condition:
Enlargement on a Coordinate Grid
Enlarge each shape from the given centre of enlargement by the scale factor shown. Write the new coordinates.
Triangle with vertices A(1, 1), B(3, 1), C(2, 3). Scale factor 2, centre at origin. New vertices: A' = ___, B' = ___, C' = ___
Rectangle with vertices P(2, 2), Q(4, 2), R(4, 6), S(2, 6). Scale factor 1/2, centre at origin. New vertices: P' = ___, Q' = ___, R' = ___, S' = ___
Finding Missing Sides in Similar Triangles
Use the scale factor to find unknown side lengths.
Triangles ABC and DEF are similar. AB = 6, BC = 9, DE = 4. Find EF.
Triangles PQR and STU are similar. PQ = 10, QR = 8, PR = 6, ST = 15. Find TU and SU.
A 2 m ruler casts a shadow of 1.5 m. At the same time, a tree casts a shadow of 9 m. How tall is the tree?
Similarity Condition Match
Match each similarity condition to its description.
Maps and Scale — Real-World Application
Apply scale factor to map reading.
A map has scale 1:250,000. Two towns are 4.5 cm apart on the map. What is the real distance in km?
The real distance between two cities is 300 km. On a map at 1:1,000,000, how far apart will they appear?
A map shows a lake that measures 3 cm × 2 cm at scale 1:50,000. What is the actual area of the lake in km²?
Proving Similarity — Worked Problems
Prove triangles are similar and find unknown lengths.
Triangle ABC has AB = 8, BC = 12, AC = 10. Triangle DEF has DE = 4, EF = 6, DF = 5. Prove they are similar.
Two triangles share a common angle at vertex X. One triangle has sides XY = 6, XZ = 9. The other has sides XA = 4, XB = 6. Are they similar? Which condition applies?
If XY and XA correspond, find the scale factor. What is XZ if XB = 6?
Identifying the Scale Factor
Circle the correct scale factor for each transformation.
Shape with side 5 cm mapped to shape with side 15 cm
Triangle with sides 8, 12, 16 mapped to triangle 2, 3, 4
Map distance 6 cm represents 300 km
Scale Drawings — Create Your Own
Create a scale drawing of your room or a local map.
Choose a room in your home. Measure its length and width. Decide on a suitable scale. Draw the room to scale, labelling all dimensions.
Add furniture to your scale drawing (estimate sizes). What are the scale dimensions of each piece?
Calculate the real area of your room from your scale drawing. Compare to an estimate from memory.
Sort by Scale Factor
Sort these transformations from reduction to enlargement.
Similarity — Applications in Design
Apply similar figures to real design contexts.
An architect plans a house at 1:100 scale. The design shows a room 4.5 cm × 3 cm. What are the actual room dimensions? What is the actual floor area?
The architect wants to print the plan at 1:50 scale instead. How large will the room appear on the new plan? How does the area on paper change?
A model car uses scale 1:18. The model is 25 cm long. How long is the real car?
Shadow and Height — Indirect Measurement
Use similar triangles to measure heights indirectly.
A 1.5 m person casts a 2 m shadow. At the same time, a tree casts a 12 m shadow. Calculate the height of the tree using similar triangles.
A telegraph pole casts a shadow of 8 m. A metre ruler held vertically casts a shadow of 0.4 m. How tall is the pole?
Explain why this method requires both shadows to be measured at the same time.
Similarity Investigation
Explore similarity and scale in your environment.
- 1Find a photo of yourself or a family member. Use a ruler to measure face proportions in the photo. How would these change at different print sizes? Are all prints similar?
- 2Research how Google Maps handles different zoom levels. How does the scale factor change as you zoom in and out? What happens to distances and areas on screen?
- 3Measure the shadow of a vertical stick and a tall nearby object (building, tree) at the same time of day. Use similar triangles to calculate the height of the tall object.
Similarity — Comprehensive Review
Demonstrate complete mastery of similarity and enlargement.
List three conditions that prove triangles are similar. Give an example of each.
A photo is 12 cm × 8 cm. An enlargement has area 4 times the original. What are the dimensions of the enlargement?
Two similar prisms have a linear scale factor of 2.5. If the smaller prism has volume 64 cm³ and surface area 96 cm², find the volume and surface area of the larger prism.
Explain in your own words why knowing the scale factor is enough to determine all ratios between measurements of two similar shapes.
Enlargement from a Non-Origin Centre
Enlarge shapes from a centre other than the origin.
Centre of enlargement C = (2, 1), scale factor k = 2. Point P = (5, 4). Find P'.
Centre C = (0, 2), k = 3. Triangle with vertices A(1, 2), B(3, 2), C_vertex(2, 4). Find A', B', C'.
Draw the original triangle and its image on a coordinate grid. Connect each vertex to its image. What do you notice?
Similarity in Triangles — Identifying Corresponding Parts
Identify corresponding parts of similar triangles.
△ABC ~ △DEF. Angle A = 50°, Angle B = 70°. Write all 6 equal angle pairs.
AB = 10, BC = 8, AC = 6, DE = 15. Find EF and DF.
What is the scale factor from △ABC to △DEF?
What is the ratio of the perimeters? What is the ratio of the areas?
Similar Figures in Context
Apply similarity to practical measurement problems.
A 5-metre ladder leans against a wall, with its foot 2 m from the base. A shorter ladder leans at the same angle, with its foot 0.8 m from the base. How long is the shorter ladder?
At 3 pm, a 2 m fence post casts a shadow of 1.4 m. How long is the shadow of an 8.5 m building at the same time?
A window has dimensions 1.2 m × 0.9 m. An architect's drawing shows this window as 3 cm × 2.25 cm. What is the scale of the drawing?
Designing with Similarity
Apply scale and similarity to a real design task.
You are designing a poster for a school event. The original image is 10 cm × 8 cm. You want the poster to be 60 cm × 48 cm. What is the scale factor? Is the shape preserved?
If the original image has a circular logo with diameter 2 cm, how large will the logo be on the poster?
The poster is printed on paper that is 60 cm × 90 cm. What fraction of the paper is used by the image?
Similarity — Connecting to Other Topics
Explore how similarity connects to the rest of Year 9 mathematics.
Explain how trig ratios (sin, cos, tan) depend on similarity. Why does sin 30° always equal 0.5 regardless of the triangle's size?
How does the scale factor connect to percentage increase? If a shape is enlarged by scale factor 1.2, what percentage larger is it?
Coordinate geometry connection: if a triangle with vertices at (0,0), (4,0), (0,3) is enlarged by k = 2 from the origin, write the new coordinates. What happens to the distances between vertices?
Similarity — Final Capstone Task
Demonstrate complete mastery of similarity and scale.
A model of a ship is made at scale 1:400. The model is 75 cm long. (a) How long is the real ship? (b) The model has surface area 900 cm². What is the real ship's surface area in m²? (c) The model's hull volume is 500 cm³. What is the real hull volume in m³?
Two similar triangles share a common vertex. The outer triangle has sides in the ratio 3:4:5 and the inner triangle has hypotenuse 5 cm. The outer triangle's hypotenuse is 15 cm. Find all sides of both triangles.
A photograph 8 cm × 5 cm is enlarged. The new diagonal is 34 cm. Find the scale factor and the new dimensions.
Similarity in Your World
Explore similarity and scale beyond the worksheet.
- 1Using Google Earth or Google Maps, find your school. Note the scale bar. Calculate the actual length of your school building from the satellite image.
- 2Find a map of Australia and measure the distance between two cities. Compare your calculation to the actual distance. How accurate is your map-reading?
- 3Collect packaging from 3 similar products in different sizes (small, medium, large). Measure a key dimension of each. Are they truly similar (proportional)? Calculate the scale factors.
Ratio and Proportion in Similar Shapes
Apply ratio and proportion reasoning to similarity.
A map has scale 1:200,000. Two cities are 6.4 cm apart on the map. What is the real distance? Give answer in km.
A room in a drawing is 7.5 cm × 5 cm. The real room is 6 m × 4 m. What is the scale? Write it as a ratio.
A blueprint shows a wall at 4.5 cm. The scale is 1:80. How long is the real wall in metres?
Similar Triangles — Proof Writing
Write formal proofs that two triangles are similar.
Triangles ABC and XYZ: Angle A = Angle X = 65°, Angle B = Angle Y = 80°. Prove the triangles are similar. Write a formal proof.
Triangles PQR and STU: PQ/ST = QR/TU = PR/SU = 3/5. Prove similarity and state the scale factor.
Enlargement — Identify the Image
Circle the correct image after the given enlargement.
Square with side 4 cm, scale factor 1.5. Image side length:
Rectangle 6 × 4 cm, scale factor 2/3. Image dimensions:
Triangle with area 20 cm², scale factor 3. Image area:
Similarity — Cross-Topic Application
Apply similarity to problems involving other Year 9 topics.
Two similar right triangles have legs 3 and 4 (small) and 6 and 8 (large). (a) Find the scale factor. (b) Find the hypotenuse of each using Pythagoras. (c) Find the area of each.
A similar triangle is drawn on a coordinate grid. Small triangle: A(0,0), B(3,0), C(0,4). Scale factor 2 from origin. Find coordinates of A', B', C'. Verify the perimeter scales by 2 and area by 4.
Similarity — Learning Reflection
Reflect on your learning about similarity and enlargement.
Before this worksheet, what did you know about similar shapes and scale? What is new?
Explain the three similarity conditions for triangles in your own words. Give a visual example of each.
What is the most useful real-world application of similarity you encountered? Explain why.
Write one question about similarity that you can now answer that you could not before.
Indirect Measurement Using Similar Triangles
Use similar triangles to find heights and distances.
A flagpole casts a shadow of 10 m. A nearby 1.8 m person casts a shadow of 1.2 m. How tall is the flagpole?
A tree's shadow extends 15 m. A 2 m vertical stick at the same time casts a 2.5 m shadow. Find the tree's height.
Why must both measurements be taken at the same time of day? What changes if the measurements are taken at different times?
Scale Factor Relationships
Match each scale factor scenario to the correct change in measurement.
Similarity — Problem Synthesis
Solve problems that combine multiple similarity skills.
Rectangle A is 8 × 5 cm. Rectangle B is similar with width 20 cm. (a) Find the scale factor. (b) Find the height of B. (c) Find the ratio of their perimeters. (d) Find the ratio of their areas.
A photographer takes a portrait photo 12 × 8 cm. She wants to display it at 30 × 20 cm and also at a wallet size. If the wallet width is 6 cm, what is the wallet height? Is the wallet photo similar to the original?
Similarity — Create and Solve
Design your own similarity problems.
Create a similar triangle problem where the answer for the unknown side is exactly 10 cm. Write the problem and solution.
Create a map scale problem where the answer is exactly 1 km. Write the problem and solution.
Create a problem involving similar 3D shapes where you find the volume of one given the other.
Similarity — Quick-Fire Review
Circle the correct answer for each quick question.
All similar shapes have:
Scale factor from small to large is k. Area of large = area of small × :
Two triangles with equal corresponding angles are:
A photo enlarged by k = 3: perimeter multiplies by:
Similarity and Scale — Year 9 Synthesis
Synthesise all your similarity knowledge.
Write a 5-point summary of the key ideas about similarity in Year 9 mathematics.
Give two examples of similarity in everyday life that you did not know about before this worksheet.
Create a mind map (on paper) connecting: similar triangles, scale factor, maps, photography, area, volume, and the golden ratio.
Similarity — Extending to 3D
Apply similarity concepts to 3D shapes.
Two similar rectangular prisms: small has dimensions 2 × 3 × 4 cm. The large prism has scale factor k = 5. Find the dimensions, SA, and volume of the large prism.
A manufacturer makes bottles in three similar sizes: small (V = 200 mL), medium (V = 675 mL), large (V = 1600 mL). Are these sizes truly similar? Find the linear scale factors between them.
Similarity — Teach It Back
Teach similarity to an imaginary student.
Explain what similar shapes are, what the scale factor means, and how it affects lengths, areas, and volumes. Assume your audience is a Year 7 student.
Give a worked example of finding a missing side using similar triangles. Show every step.
Give a real-world example of similarity that a Year 7 student would find interesting. Explain the maths.
Scale and Similarity — Investigation
Explore scale and similarity through practical investigation.
- 1Choose a room in your home. Draw it to scale (e.g. 1:50). Measure real items and convert them to scale for the drawing. Check by comparing the drawing's area to the real room's area.
- 2Find an old map of Australia or your city. Measure distances between landmarks and convert using the given scale. Compare your calculations to a modern online map.
- 3Research how architects create building information models (BIM). How do they use scale and similarity digitally?
Similarity — Masterclass Problem
Solve this extended similarity problem.
An artist creates a mural that is a scaled-up version of a small painting. The painting is 30 cm × 20 cm and has a central circle with diameter 10 cm. The mural is 3 m × 2 m. (a) What is the scale factor? (b) What is the diameter of the circle on the mural? (c) What fraction of the painting's area is the circle? Does this fraction change on the mural?
Similarity — Complete Year 9 Reflection
Wrap up your learning about similarity with a final written reflection.
List 5 key vocabulary terms from this worksheet and define each.
Describe one thing about similarity that surprised you or that you found particularly interesting.
Rate your confidence with similarity on a scale of 1–10 and explain your rating.
What would you explore further if you had more time?
Similarity — Key Terms Review
Match each term to its definition.
Similarity — Final Exam Preparation
Prepare for a Year 9 maths exam on similarity and scale.
Two triangles: Triangle A has sides 6, 8, 10 and Triangle B has sides 9, 12, 15. Prove they are similar. Find the scale factor and the ratio of their areas.
An A4 page is 29.7 cm × 21 cm. An A3 page is 42 cm × 29.7 cm. Are these pages similar? Justify your answer with calculations.
A model satellite dish is built at scale 1:20. The real dish has area 12.56 m². What area does the model dish have in cm²?
Similarity — Self-Assessment and Next Steps
Complete a final self-assessment and plan your next learning steps.
List three similarity skills you are confident with.
List one similarity skill you want to practise more. How will you practise it?
Identify one connection between similarity and another Year 9 topic (e.g. trig, coordinate geometry). Explain the connection.