GCSE Right-Angled Triangles Dr J Frost 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.

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) 

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

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

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

6 8 x x x 6 4 “To learn secret way of ninja, find x you must.” 1 1 x x ? ? ? ? ? The Wall of Triangle Destiny

Exercise x 1 x = y Give your answers in both surd form and to 3 significant figures. x =  51 = x x =  29 = 5.39 x = 6  5 = y x x =  43 = x x =  3 = Find the height of this triangle x x x = 81 – x 2 x = 4 7  ? ? ? ? ? ? ? ?

Areas of isosceles triangles To find the area of an isosceles triangle, simplify split it into two right-angled triangles 3232 Area =  3 4 Area = 60 ? ? ? ?

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

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.

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

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

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

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 ? ? ?

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 ? ??

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 ??

60 ° x 12 30° 4 x More Examples ? ?

Exercise a b c d e f ? ? ? ? ? ? ? ? ?

x y θ

30 ° 4 x RECAP: Find x ?

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

What is the missing angle?

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

x 40 ° 60 ° 3m Find x 3.19m

Exercises GCSE questions on provided worksheet

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

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

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 ?

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 x =  14 = 3.74 ? x x =  45 = x x =  28 = 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) = x = x =  51 = 7.14 ? ? ? ? ? x x 11 22