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# Similar shapes

Angles and shapes · Congruence and similarity

<!-- Speaker note: 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). About 40 minutes (Predict and Run 8 min · Investigate 10 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** similar shapes are the same shape but can be different sizes, like a photo and its enlargement. Find the scale factor, use it to work out missing sides, and learn the conditions that make two triangles similar.

- 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.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need congruent triangles (last lesson) and multiplying by decimals. The one idea: similar means every length times the same scale factor, with the angles unchanged. The common mistake is adding a fixed amount to every side (Mia's mistake). Similarity conditions for triangles: AA, SSS (all sides in one ratio) and SAS (two sides in one ratio, the included angle equal). Only AA is practised much here; SSS appears in Practise. The link to transformations: translations, reflections and rotations give congruent shapes; an enlargement gives a similar shape. Wider: measure something tall outside with shadows and a metre ruler. -->

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## Why it matters

> 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.

<!-- Speaker note: Two shapes are **similar** when one is an **enlargement** of the other: the same shape, maybe a different size. What changes when a shape is enlarged, and what stays the same? -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**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

<!-- Speaker note: What happens: They stay the same. Doubling every side makes a bigger triangle of exactly the same shape. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what doubling does to a corner's angle. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fangles%2Fsimilar-shapes%2Fstep-2&code=widget%3A+similar%0Ashape%3A+triangle%0Asides%3A+3%2C+4%2C+5%0Aslider+k%3A+0.5%2C+3%2C+1%2C+0.5%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/angles/similar-shapes/?full=step-2#step-2)

<!-- Speaker note: Then try a scale factor less than 1. What happens to the copy? -->

<!-- Speaker note: Teacher: ask what each copy side ÷ original side gives. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* They stay the same. Doubling every side makes a bigger triangle of exactly the same shape.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## A missing side

Here is the routine.

$$\text{original } 2, 3, 4;\ \text{copy shortest} = 5$$

1) $k = 5 \div 2 = 2.5$ — Scale factor: copy ÷ original.
2) $4 \times 2.5 = 10$ — Multiply the original side by k.
3) $10 \text{ cm}$ — Check: 10 ÷ 4 = 2.5 as well.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write original 2, 3, 4;copy shortest = 5 and say "Match the sides: shortest with shortest." Line 1: write k = 5 ÷ 2 = 2.5 and say "Scale factor: copy ÷ original." Line 2: write 4 × 2.5 = 10 and say "Multiply the original side by k." Line 3: write 10 cm and say "Check: 10 ÷ 4 = 2.5 as well." -->

<!-- Speaker note: Teacher: ask Mia whether her photo would look stretched. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A missing side

**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: 12

<!-- Speaker note: k = 6 ÷ 4 = 1.5, so the longest side is 8 × 1.5 = 12 cm. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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.
```

* Answer: line 2

<!-- Speaker note: Line 2: an enlargement multiplies. k = 7 ÷ 4 = 1.75, so the 6 cm side becomes 6 × 1.75 = 10.5 cm. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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

* Answer: Yes, by AA: the third angles are both 65°

<!-- Speaker note: 180 − 40 − 75 = 65 in both, so all three angles match: the same shape (AA). -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Rectangles and shadows



- **Similar, not congruent:** 
- **Not similar:** 
- **The tree's shadow:** 
- **Squares and rectangles:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/similar-shapes/?full=step-4#step-4)

<!-- Speaker note: What this step is for: Rectangles can have equal angles and still not be similar; similar triangles measure a tree from shadows; all squares are similar but not all rectangles. Hints, in order: Multiply both the width and the height by the same number. For students who need more support: A table with original sides in one column and copy sides in the next, with a ÷ column for k. For early finishers: A similar triangle has scale factor 3. How many times bigger is its area? Test it with the 3, 4, 5 triangle. The hard part, on purpose: Matching the sides. Ask students to say which side matches which (shortest with shortest) before they divide. When students get stuck: A student adds the same amount to every side. Ask: Would the picture look the same shape? Fix: Find k by dividing, then multiply every side by k. / A student divides the wrong way (original ÷ copy). Ask: Is the copy bigger or smaller? Should k be more or less than 1? Fix: k = copy ÷ original. / A student says all rectangles are similar because the angles match. Ask: Is a 4 by 2 rectangle the same shape as a 4 by 3? Fix: For shapes other than triangles, check the sides' ratios too. -->

<!-- Speaker note: Teacher: ask why two rectangles with all 90° angles may not be similar. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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: 3

<!-- Speaker note: 15 ÷ 5 = 21 ÷ 7 = 27 ÷ 9 = 3 (SSS similarity). -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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: 3

<!-- Speaker note: k = 0.5, so 6 × 0.5 = 3 cm. A scale factor less than 1 makes the copy smaller. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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

* Answer: An enlargement

<!-- Speaker note: Translations, reflections and rotations keep every length, so they give congruent shapes. Only an enlargement (k ≠ 1) changes the size. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## 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

* Answer: Yes, by ASA

<!-- Speaker note: Two angles and the side between them: ASA. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: find the scale factor

Then say what the copy's other two sides are, and how you know the angles match.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/angles/similar-shapes/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Working back from one copy side to k shows they can find a scale factor and use it. Hints, in order: Which original side matches y? / k = the copy side ÷ the matching original side. -->

<!-- Speaker note: Teacher: ask which original side matches y. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**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?**

* k = 5 ÷ 2 = 2.5, so 3 × 2.5 = 7.5 m.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: This covers AC9M8SP01 nodes 2 and 3. Where it leads: proving quadrilateral properties with congruent triangles (next lesson), congruence algorithms, and trigonometry in Year 10 (which rests on similar right-angled triangles). -->