Introduction to Pythagoras’ Theorem Lesson

Introduction to Pythagorean Theorem

1. Pythagoras’ Theorem Objectives Know the Pythagoras’ Theorem Understand how to use the theorem to find missing lengths Be able to solve problems using Pythagoras’ Theorem

2. For any right-angled triangle with hypotenuse .         Hypotenuse (the longest side) ! Take note Pythagoras’ Theorem Note : notice that it is the longest side that’s on its own on one side of the equation. The (squared) shorter sides are the ones that are added.

3.       Step 1: Determine the hypotenuse. Step 2: Form an equation   Step 3: Solve the equation to find the unknown side. or 4.47 to 2dp   The hypotenuse appears on its own. Example ? ? Reveal >

4.       Step 1: Determine the hypotenuse. Step 2: Form an equation   Step 3: Solve the equation to find the unknown side. or 6.32 to 2dp   The hypotenuse appears on its own. Further Example ? ? Reveal >

5. Test Your Understanding Work out the missing values.

6. Harder Questions There’s a variety of ways in which Pythagoras questions could get harder: 6 3   4 Multiple triangles chained together. A B Adding lines to form right-angled triangles that weren’t originally there. Area? 2 2 2 3   7 9 C Requiring algebraic manipulation.      

7. E xample 1 Calculate the length DB

8. Test Your Understanding Calculate the length of x.

9. Pythagorean Triples Pythagorean Triples are three positive integers that satisfy the Pythagorean Theorem.

10. ✎ Your Task Worksheet

11. Challenge Task

12. Plenary: Give your answer to 3 s.f. Work out the length BC. Work out the value of y. and cm. Find the sides of the triangle.  

13. Plenary: Give your answer to 3 s.f. Work out the length BC. Work out the value of y. and cm. Find the sides of the triangle.   Ans: 27.3 mm Ans: 3.63 cm

14. Learning Outcomes I k now the Pythagoras’ Theorem I u nderstand how to use the theorem to find missing lengths I am able to solve problems using Pythagoras’ Theorem

15. Pythagoras’ Theorem in 3D Objective Be able to apply Pythagoras’ Theorem in problems involving 3D.

16. Calculate the length BH. A H G F E D C B 10cm 4cm 8cm 4cm 10cm H F E FH 2 = 4 2 + 10 2 FH 2 = 16 + 100 FH 2 = 116 FH = √116 Pythagoras’ Theorem in 3D

17. Calculate the length BH. A H G F E D C B 10cm 4cm 8cm 8cm √116cm H B F BH 2 = 8 2 + (√116) 2 BH 2 = 64 + 116 BH 2 = 180 BH = √180 BH = 13.4cm (3 s.f. ) Pythagoras’ Theorem in 3D

18. Pythagoras’ Theorem in 3D – Fast Way!

19. Pythagoras’ Theorem in 3D – Fast Way!

20. A square based pyramid has a base with side lengths 5cm and sloping sides of length 13cm. It has a vertex O and the midpoint of the base is at M vertically below O. Calculate the lengths: i ) AM and ii) OM A B D C O M 5cm 13cm 13cm O M A We can’t calculate AM since we don’t know OM yet! We need to work out AC first. Pythagoras’ Theorem in 3D

21. A B D C O M 5cm 13cm 5cm 5cm D C A AC 2 = 5 2 + 5 2 AC 2 = 25 + 25 AC 2 = 50 AC= Therefore, AM =   Pythagoras’ Theorem in 3D

22. 13cm   M O A 13 2 = + OM 2 169 = + OM 2 = OM 2 = OM 2 OM = 12.5cm (3 s.f. )   A B D C O M 5cm 13cm Pythagoras’ Theorem in 3D

23. Worksheet ✎ Your Task

24. Plenary 1 Ans: 34.4

25. Plenary 2 Ans: 7.84