--- title: "Pythagoras' theorem" subtitle: "Area, volume, circles and Pythagoras · Pythagoras · Lesson 9" --- # Pythagoras' theorem **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** Builders check a corner is square with a 3-4-5 triangle: 3 m along one wall, 4 m along the other, and if the diagonal is exactly 5 m, the corner is a right angle. People have used this for over three thousand years. :::teacher-note **In one sentence:** Students build right-angled triangles with squares on each side, see that the two smaller squares exactly make the biggest, and use a² + b² = c² to find the hypotenuse (AC9M8M06, node 1). **Students should be able to say:** I can find the hypotenuse of a right-angled triangle. I can explain Pythagoras' theorem with squares on the sides. I can check that an answer is sensible (longer than each short side, shorter than both together). **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 Is the hypotenuse shorter or longer than going round the corner, 3 + 4? :::keep-together **A right-angled triangle has sides 3 cm and 4 cm at the right angle. How long is the hypotenuse?** ☐ 5 cm ☐ 7 cm ☐ 12 cm ☐ 25 cm My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 5 cm. 3² + 4² = 9 + 16 = 25 = 5². Adding the sides (7) is too long: the straight route across is shorter than going round the corner. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Square, add, root :::keep-together ### Square, add, root Here is the method. 1. $c \text{ is opposite the right angle}$ — Find the hypotenuse first. 2. $a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100$ — Square the two shorter sides and add. 3. $c = \sqrt{100} = 10 \text{ cm}$ — Take the square root. 4. $6 < 10 < 6 + 8$ — Check: longer than each shorter side, shorter than both together. Now try one: sides 12 cm and 16 cm. **A right-angled triangle has sides 12 cm and 16 cm at the right angle. How long is the hypotenuse, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 20 12² + 16² = 144 + 256 = 400, so c = √400 = 20 cm. ::: :::keep-together ### Find the mistake **Leah finds the hypotenuse. Click the line where her method goes wrong.** ``` 1 A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse. 2 The hypotenuse is the two shorter sides added together. 3 6 + 8 = 14 4 So the hypotenuse is 14 cm. ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: it is the squares that add: 36 + 64 = 100, so c = 10 cm. 14 cm would be longer than going round the corner. ::: :::keep-together ### The hypotenuse **Which side of a right-angled triangle is the hypotenuse?** ☐ The side along the bottom ☐ The side opposite the right angle, the longest ☐ Either side at the right angle ☐ The shortest side ::: :::teacher-note **Answer:** The side opposite the right angle, the longest The hypotenuse faces the right angle and is always the longest side. ::: :::typical ## One change at a time **5 and 12.** Each task changes one thing. **9 and 12.** **Not a whole number.** **Always the longest.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## The hypotenuse :::keep-together ### 8 and 15 **The sides at the right angle are 8 cm and 15 cm. Hypotenuse, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 17 64 + 225 = 289, √289 = 17 cm. ::: :::keep-together ### A decimal **The sides at the right angle are 2 cm and 3 cm. Hypotenuse, in cm, to two decimal places?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 3.61 4 + 9 = 13, √13 ≈ 3.61 cm. ::: :::keep-together ### A shortcut across **A park is a rectangle 30 m by 40 m. How long is the path straight across, corner to corner, in m?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 50 The diagonal is a hypotenuse: 900 + 1600 = 2500, √2500 = 50 m. ::: :::keep-together ### Review: roots **Review: work out $\sqrt{144}$.** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 12 × 12 = 144. ::: ## Make: a hypotenuse of 10 Then: is 6-8-10 a new triangle, or 3-4-5 made bigger? Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ## Exit ticket :::keep-together **A right-angled triangle has sides 10 cm and 24 cm at the right angle. Find the hypotenuse.** {{lines-3}} ::: :::teacher-note **Answer:** 100 + 576 = 676, √676 = 26 cm. :::