1. GCSE Right-Angled Triangles Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 2 nd March 2014 Learning Objectives: To be able to find missing sides and missing angles in right-angled triangles and 3D shapes.

2. RECAP: Pythagoras’ Theorem For any right-angled triangle with longest side c. a 2 + b 2 = c 2 c b a Hypotenuse (the longest side) 

3. 2 4 x Step 1: Determine the hypotenuse. Step 2: Form an equation 2 2 + 4 2 = x 2 Step 3: Solve the equation to find the unknown side. x 2 = 4 + 16 = 20 x = √ 20 = 4.47 to 2dp The hypotenuse appears on its own. Example

4. If you’re looking for the hypotenuse  Square root the sum of the squares If you’re looking for another side  Square root the difference of the squares 3 5 h x 4 7 ? ? Pythagoras Mental Arithmetic

5. 12 5 h ? 4 y 10 ? x 2 9 ? 1 2 q ? Pythagoras Mental Arithmetic

6. 6 8 x 42 55 x x 6 4 “To learn secret way of ninja, find x you must.” 1 1 x x 10 12 1 2 3 4 5 ? ? ? ? ? The Wall of Triangle Destiny

7. Exercise 1 6 8 x 1 x = 10 2 7 10 y Give your answers in both surd form and to 3 significant figures. x =  51 = 7.14 3 5 2 x x =  29 = 5.39 x = 6  5 = 13.4 18 y 12 4 6 34 x x =  43 = 6.56 5 x 1 1 1 x =  3 = 1.73 6 7 13 10 Find the height of this triangle. 12 9 x x x 2 + 49 = 81 – x 2 x = 4 7  ? ? ? ? ? ? ? ?

8. Areas of isosceles triangles To find the area of an isosceles triangle, simplify split it into two right-angled triangles. 1 11 10 13 12 3232 Area =  3 4 Area = 60 ? ? ? ?

9. Exercise 2 Determine the area of the following triangles. 6 5 Area = 12 ? 5 4 4 Area = 2  12 = 4  3 = 6.93 ? 4 16 17 Area = 120 ? 17 1.6 1 Area = 0.48 ? 1 12 Area = 40.2 ? 7 1 2 3 4 5

10. x y θ (a,b) r When I was in Year 9 I was trying to write a program that would draw an analogue clock. I needed to work out between what two points to draw the hour hand given the current hour, and the length of the hand.

11. 30° 4 x y Given a right-angled triangle, you know how to find a missing side if the two others are given. But what if only one side and an angle are given? Trigonometry

12. 30° hypotenuse adjacent opposite Names of sides relative to an angle ? ? ?

13. 60° x y z HypotenuseOppositeAdjacent xyz √211 cab 45° 1 √2 1 20° a c b ??? ??? ??? Names of sides relative to an angle

14. θ o h a “soh cah toa” sin, cos and tan give us the ratio between pairs of sides in a right angle triangle, given the angle. Sin/Cos/Tan ? ? ?

15. Example 45 opposite adjacent Looking at this triangle, how many times bigger is the ‘opposite’ than the ‘adjacent’ (i.e. the ratio) Ratio is 1 (they’re the same length!) Therefore: tan(45) = 1 ? ??

16. 40 ° 4 x 20 ° 7 x Step 1: Determine which sides are hyp/adj/opp. Step 2: Work out which trigonometric function we need. More Examples ??

17. 60 ° x 12 30° 4 x More Examples ? ?

18. Exercise 3 15 1 22 a b 20 10 4 c d e f 2 3 4 ? ? ? ? ? ? ? ? ?

19. x y θ

20. 30 ° 4 x RECAP: Find x ?

21. 3 5 But what if the angle is unknown? ? ? We can do the ‘reverse’ of sin, cos or tan to find the missing angle.

22. What is the missing angle?

26. 2 3 θ 1 3 “To learn secret way of math ninja, find θ you must.” 1 1 θ 6 θ 8 1 2 3 4 θ ? ? ? ?

27. x 40 ° 60 ° 3m Find x 3.19m

28. Exercises GCSE questions on provided worksheet

29. 3D Pythagoras The strategy here is to use Pythagoras twice, and use some internal triangle in the 3D shape. 1 1 1 √2√2 √3√3 Determine the length of the internal diagonal of a unit cube. ? ? Click to Bro- Sketch

30. Test Your Understanding The strategy here is to use Pythagoras twice, and use some internal triangle in the 3D shape. 4 3 12 13 Determine the length of the internal diagonal of a unit cube. ?

31. Test Your Understanding 2 Determine the height of this right* pyramid. 2 2 * A ‘right pyramid’ is one where the top point is directly above the centre of the base, i.e. It’s not slanted. 22 ?

32. Exercise 4 Determine the length x in each diagram. Give your answer in both surd for and as a decimal to 3 significant figures. x 1 2 3 x =  14 = 3.74 ? x 4 5 2 x =  45 = 6.71 2 2 2 22 2 2 x x =  28 = 5.29 1 1 1 x Hint: the centre of a triangle is 2/3 of the way along the diagonal connecting a corner to the opposite edge. x =  (2/3) = 0.816 13 6 8 x = 12 8 6 4 x =  51 = 7.14 ? ? ? ? ? x x 1 2 3 4 11 22