--- title: "Finding a side" subtitle: "Area, volume, circles and Pythagoras · Pythagoras · Lesson 10" --- # Finding a side **Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}} > **Key question:** A 5 m ladder leans against a wall with its foot 3 m out. How high up the wall does it reach? Screens, ramps, roofs and ladders all hide a right-angled triangle. :::teacher-note **In one sentence:** Students use Pythagoras to find the hypotenuse or a shorter side (subtracting the squares), with non-whole answers to two decimal places and a sense check, in contexts such as ladders, screens and ramps (AC9M8M06, node 2). **Students should be able to say:** I can find a shorter side of a right-angled triangle. I can decide whether to add or subtract the squares. I can solve problems with ladders, screens and ramps, and check my answer makes sense. **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 Which side of the triangle is the ladder? :::keep-together **A 5 m ladder leans against a wall. Its foot is 3 m from the wall. How high up the wall does it reach?** ☐ 8 m ☐ 4 m ☐ About 5.83 m ☐ 2 m My prediction, and why: {{lines-3}} ::: :::teacher-note **What happens:** 4 m. The ladder is the hypotenuse: 5² − 3² = 25 − 9 = 16, so the height is 4 m. 5.83 m adds the squares, which would make the wall taller than the ladder. ::: ## Run Try it on the board in the tutorial. Was your prediction right? What was different? {{lines-2}} ## Add or subtract? :::keep-together ### Add for c, subtract for a side Here are both kinds. 1. $c^2 = 6^2 + 8^2 = 100,\ c = 10$ — Missing the hypotenuse: add the squares. 2. $b^2 = 10^2 - 6^2 = 100 - 36 = 64,\ b = 8$ — Missing a shorter side: subtract from the hypotenuse's square. 3. $b^2 = 7^2 - 4^2 = 33,\ b = \sqrt{33} \approx 5.74$ — Not a perfect square: between 5 and 6 (25 and 36), about 5.74. 4. $5.74 < 7$ — Check: the hypotenuse must still be the longest. Now try one: hypotenuse 13 cm, one side 5 cm. **A right-angled triangle has hypotenuse 13 cm and one shorter side 5 cm. How long is the other side, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 12 13² − 5² = 169 − 25 = 144, √144 = 12 cm. ::: :::keep-together ### Find the mistake **Ken finds a shorter side. Click the line where his working goes wrong.** ``` 1 A right-angled triangle has hypotenuse 10 cm and one shorter side 6 cm. Find the other side. 2 b² = 10² + 6² 3 = 100 + 36 = 136 4 b = √136 ≈ 11.66 cm ``` Line: {{field-15}} Why is it wrong? {{lines-2}} ::: :::teacher-note **Answer:** line 2 Line 2: 10 cm is the hypotenuse, so subtract: b² = 100 − 36 = 64, b = 8 cm. His 11.66 cm is longer than the hypotenuse, which is impossible. ::: :::keep-together ### A screen **A rectangular screen is 48 cm wide and 36 cm high. How long is its diagonal? Write your steps.** Answer: {{field-15}} ::: :::teacher-note **Answer:** The diagonal is the hypotenuse of a right-angled triangle with sides 48 and 36 (the corners of a rectangle are right angles); $48^2 + 36^2 = 2304 + 1296 = 3600$ (for the hypotenuse, add the squares); $\sqrt{3600} = 60$ cm (take the square root) 48² + 36² = 3600, √3600 = 60 cm. ::: :::typical ## One change at a time **On the board.** Each task changes one thing. **A ramp.** **Not a whole number.** **A shortcut.** **Too long.** Work in the tutorial. Plan first: what will you change, and how will you test it? {{box-4}} ::: ## Any side :::keep-together ### A shorter side **Hypotenuse 25 cm, one side 7 cm. The other side, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 24 625 − 49 = 576, √576 = 24 cm. ::: :::keep-together ### A diagonal **A square has sides of 5 cm. How long is its diagonal, in cm, to two decimal places?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 7.07 25 + 25 = 50, √50 ≈ 7.07 cm. ::: :::keep-together ### A ladder **A 10 m ladder reaches 8 m up a wall. How far is its foot from the wall, in m?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 6 100 − 64 = 36, √36 = 6 m. ::: :::keep-together ### Review: the hypotenuse **Review: the sides at the right angle are 9 cm and 12 cm. Hypotenuse, in cm?** Answer: {{field-15}} ::: :::teacher-note **Answer:** 15 81 + 144 = 225, √225 = 15 cm. ::: ## Make: design a ramp Then: real ramps should rise no more than 1 m for every 14 m along. Is this ramp too steep? 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 hypotenuse 17 cm and one side 8 cm. Find the other side.** {{lines-3}} ::: :::teacher-note **Answer:** 289 − 64 = 225, √225 = 15 cm. :::