Unlocking the Secrets of Right-Angled Triangles with Dr. J. Frost
Dive into the world of right-angled triangles with Dr. J. Frost to master finding missing sides and angles using Pythagoras' Theorem. Learn mental arithmetic shortcuts and tackle exercises to enhance your skills. Explore areas of isosceles triangles and sharpen your geometry knowledge for GCSE success.
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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 3D shapes. Last modified: 2nd March 2014
RECAP: Pythagoras Theorem Hypotenuse (the longest side) For any right-angled triangle with longest side c. a2 + b2 = c2 c b a
Example Step 1: Determine the hypotenuse. x Step 2: Form an equation 2 The hypotenuse appears on its own. 22 + 42 = x2 4 Step 3: Solve the equation to find the unknown side. x2 = 4 + 16 = 20 x = 20 = 4.47 to 2dp
Pythagoras Mental Arithmetic We ve so far written out the equation ?2+ ?2= ?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 34 ? 32+ 52= 72 42= 33 ? = ? =
Pythagoras Mental Arithmetic 10 h 12 4 5 y 122+ 52= 13 ? ? 102 42= = ? = 84 9 q 1 x 2 2 22+ 12= ? ? = 5 ? 92 22= ? = 77
The Wall of Triangle Destiny ? ? Answer: ? = 5 2 3 ? Answer: ? = ?? 42 1 6 1 1 x x x 6 4 x x 4 55 ?? ? Answer: ? = 12 8 10 ???? ? Answer: ? = ?? ? Answer: ? = 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 18 x 12 6 y 10 8 ? x = 6 5 = 13.4 ? Find the height of this triangle. x = 10 5 2 x 12 ? 6 10 y x 4 3 x = 43 = 6.56 ? 9 7 ? 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 = 3 4 Area = 60 ?
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 = 2 12 = 4 3 = 6.93 ? Area = 0.48
y (a,b) When I was in Year 9 I was trying to write a program that would draw an analogue clock. r x I needed to work out between what two points to draw the hour hand given the current hour, and the length of the hand.
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 Opposite Adjacent x 60 x ? y ? z ? z y 1 2 2 ? 1 ? 1 ? 45 1 c 20 c ? a ? b ? 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. ? sin ? = ? h o ? cos ? = ? a ? ? tan ? = ? 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. Find ? (to 3sf) 20 7 4 40 x x ? ? ? = 3.06 ? = 2.39
More Examples ? = 24? x 60 12 x 4 30 ? = 8 ?
Exercise 3 Find ?, giving your answers to 3??. Please copy the diagrams first. 1 ? ? ? = 16.9 ? = 20.3 c b 22 a 15 ? ?? ?? ?? ? ? = 14.1 ? 20 ? f e d ? ? = 11.0 ? ?? ?? 4 ? ? = 7.00 ? ? ? = 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? ?.??? ? 3 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? ??? ????? = ???.?? ? ? ? = ? ? ? ? Find ?. 4 ?? ? + ?
y ?,? ? x So what is ?,? ?
RECAP: Find x 4 ? ? = tan30= 4 3 ?? 6.93 4 30 x
But what if the angle is unknown? 5 3 ? sin ? =3 ? 5 ?? ? = sin 13 = 36.9 ? 5 We can do the reverse of sin, cos or tan to find the missing angle.
What is the missing angle? ? ? ? cos 15 sin 15 cos 14 cos 14 4 4 5 5
What is the missing angle? ? ? ? cos 11 tan 11 sin 12 tan 12 2 2
What is the missing angle? ? ? ? cos 13 sin 13 tan 13 sin 15 5 5 5 3
What is the missing angle? ? ? ? cos 12 sin 12 sin 13 tan 12 3 3 2 3
The Wall of Trig Destiny ? = 45 ? 2 3 1 1 1 1 ? = 48.59 ? 4 2 3 6 8 ? = 70.53 ? 3 ? = 33.7 ? To learn secret way of math ninja, find you must.
3.19m 60 3m 40 x Find x
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. Determine the length of the internal diagonal of a unit cube. 1 3 ? 1 2 ? Click to Bro- 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 1 x 3 1 13 2 x x 2 2 2 6 3 8 2 2 x = 14 = 3.74 ? ? x = 12 x = 28 = 5.29 ? 2 2 4 x 4 8 1 1 x 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 ?
