Right Angled Triangles

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Pythagoras Theorem

Pythagoras' theorem: For a right-angled triangle with hypotenuse×c×and shorter sides×a×,×b×: a² + b² = c².
The hypotenuse is the longest side, opposite the right angle.
To find the hypotenuse: c = √(a² + b²) (add squares).
To find a shorter side: a = √(c² – b²) (subtract squares).
If the hypotenuse is shorter than another side, you have made a mistake.
Pythagoras can be used in any shape that can be split into right-angled triangles (e.g., diagonal of a rectangle).
In multi-step problems, leave intermediate answers as exact (e.g., √63) until the final step.

Right-angled triangle

SOHCAHTOA

SOHCAHTOA is a mnemonic for trig ratios in right-angled triangles.
sin θ = Opposite / Hypotenuse (SOH).
cos θ = Adjacent / Hypotenuse (CAH).
tan θ = Opposite / Adjacent (TOA).
Label sides relative to angle θ: H (hypotenuse), O (opposite), A (adjacent).
To find a missing side: choose the correct ratio, substitute known values, and rearrange.
To find a missing angle: use inverse trig functions (e.g., θ = sin⁻¹(O/H)).
Ensure calculator is in degree mode (D or Deg).

SOHCAHTOA

Angles of Elevation & Depression

Angle of elevation: angle above horizontal when looking up at an object.
Angle of depression: angle below horizontal when looking down at an object.
These angles are measured from the horizontal to the line of sight.
Right-angled trigonometry (often tan) is used to find distances or heights.
Draw a clear diagram; use alternate angles if needed (e.g., angle of depression equals angle at base).

Angles of elevation and depression

Exact Trig Values

Know exact values for 0°, 30°, 45°, 60°, 90°.
sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1.
cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, cos 90° = 0.
tan 0° = 0, tan 30° = √3/3, tan 45° = 1, tan 60° = √3, tan 90° undefined.
Use special triangles (45°-45°-90° and 30°-60°-90°) to derive these values.
In non-calculator questions, substitute exact values and simplify (e.g., cos45° = √2/2).
Sketch the triangles or table at the start of the exam for quick reference.

Triangle for 30° and 60°

Problem Solving with Right-Angled Triangles

Identify right-angled triangles within composite shapes (e.g., quadrilaterals, prisms, circles).
The shortest distance from a point to a line is the perpendicular distance; form a right-angled triangle.
In multi-step problems, find intermediate lengths using Pythagoras or trig before final answer.
Round final answers as instructed: sides to 3 significant figures, angles to 1 decimal place.
Check reasonableness: hypotenuse is longest side; angles in a triangle sum to 180°.

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Câu hỏi luyện tập

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1.In a right-angled triangle, the side opposite the right angle is called the
Easy
- Ahypotenuse
- Badjacent
- Copposite
- Dbase
2.What is the value of sin 30°?
Easy
- A0
- B1/2
- C√2/2
- D√3/2
3.Which trigonometric ratio is defined as opposite/hypotenuse?
Easy
- Asine
- Bcosine
- Ctangent
- Dsecant
4.In a right-angled triangle with sides 3 cm and 4 cm, the length of the hypotenuse is
Easy
- A5 cm
- B7 cm
- C1 cm
- D25 cm
5.A right-angled triangle has hypotenuse 10 cm and one leg 6 cm. The length of the other leg is
Medium
- A8 cm
- B4 cm
- C√136 cm
- D16 cm
6.In triangle ABC, angle B = 90°, AB = 5 cm, BC = 12 cm. Find AC.
Medium
- A13 cm
- B7 cm
- C17 cm
- D√119 cm
7.Calculate the angle x in a right-angled triangle where the opposite side is 5 cm and the adjacent side is 12 cm. Give your answer correct to 1 decimal place.
Medium
- A22.6°
- B67.4°
- C24.6°
- D65.4°
8.A ladder 5 m long leans against a vertical wall. The foot of the ladder is 3 m from the wall. How high up the wall does the ladder reach?
Medium
- A4 m
- B2 m
- C√34 m
- D8 m

Thêm câu hỏi

9.In a right-angled triangle, the hypotenuse is 13 cm and one leg is 5 cm. Find the length of the other leg.
Medium
- A12 cm
- B8 cm
- C√194 cm
- D18 cm
10.Find the exact value of cos 60°.
Easy
- A1/2
- B√3/2
- C√2/2
- D0
11.The angle of elevation of the top of a tree from a point 20 m away from its base is 30°. Find the height of the tree. Give your answer as an exact value.
Medium
- A20/√3 m
- B20√3 m
- C10 m
- D40 m
12.A right-angled triangle has sides of length x, x+1, and x+2. Find x.
Hard
- A3
- B1
- C2
- D4
13.In triangle PQR, angle Q = 90°, PQ = 8 cm, QR = 15 cm. Find angle PRQ correct to 1 decimal place.
Medium
- A28.1°
- B61.9°
- C32.0°
- D58.0°
14.The diagonal of a rectangle is 13 cm and one side is 5 cm. Find the perimeter of the rectangle.
Medium
- A34 cm
- B26 cm
- C30 cm
- D28 cm
15.A ship sails 30 km due east then 40 km due north. How far is it from its starting point?
Medium
- A50 km
- B70 km
- C10 km
- D√2500 km
16.In a right-angled triangle, the side opposite angle θ is 7 cm and the hypotenuse is 25 cm. Find sin θ.
Easy
- A7/25
- B24/25
- C7/24
- D25/7
17.In triangle ABC, angle B = 90°, AB = 8 cm, BC = 15 cm. Calculate the length of AC.
Medium
- A17 cm
- B12.7 cm
- C23 cm
- D7 cm
18.A ladder 6.5 m long leans against a vertical wall. The foot of the ladder is 2.5 m from the wall. How high up the wall does the ladder reach?
Medium
- A6 m
- B6.5 m
- C5 m
- D7 m
19.In a right-angled triangle, the hypotenuse is 17 cm and one leg is 8 cm. Find the length of the other leg.
Medium
- A15 cm
- B9 cm
- C18.8 cm
- D12 cm
20.In triangle PQR, angle Q = 90°, PQ = 12 cm, QR = 5 cm. Calculate angle PRQ correct to 1 decimal place.
Medium
- A22.6°
- B67.4°
- C24.0°
- D65.0°
21.The diagonal of a rectangle is 15 cm and one side is 9 cm. Find the perimeter of the rectangle.
Medium
- A42 cm
- B48 cm
- C36 cm
- D30 cm
22.A ship sails 50 km due east then 120 km due north. How far is it from its starting point?
Medium
- A130 km
- B170 km
- C70 km
- D140 km
23.In a right-angled triangle, the side opposite angle θ is 9 cm and the hypotenuse is 41 cm. Find sin θ.
Easy
- A9/41
- B40/41
- C9/40
- D41/9
24.What is the exact value of tan 60°?
Easy
- A√3
- B1/√3
- C√3/2
- D1
25.The angle of elevation of the top of a building from a point 30 m away from its base is 45°. Find the height of the building.
Medium
- A30 m
- B30√2 m
- C30/√2 m
- D15√2 m
26.In the diagram, a right-angled triangle has sides of length x, x+2, and x+4. Find x.
Medium
- A6
- B8
- C10
- D12
27.A right-angled triangle has legs of length 7 cm and 24 cm. Calculate the length of the hypotenuse.
Easy
- A25 cm
- B31 cm
- C23 cm
- D26 cm
28.In triangle ABC, angle B = 90°, AB = 3√2 cm, BC = 3√2 cm. Find angle ACB.
Medium
- A45°
- B30°
- C60°
- D90°
29.From the top of a lighthouse 80 m high, the angle of depression of a boat is 30°. How far is the boat from the foot of the lighthouse? (Give your answer correct to 3 significant figures.)
Medium
- A139 m
- B46.2 m
- C160 m
- D92.4 m
30.Find the exact value of sin 45° × cos 45°.
Medium
- A1/2
- B√2/2
- C1/4
- D1
31.A rectangle has length 12 cm and width 5 cm. Calculate the angle between the diagonal and the longer side.
Medium
- A22.6°
- B67.4°
- C24.0°
- D65.0°
32.In triangle ABC, angle C = 90°, AC = 8 cm, BC = 15 cm. Find the length of the altitude from C to AB.
Hard
- A7.06 cm
- B6.67 cm
- C8.00 cm
- D9.00 cm
33.In a right-angled triangle, which side is always the longest?
Easy
- AOpposite
- BAdjacent
- CHypotenuse
- DBase
34.What is the exact value of sin 45°?
Easy
- A1/2
- B√2/2
- C√3/2
- D1
35.Which trigonometric ratio is defined as adjacent/hypotenuse?
Easy
- ASine
- BCosine
- CTangent
- DCotangent
36.A right-angled triangle has sides of length 5 cm and 12 cm. What is the length of the hypotenuse?
Easy
- A13 cm
- B17 cm
- C7 cm
- D√119 cm
37.In a right-angled triangle, the hypotenuse is 15 cm and one leg is 9 cm. Find the length of the other leg.
Easy
- A12 cm
- B6 cm
- C√306 cm
- D24 cm
38.In triangle ABC, angle B = 90°, AB = 6 cm, BC = 8 cm. Find AC.
Easy
- A10 cm
- B14 cm
- C2 cm
- D√28 cm
39.In a right-angled triangle, the side opposite angle θ is 8 cm and the adjacent side is 15 cm. Find tan θ.
Easy
- A8/15
- B15/8
- C8/17
- D15/17
40.A ladder 10 m long leans against a vertical wall. The foot of the ladder is 6 m from the wall. How high up the wall does the ladder reach?
Medium
- A8 m
- B4 m
- C√136 m
- D16 m
41.In a right-angled triangle, the longest side is called the
Easy
- Ahypotenuse
- Bopposite
- Cadjacent
- Daltitude
42.What is the exact value of sin 60°?
Easy
- A1/2
- B√3/2
- C√2/2
- D1
43.Match each trigonometric ratio to its correct definition.
Easy
- sin θ
- cos θ
- tan θ
- opposite / hypotenuse
- adjacent / hypotenuse
- opposite / adjacent
44.Order the following steps to find a missing side using SOHCAHTOA.
Medium
- Label the sides H, O and A
- Identify which ratio to use
- Substitute values into the formula
- Rearrange and solve
45.Which of the following statements about a right-angled triangle are true? (Select all that apply)
Medium
- AThe hypotenuse is the longest side.
- BPythagoras' theorem can be used on any triangle.
- CThe side opposite the right angle is the hypotenuse.
- DSOHCAHTOA can only be used in right-angled triangles.
46.The exact value of sin 45° is \frac{\√{2}}{2}.
Easy
True or false?
47.Which of the following are exact values of trigonometric functions? (Select all that apply)
Medium
- Asin 60° = \frac{\√{3}}{2}
- Bcos 45° = \frac{1}{\√{2}}
- Ctan 30° = \frac{1}{\√{3}}
- Dsin 90° = 0
48.What is the exact value of sin 30°?
Easy
- A1/2
- B√3/2
- C√2/2
- D1
49.In the diagram, a right-angled triangle has legs of length 6 cm and 8 cm. What is the length of the hypotenuse?
Easy
- A10 cm
- B12 cm
- C14 cm
- D2√7 cm
50.In a right-angled triangle ABC, angle B = 90°, AB = 5 cm, BC = 12 cm. What is the length of AC?
Easy
- A7 cm
- B13 cm
- C17 cm
- D√119 cm
51.A right-angled triangle has legs of length 5 cm and 12 cm. Calculate the length of the hypotenuse.
Easy
52.Write down the exact value of cos 60°.
Easy
53.A right-angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the length of the other leg.
Easy
54.In a right-angled triangle, the side opposite angle θ is 7 cm and the adjacent side is 24 cm. Find the value of tan θ.
Medium
55.Complete the sentence.
Easy
In a right-angled triangle, the side opposite the right angle is called the ____.
56.Write down the exact value of cos 30°.
Easy
57.A right-angled triangle has legs of length 8 cm and 15 cm. Calculate the length of the hypotenuse.
Easy
58.A right-angled triangle has legs of length 9 cm and 12 cm. Calculate the length of the hypotenuse.
Easy
59.Complete the sentence using one of the words: opposite, adjacent, hypotenuse.
Easy
In a right-angled triangle, the side opposite the right angle is called the ____. The side next to the angle θ is called the ____. The side opposite the angle θ is called the ____.
60.Write down the exact value of tan 60°.
Easy
61.A ladder 5.5 m long leans against a vertical wall. The foot of the ladder is 3 m from the wall. How high up the wall does the ladder reach? Give your answer correct to 1 decimal place.
Medium
62.A right-angled triangle has an angle of 30° and an adjacent side of length 10 cm. Use exact trig values to find the length of the opposite side.
Medium
tan 30° = ____ / ____. The exact value of tan 30° is ____. Therefore the opposite side length is ____ cm.
63.In triangle ABC, angle B = 90°, AB = 8 cm, BC = 15 cm. Calculate angle ACB correct to 1 decimal place.
Medium
64.A right-angled triangle has legs of length 5 cm and 12 cm. Calculate the length of the hypotenuse.
Easy
- A13 cm
- B14 cm
- C15 cm
- D12 cm
65.A right-angled triangle has a hypotenuse of 13 cm and one leg of 5 cm. Find the length of the other leg.
Easy
- A11 cm
- B14 cm
- C12 cm
- D13 cm
66.In a right-angled triangle, the side opposite angle θ is 7 cm and the adjacent side is 24 cm. Find the value of tan θ.
Medium
- A2.2917
- B-0.7083
- C0.2917
- D1.2917
67.A right-angled triangle has legs of length 8 cm and 15 cm. Calculate the length of the hypotenuse.
Easy
- A19 cm
- B16 cm
- C17 cm
- D18 cm
68.A right-angled triangle has legs of length 9 cm and 12 cm. Calculate the length of the hypotenuse.
Easy
- A14 cm
- B17 cm
- C15 cm
- D16 cm
69.A ladder 5.5 m long leans against a vertical wall. The foot of the ladder is 3 m from the wall. How high up the wall does the ladder reach? Give your answer correct to 1 decimal place.
Medium
- A3.6 m
- B5.6 m
- C6.6 m
- D4.6 m

Ghi chú ôn tập

## Right-Angled Triangles (Pythagoras & Trigonometry) ### Pythagoras Theorem - For a right-angled triangle with hypotenuse **c** and shorter sides **a**, **b**: \[ a^2 + b^2 = c^2 \] - **Finding hypotenuse**: \( c = \sqrt{a^2 + b^2} \) - **Finding a shorter side**: \( a = \sqrt{c^2 - b^2} \) - The hypotenuse is always opposite the right angle and is the longest side. ### SOHCAHTOA Label sides relative to angle θ: **H** (hypotenuse), **O** (opposite), **A** (adjacent). - \( \sin \theta = \frac{O}{H} \) - \( \cos \theta = \frac{A}{H} \) - \( \tan \theta = \frac{O}{A} \) - **To find a side**: Identify known and wanted sides, choose correct ratio, substitute, solve. - **To find an angle**: Use inverse trig functions (e.g., \( \theta = \sin^{-1}(O/H) \)). - Only works in right-angled triangles. ### Angles of Elevation & Depression - **Angle of elevation**: measured upwards from horizontal. - **Angle of depression**: measured downwards from horizontal. - Use right-angled trigonometry (often tan) to find heights or distances. ### Exact Trig Values | θ | 0° | 30° | 45° | 60° | 90° | |---|----|-----|-----|-----|-----| | sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 | | cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 | | tan θ | 0 | √3/3 | 1 | √3 | undefined | - Use special triangles (45°-45°-90°, 30°-60°-90°) to derive values. - Common in non-calculator questions. ### Tips - Always ensure calculator is in **degree mode**. - When finding a shorter side with Pythagoras, subtract the smaller square from the larger. - Leave answers in exact form (surd) until final step to avoid rounding errors. - The shortest distance from a point to a line is the perpendicular distance.

Pythagoras' theorem: For a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
SOHCAHTOA: Mnemonic for trig ratios: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.
Angle of elevation: The angle measured upwards from the horizontal to an object.
Angle of depression: The angle measured downwards from the horizontal to an object.
Exact trig values: Known exact values of sin, cos, and tan for angles 0°, 30°, 45°, 60°, 90° (often expressed as fractions and surds).

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