1. 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

2. 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

3. 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
= x2
The hypotenuse appears on its own. 4
Step 3: Solve the equation to find the unknown side.
x2 = = 20
x = √20 = 4.47 to 2dp

4. 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= = 34 ?
𝑥= − 4 2 = 33 ?

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

6. 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.”

7. 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
x = 81 – x2
x = 4 ? 5 1
x = 29 = 5.39 ? ?
x = 3 = 1.73

8. 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 ?

9. 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

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

11. 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

12. “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”

13. One way to remember this…

14. 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

15. 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 ?

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

17. 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

18. 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 − =36.9° ? ?
We can do the ‘reverse’ of sin, cos or tan to find the missing angle.

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

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

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

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

23. 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.”

24. 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

25. 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

26. 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.

27. 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 ?