GCSE RightAngled Triangles Dr J Frost jfrosttiffin kingston

RECAP: Pythagoras’ Theorem ! Hypotenuse (the longest side) c a b For any right-angled

Example Step 1: Determine the hypotenuse. x 2 Step 2: Form an equation 22

Pythagoras Mental Arithmetic If you’re looking for the hypotenuse Square root the sum of

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

The Wall of Triangle Destiny 2 ? ? 3 42 1 1 1 5

Exercise 1 Give your answers in both surd form and to 3 significant figures.

Areas of isosceles triangles To find the area of an isosceles triangle, simplify split

Exercise 2 Determine the area of the following triangles. 1 5 5 3 17

y (a, b) When I was in Year 9 I was trying to write

Trigonometry Given a right-angled triangle, you know how to find a missing side if

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

Names of sides relative to an angle Hypotenuse x Opposite Adjacent 60° z y

Sin/Cos/Tan sin, cos and tan give us the ratio between pairs of sides in

Example Looking at this triangle, how many times bigger is the ‘opposite’ than the

More Examples Step 1: Determine which sides are hyp/adj/opp. Step 2: Work out which

But what if the angle is unknown? 5 3 ? ? We can do

The Wall of Trig Destiny 2 1 1 1 θ 2 1 θ ?

Exercises GCSE questions on provided worksheet

3 D Pythagoras The strategy here is to use Pythagoras twice, and use some

Test Your Understanding The strategy here is to use Pythagoras twice, and use some

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

Exercise 4 Determine the length x in each diagram. Give your answer in both

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

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

Example Step 1: Determine the hypotenuse. x 2 Step 2: Form an equation 22 4 + 42 = x 2 The hypotenuse appears on its own. Step 3: Solve the equation to find the unknown side. x 2 = 4 + 16 = 20 x = √ 20 = 4. 47 to 2 dp

Pythagoras Mental Arithmetic 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 5 x 4 ? ?

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

The Wall of Triangle Destiny 2 ? ? 3 42 1 1 1 5 6 x x x 6 4 x 55 x 12 8 ? 4 10 ? “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 1 x 7 6 y 8 ? ? 5 2 x 13 10 x = 6 5 = 13. 4 x = 10 Find the height of this triangle. ? 6 12 10 y 4 7 ? ? 2 5 ? x = 29 = 5. 39 x N 9 7 1 6 x 3 x = 43 = 6. 56 x = 51 = 7. 14 3 13 18 12 x 1 x x 2 + 49 = 81 – x 2 x=4 ? 1 ? 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 13 12 ? 1 1 3 ? 2 10 1 Area = 60 ? Area = 3 ? 4

Exercise 2 Determine the area of the following triangles. 1 5 5 3 17 17 5 12 7 12 6 Area = 12 ? 2 16 Area = 120 ? 4 4 4 1 1 4 1. 6 Area = 2 12 = 4 3? = 6. 93 Area = 0. 48 ? Area = 40. 2 ?

y (a, b) 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. θ r x

Trigonometry 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? x 4 30° y

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

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

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

Example 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 ?

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

More Examples 60 ° ? x 12 x 4 30° ?

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

RECAP: Find x ? 4 30 ° x

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

What is the missing angle?

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

3. 19 m 40° x Find x 60° 3 m

Exercises GCSE questions on provided worksheet

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

Test Your Understanding The strategy here is to use Pythagoras twice, and use some internal triangle in the 3 D 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. 1 x 1 3 13 N 1 2 x 2 3 x = 14 =? 3. 74 2 6 8 4 x = 45 =? 6. 71 x 2 2 2 x = 28 =? 5. 29 N 2 x 8 5 2 2 2 x = 12? 4 x 2 4 x = 51 =? 7. 14 1 x 1 6 1 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