GCSE Right-Angled Triangles Dr J Frost (jfrosttiffin.kingston.sch.uk) Last modified: 2nd March 2014 Learning Objectives: To be able to find missing sides and missing angles in right-angled triangles and 3D shapes. For any right-angled
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GCSE Right-Angled Triangles Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 2nd March 2014 Learning Objectives: To be able to find missing sides and missing angles in right-angled triangles and 3D shapes.<br>
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For any right-angled triangle with longest side c.
a2 + b2 = c2 c b a !<br>
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2 4 x Step 1: Determine the hypotenuse. Step 2: Form an equation 22 + 42 = x2 Step 3: Solve the equation to find the unknown side. x2 = 4 + 16 = 20
x = √20 = 4.47 to 2dp<br>
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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 ? ?<br>
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12 5 h ? 4 y 10 ? x 2 9 ? 1 2 q ?<br>
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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 ? ? ? ? ?<br>
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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 = 65 = 13.4 18 y 12 4 6 3 4 x x = 43 = 6.56 5 x 1 1 1 x = 3 = 1.73 6 7 13 13 10 Find the height of this triangle. 12 9 x x x2 + 49 = 81 – x2
x = 4 7 N ? ? ? ? ? ? ? ?<br>
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To find the area of an isosceles triangle, simplify split it into two right-angled triangles. 1 1 1 10 13 13 12 3
2 Area = 3
4 Area = 60 ? ? ? ?<br>
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Determine the area of the following triangles. 6 5 Area = 12 ? 5 4 4 Area = 212 = 43 = 6.93 ? 4 16 17 Area = 120 ? 17 1.6 1 Area = 0.48 ? 1 12 12 Area = 40.2 ? 7 1 2 3 4 5<br>
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Frost Childhood Story 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.<br>
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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?<br>
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30° hypotenuse adjacent opposite ? ? ?<br>
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60° x y z 45° 1 √2 1 20° a c b ? ? ? ? ? ? ? ? ?<br>
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sin, cos and tan give us the ratio between pairs of sides in a right angle triangle, given the angle. ? ? ?<br>
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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 ? ? ?<br>
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Step 1: Determine which sides are hyp/adj/opp.
Step 2: Work out which trigonometric function we need. ? ?<br>
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? ?<br>
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1 a b c d e f 2 3 4 ? ? ? ? ? ? ? ? ?<br>
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Frost Childhood Story x y θ<br>
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?<br>
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? ? We can do the ‘reverse’ of sin, cos or tan to find the missing angle.<br>
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What is the missing angle?<br>
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What is the missing angle?<br>
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What is the missing angle?<br>
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What is the missing angle?<br>
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The Wall of Trig Destiny 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 ? ? ? ?<br>
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Real-World Example Find x 3.19m<br>
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GCSE questions on provided worksheet<br>
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The strategy here is to use Pythagoras twice, and use some internal triangle in the 3D shape. 1 1 1 √2 √3 Determine the length of the internal diagonal of a unit cube. ? ? Click to Bro-Sketch<br>
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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. ?<br>
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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 ?<br>
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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 2 2 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 N1 N2<br>
