GCSE Right-Angled Triangles Skipton Girls’ High School Learning Objectives: To be able to find missing sides and missing angles in right-angled triangles and 3D shapes. Last modified: 2nd March 2014

For any right-angled triangle with longest side c. Pythagoras’ Theorem ! Hypotenuse (the longest side) For any right-angled triangle with longest side c. a2 + b2 = c2 c b a

x 4 2 22 + 42 = x2 x2 = 4 + 16 = 20 x = √20 = 4.47 to 2dp Example Step 1: Determine the hypotenuse. x Step 2: Form an equation 2 22 + 42 = x2 The hypotenuse appears on its own. 4 Step 3: Solve the equation to find the unknown side. x2 = 4 + 16 = 20 x = √20 = 4.47 to 2dp

h 7 3 x 5 4 Pythagoras Mental Arithmetic ℎ= 3 2 + 5 2 = 34 We’ve so far written out the equation 𝑎 2 + 𝑏 2 = 𝑐 2 , filled in our information, and rearranged to find the missing side. But it’s helpful to be able to do it in our heads sometimes! 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 h 7 3 x 5 4 ℎ= 3 2 + 5 2 = 34 ? 𝑥= 7 2 − 4 2 = 33 ?

10 h 12 4 5 y 9 q 1 x 2 2 Pythagoras Mental Arithmetic ? ? ℎ= 12 2 + 5 2 =13 𝑦= 10 2 − 4 2 = 84 9 q 1 x 2 2 𝑥= 9 2 − 2 2 = 77 ? 𝑞= 2 2 + 1 2 = 5 ?

The Wall of Triangle Destiny Answer: 𝐱= 𝟐 ? 5 2 3 Answer: 𝐱=𝟏𝟎 ? 42 1 6 1 1 x x x 6 4 x x 4 55 12 Answer: 𝐱= 𝟐𝟎 ? 8 10 Answer: 𝐱= 𝟒𝟕𝟖𝟗 ? Answer: 𝐱= 𝟒𝟒 ? “To learn secret way of ninja, find x you must.”

Exercise 1 Give your answers in both surd form and to 3 significant figures. 4 7 1 13 13 x 18 12 6 y 10 8 x = 65 = 13.4 ? Find the height of this triangle. ? x = 10 5 2 x 6 12 ? 10 y N x 4 3 7 ? x = 43 = 6.56 9 x = 51 = 7.14 ? 7 1 6 3 x x x 2 1 x2 + 49 = 81 – x2 x = 4 ? 5 1 x = 29 = 5.39 ? ? x = 3 = 1.73

Areas of isosceles triangles To find the area of an isosceles triangle, simplify split it into two right-angled triangles. 13 1 1 13 3 2 ? 12 ? 10 1 Area = 60 ? Area = 3 4 ?

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

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

Names of sides relative to an angle Hypotenuse Opposite Adjacent x y z √2 1 c a b x ? ? ? 60° z y ? ? ? 1 √2 45° 1 ? ? ? c 20° a b

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

One way to remember this…

tan(45) = 1 Example ? opposite ? ? 45 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: opposite tan(45) = 1 ? ? 45 adjacent

Find 𝑥 (to 3sf) More Examples ? ? 20 ° 7 x 40 ° 4 x 𝑥=3.06 𝑥=2.39 Step 1: Determine which sides are hyp/adj/opp. Step 2: Work out which trigonometric function we need. Find 𝑥 (to 3sf) 20 ° 7 x 40 ° 4 x 𝑥=3.06 ? 𝑥=2.39 ?

More Examples 60 ° x 12 𝑥=24 ? 30° 4 x 𝑥=8 ?

Exercise 3 1 Find 𝑥, giving your answers to 3𝑠𝑓. Please copy the diagrams first. 𝟕𝟎° 15 𝒙 𝟒𝟎° 22 𝒙 𝑥=16.9 ? 𝟖𝟎° 20 𝒙 𝑥=20.3 ? a b c 𝑥=14.1 ? 𝟕𝟎° 4 𝒙 𝟕𝟎° 𝒙 𝟒 𝟓𝟓° 10 𝒙 f d e 𝑥=11.0 ? 𝑥=7.00 ? 𝑥=11.7 ? 2 I put a ladder 1.5m away from a tree. The ladder is inclined at 70° above the horizontal. What is the height of the tree? 𝟒.𝟏𝟐𝒎 Ship B is 100m east of Ship A, and the bearing of Ship B from Ship A is 30°. How far North is the ship? 𝟏𝟎𝟎÷ 𝐭𝐚𝐧 𝟑𝟎 =𝟏𝟕𝟑.𝟐𝒎 Find 𝑥. 𝒙= 𝟏 𝟑 −𝟏 ? 3 ? 𝟑𝟎° 𝒙 𝒙+𝟏 ? 4

We can do the ‘reverse’ of sin, cos or tan to find the missing angle. But what if the angle is unknown? 𝒂 3 5 sin 𝑎 = 3 5 𝑆𝑜 𝑎= sin −1 3 5 =36.9° ? ? We can do the ‘reverse’ of sin, cos or tan to find the missing angle.

What is the missing angle? Quiz What is the missing angle? 𝟓 𝒂 𝟒 cos −1 4 5 cos −1 5 4 cos −1 4 5 sin −1 5 4

What is the missing angle? 𝒂 𝟏 𝟐 cos −1 1 2 sin −1 2 tan −1 2 tan −1 1 2

What is the missing angle? 𝟓 𝟑 𝒂 cos −1 3 5 sin −1 3 5 tan −1 3 5 sin −1 5 3

What is the missing angle? 𝟑 𝒂 𝟐 cos −1 2 3 sin −1 2 3 sin −1 3 2 tan −1 2 3

The Wall of Trig Destiny ? 𝜃=45° 2 3 1 1 1 1 θ θ ? 𝜃=48.59° 4 2 3 6 θ 8 ? 𝜃=70.53° 3 θ ? 𝜃=33.7° “To learn secret way of math ninja, find θ you must.”

3D Pythagoras- not usually tested but a higher level 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. 1 ? √3 1 √2 ? Click to Sketch 1

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. 12 13 ? 4 3

Test Your Understanding Determine the height of this right* pyramid. 2 2 ? 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.

Exercise 4 Determine the length x in each diagram. Give your answer in both surd for and as a decimal to 3 significant figures. 2 2 1 x 3 N1 1 13 2 x x 2 2 2 6 3 8 2 2 x = 14 = 3.74 ? x = 12 ? x = 28 = 5.29 ? 2 4 N2 x 4 8 1 x 1 x 5 6 2 4 1 x = 51 = 7.14 ? x = 45 = 6.71 ? 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 ?