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# Right, acute or obtuse?

Area, volume, circles and Pythagoras · Pythagoras

<!-- Speaker note: Students open and close the angle between two sides to see that c² = a² + b² only at 90°, then use the comparison to classify triangles from their side lengths as right-angled, acute or obtuse, and hunt for Pythagorean triples (AC9M8M06, node 3). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** pythagoras works backwards: square the two shorter sides and add, then compare with the longest side squared. Equal: right-angled. The longest square is bigger: obtuse. Smaller: acute.

- I can tell from three side lengths whether a triangle is right-angled, acute or obtuse.
- I can explain why the longest side must be c in the test.
- I can find and check Pythagorean triples.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need Pythagoras (previous two lessons) and acute, right and obtuse triangles (Classifying triangles). The one idea: compare a² + b² with the longest side squared; equal means right-angled, and the bigger angle opposite c pushes c² up. The common mistake is testing with a shorter side as c (Ava's mistake). The test is the converse of Pythagoras' theorem: if a² + b² = c², the angle opposite c is a right angle. It is a separate fact from the theorem (which starts from a right angle); the board's angle slider shows why it holds. The board's angle slider keeps a and b fixed and works out c, so students see c² change continuously. -->

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## Why it matters

> Given only three lengths, can you tell whether a triangle has a right angle, without drawing it? Builders do exactly this when they mark out a square corner with a tape measure.

<!-- Speaker note: Pythagoras' theorem only works for right-angled triangles, and that is what makes it a test. Take the three sides of any triangle, call the **longest** one $c$, and compare: | If… | the angle opposite $c$ is… | the triangle is… | |---|---|---| | $a^2 + b^2 = c^2$ | 90° | right-angled | | $a^2 + b^2 < c^2$ | more than 90° | obtuse | | $a^2 + b^2 > c^2$ | less than 90° | acute | Whole numbers that fit exactly, like 3, 4, 5, are called **Pythagorean triples**. -->

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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 4 cm, 5 cm and 6 cm. Is it right-angled, acute or obtuse?**

- Acute
- Obtuse
- Right-angled

<!-- Speaker note: What happens: Acute. 4² + 5² = 16 + 25 = 41, and 6² = 36. The two squares are more than the longest one, so the biggest angle is less than 90°. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a vote; most guess right-angled because the numbers are consecutive. -->

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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%2Fmeasurement%2Fright-or-not%2Fstep-2&code=widget%3A+pythag%0Aa%3A+3%0Ab%3A+4%0Aslider+t%3A+60%2C+120%2C+70%2C+5%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/measurement/right-or-not/?full=step-2#step-2)

<!-- Speaker note: **Open the angle** and **Close the angle** change the corner between a and b. -->

<!-- Speaker note: Teacher: ask what happens to c as the angle opens, and to c². -->

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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?

* Acute. 4² + 5² = 16 + 25 = 41, and 6² = 36. The two squares are more than the longest one, so the biggest angle is less than 90°.

<!-- 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>

## The test

Here is the test, step by step.

Now try one: sides 6, 8 and 11.

$$c = 13 \text{ (the longest)}$$

1) $5^2 + 12^2 = 25 + 144 = 169$ — Square the other two and add.
2) $13^2 = 169$ — Square the longest.
3) $169 = 169 \Rightarrow \text{right-angled}$ — Equal: right-angled. Less: obtuse. More: acute.

<!-- 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 c = 13 (the longest) and say "Choose the longest side as c." Line 1: write 5^2 + 12^2 = 25 + 144 = 169 and say "Square the other two and add." Line 2: write 13^2 = 169 and say "Square the longest." Line 3: write 169 = 169 right-angled and say "Equal: right-angled. Less: obtuse. More: acute." -->

<!-- Speaker note: Teacher: build the three-row table on the board from the Run observations. -->

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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>

## The test

**A triangle has sides 6 cm, 8 cm and 11 cm. Is it right-angled, acute or obtuse?**

- Right-angled
- Acute
- Obtuse

* Answer: Obtuse

<!-- Speaker note: 6² + 8² = 100 is less than 11² = 121: the angle opposite 11 is more than 90°, obtuse. -->

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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

**Ava tests whether a triangle is right-angled. Click the line where her method goes wrong.**

```
 1  Is a triangle with sides 5, 13 and 12 right-angled?
 2  5² + 13² = 25 + 169 = 194
 3  12² = 144
 4  194 is not 144, so it is not right-angled.
```

* Answer: line 2

<!-- Speaker note: Line 2: c must be the longest side, 13. Test 5² + 12² = 25 + 144 = 169 = 13²: it is right-angled. -->

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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>

## Triples

**3, 4, 5 is a Pythagorean triple. Which of these is also one?**

- 3, 4, 6
- 6, 8, 10
- 4, 5, 6
- 5, 6, 7

* Answer: 6, 8, 10

<!-- Speaker note: 6, 8, 10 is 3, 4, 5 doubled: 36 + 64 = 100 = 10². -->

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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>

## One change at a time



- **Acute on the board:** Each task changes one thing.
- **A triple?:** 
- **Just obtuse:** 
- **Not a triangle:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/right-or-not/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: an acute triangle on the board, a triple to test, the smallest obtuse side, then the edge case of lengths that make no triangle at all. Hints, in order: Close the angle. Watch c² fall below 25. For students who need more support: Use the decision table and a list of squares to 20²; check with the board. For early finishers: Euclid's recipe: for whole numbers m > n, the numbers m² − n², 2mn and m² + n² make a triple. Test it for m = 2, n = 1 and m = 3, n = 2. The hard part, on purpose: Remembering which way the inequality goes. Ask students to open the angle on the board and say what happened to c² before they use the rule, instead of giving them the table. When students get stuck: A student uses a shorter side as c (Ava's mistake). Ask: Which side is opposite the biggest angle? Fix: c is always the longest side. / A student mixes up acute and obtuse. Ask: If the angle opens wider, does the opposite side get longer or shorter? Fix: Wider angle, longer c, bigger c²: obtuse. / A student compares the sides, not the squares. Ask: What does the theorem say: sides or squares? Fix: Square all three first. -->

<!-- Speaker note: Teacher: for 2, 3, 6, ask students to try to build it with straws or strips. -->

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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>

## Right

**Sides 9, 12 and 15?**

- Obtuse
- Right-angled
- Acute

* Answer: Right-angled

<!-- Speaker note: 81 + 144 = 225 = 15²: right-angled (3, 4, 5 times 3). -->

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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>

## Acute

**Sides 5, 6 and 7?**

- Obtuse
- Right-angled
- Acute

* Answer: Acute

<!-- Speaker note: 25 + 36 = 61 is more than 7² = 49: acute. -->

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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>

## Obtuse

**Sides 3, 5 and 7?**

- Right-angled
- Obtuse
- Acute

* Answer: Obtuse

<!-- Speaker note: 9 + 25 = 34 is less than 7² = 49: obtuse. -->

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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: a side

**Review: hypotenuse 10 cm, one side 6 cm. The other side, in cm?**

* Answer: 8

<!-- Speaker note: 100 − 36 = 64, √64 = 8 cm. -->

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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: a new triple

Then: is the triple you found a multiple of another one?

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/right-or-not/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Searching for a Pythagorean triple that is not 3-4-5 or a multiple of it turns the test into a hunt for two squares that add to a square. Hints, in order: Try a = 5 and look along the b values. / Or a = 8: which b makes a² + b² a perfect square? -->

<!-- Speaker note: Teacher: list the triples the class finds and spot multiples. -->

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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

**Classify triangles with sides (a) 8, 15, 17 and (b) 4, 6, 8.**

* (a) 64 + 225 = 289 = 17²: right-angled. (b) 16 + 36 = 52 &lt; 64: obtuse.

<!-- 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 completes AC9M8M06. Where it leads: the converse of Pythagoras in proofs, the cosine rule (Year 11), and distances on the Cartesian plane. -->