--- title: "Similar shapes" subtitle: "Angles and shapes · Congruence and similarity · Lesson 17" --- # Similar shapes **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** How can you find the height of a tree without climbing it? Its shadow and your own shadow make two triangles of the same shape. :::teacher-note **In one sentence:** Students scale a triangle by a scale factor, find missing sides of similar shapes, and compare rectangles that are similar, congruent or neither (AC9M8SP01, nodes 2 and 3). **Students should be able to say:** I can find a scale factor from a pair of matching sides. I can use a scale factor to find a missing side. I can say whether two shapes are congruent, similar or neither, and why. **Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet. ::: ## Predict In an enlargement, every length is multiplied by the same **scale factor** $k$, and every angle stays the same. Congruent shapes are similar with $k = 1$. $$k = \frac{\text{copy length}}{\text{matching original length}}$$ Triangles are similar when any one of these holds: | Condition | What matches | |---|---| | **AA** | two angles equal (so the third is too) | | **SSS** | all three pairs of sides in the same ratio | | **SAS** | two pairs of sides in the same ratio, and the angle between them equal | Translations, reflections and rotations move a shape without changing its size, so they make **congruent** shapes. An enlargement makes a **similar** shape. :::keep-together **A triangle has sides 3 cm, 4 cm and 5 cm. A copy has every side doubled. What happens to its angles?** ☐ They stay the same ☐ They halve ☐ They double My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** They stay the same. Doubling every side makes a bigger triangle of exactly the same shape. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Scale factors :::keep-together ### A missing side Here is the routine. 1. $\text{original } 2, 3, 4;\ \text{copy shortest} = 5$ — Match the sides: shortest with shortest. 2. $k = 5 \div 2 = 2.5$ — Scale factor: copy ÷ original. 3. $4 \times 2.5 = 10$ — Multiply the original side by k. 4. $10 \text{ cm}$ — Check: 10 ÷ 4 = 2.5 as well. **A triangle has sides 4 cm, 6 cm and 8 cm. A similar triangle's shortest side is 6 cm. How long is its longest side, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 k = 6 ÷ 4 = 1.5, so the longest side is 8 × 1.5 = 12 cm. ::: :::keep-together ### Find the mistake **Mia enlarges a photo. Click the line where her working goes wrong.** ``` 1 A 4 cm by 6 cm photo is enlarged so the 4 cm side becomes 7 cm. 2 7 − 4 = 3, so 3 cm is added to every side. 3 The 6 cm side becomes 6 + 3 = 9 cm. 4 The enlargement is 7 cm by 9 cm. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: an enlargement multiplies. k = 7 ÷ 4 = 1.75, so the 6 cm side becomes 6 × 1.75 = 10.5 cm. ::: :::keep-together ### Angles only **Two triangles both have angles of 40° and 75°. Are they similar?** ☐ Yes, by AA: the third angles are both 65° ☐ No: you need the sides ☐ Only if they are the same size ☐ You cannot tell ::: :::teacher-note **Answer:** Yes, by AA: the third angles are both 65° 180 − 40 − 75 = 65 in both, so all three angles match: the same shape (AA). ::: :::typical ## Rectangles and shadows **Similar, not congruent.** **Not similar.** **The tree's shadow.** **Squares and rectangles.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Similar shapes :::keep-together ### Scale factor **A triangle's sides are 5, 7 and 9 cm. A similar triangle's matching sides are 15, 21 and 27 cm. What is the scale factor?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3 15 ÷ 5 = 21 ÷ 7 = 27 ÷ 9 = 3 (SSS similarity). ::: :::keep-together ### Smaller copies **A 10 cm by 6 cm rectangle is reduced so the 10 cm side becomes 5 cm. How long is the 6 cm side now, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3 k = 0.5, so 6 × 0.5 = 3 cm. A scale factor less than 1 makes the copy smaller. ::: :::keep-together ### Transformations **A triangle is moved by a transformation, and its image is similar but not congruent. Which transformation was it?** ☐ A rotation ☐ A translation ☐ A reflection ☐ An enlargement ::: :::teacher-note **Answer:** An enlargement Translations, reflections and rotations keep every length, so they give congruent shapes. Only an enlargement (k ≠ 1) changes the size. ::: :::keep-together ### Review: congruence **Review: two triangles have equal angles of 50° and 60°, and the sides between those angles are both 8 cm. Are they congruent?** ☐ No, they are only similar ☐ Yes, by ASA ☐ Yes, by SSS ☐ You cannot tell ::: :::teacher-note **Answer:** Yes, by ASA Two angles and the side between them: ASA. ::: ## Make: find the scale factor Then say what the copy's other two sides are, and how you know the angles match. Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A 2 m by 3 m garden bed is enlarged so the 2 m side becomes 5 m. How long is the 3 m side now?** {{lines-3}} ::: :::teacher-note **Answer:** k = 5 ÷ 2 = 2.5, so 3 × 2.5 = 7.5 m. :::