--- title: "Right, acute or obtuse?" subtitle: "Area, volume, circles and Pythagoras · Pythagoras · Lesson 11" --- # Right, acute or obtuse? **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** 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. :::teacher-note **In one sentence:** 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). **Students should be able to say:** 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. **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 Compare $4^2 + 5^2$ with $6^2$. :::keep-together **A triangle has sides 4 cm, 5 cm and 6 cm. Is it right-angled, acute or obtuse?** ☐ Acute ☐ Obtuse ☐ Right-angled My prediction, and why: {{lines-3}} ::: :::teacher-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°. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Compare the squares :::keep-together ### The test Here is the test, step by step. 1. $c = 13 \text{ (the longest)}$ — Choose the longest side as c. 2. $5^2 + 12^2 = 25 + 144 = 169$ — Square the other two and add. 3. $13^2 = 169$ — Square the longest. 4. $169 = 169 \Rightarrow \text{right-angled}$ — Equal: right-angled. Less: obtuse. More: acute. Now try one: sides 6, 8 and 11. **A triangle has sides 6 cm, 8 cm and 11 cm. Is it right-angled, acute or obtuse?** ☐ Right-angled ☐ Acute ☐ Obtuse ::: :::teacher-note **Answer:** Obtuse 6² + 8² = 100 is less than 11² = 121: the angle opposite 11 is more than 90°, obtuse. ::: :::keep-together ### 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. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: c must be the longest side, 13. Test 5² + 12² = 25 + 144 = 169 = 13²: it is right-angled. ::: :::keep-together ### 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 ::: :::teacher-note **Answer:** 6, 8, 10 6, 8, 10 is 3, 4, 5 doubled: 36 + 64 = 100 = 10². ::: :::typical ## One change at a time **Acute on the board.** Each task changes one thing. **A triple?.** **Just obtuse.** **Not a triangle.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Right, acute or obtuse :::keep-together ### Right **Sides 9, 12 and 15?** ☐ Obtuse ☐ Right-angled ☐ Acute ::: :::teacher-note **Answer:** Right-angled 81 + 144 = 225 = 15²: right-angled (3, 4, 5 times 3). ::: :::keep-together ### Acute **Sides 5, 6 and 7?** ☐ Obtuse ☐ Right-angled ☐ Acute ::: :::teacher-note **Answer:** Acute 25 + 36 = 61 is more than 7² = 49: acute. ::: :::keep-together ### Obtuse **Sides 3, 5 and 7?** ☐ Right-angled ☐ Obtuse ☐ Acute ::: :::teacher-note **Answer:** Obtuse 9 + 25 = 34 is less than 7² = 49: obtuse. ::: :::keep-together ### Review: a side **Review: hypotenuse 10 cm, one side 6 cm. The other side, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 8 100 − 36 = 64, √64 = 8 cm. ::: ## Make: a new triple Then: is the triple you found a multiple of another one? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **Classify triangles with sides (a) 8, 15, 17 and (b) 4, 6, 8.** {{lines-3}} ::: :::teacher-note **Answer:** (a) 64 + 225 = 289 = 17²: right-angled. (b) 16 + 36 = 52 < 64: obtuse. :::