Ch 1 · 6.1–6.2 Pythagoras' Theorem and Right-angled Triangles · O Level Mathematics 4024
Topic 1 of 3 in Trigonometry — O Level Mathematics 4024 — 46 lessons.
Introduction
How tall is a tree you cannot climb? How far is a boat from the foot of a cliff? Right-angled triangles answer these with two tools: Pythagoras' theorem for sides, and trigonometry for angles. (Syllabus sections 6.1 and 6.2.)
Key ideas — Pythagoras' theorem
In a right-angled triangle with hypotenuse c (the longest side, opposite the right angle):
🧑🏫 Teacher's explanation
Imagine you are standing at the corner of a rectangular park. If you walk along the two edges (the length and the width), you travel a certain distance. If you decide to cut across the middle directly to the opposite corner, you are taking the 'hypotenuse' path. Pythagoras' theorem tells us exactly how much shorter that shortcut is compared to walking the edges.
📖 Definitions
🔍 Worked example — Easy
A right-angled triangle has shorter sides of length 3 cm and 4 cm. Calculate the length of the hypotenuse.
🔍 Worked example — Standard
The hypotenuse of a right-angled triangle is 13 cm and one side is 5 cm. Find the length of the remaining side.
🔍 Worked example — Exam-style
A ladder of length 5 m leans against a vertical wall. The foot of the ladder is 1.5 m from the base of the wall. Calculate the height the ladder reaches up the wall, correct to 1 decimal place.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — The trigonometric ratios
Label the sides relative to the angle θ: opposite, adjacent, hypotenuse.
🧑🏫 Teacher's explanation
Imagine you are standing at the base of a tall tree. If you look up at the top, the distance from your eyes to the top is the hypotenuse. The height of the tree is the 'opposite' side relative to your line of sight, and the distance along the ground to the tree is the 'adjacent' side. Trigonometry is simply the study of the fixed relationships between these lengths and the angle you are looking up at.\n\nBecause the ratios between the sides of a right-angled triangle are always the same for a given angle, we don't need to know the actual size of the triangle to find missing parts. If we know one angle and one side, we can use these ratios to 'unlock' the rest of the triangle. It is like having a blueprint where the proportions are locked in place by the angles.
📖 Definitions
🔍 Worked example — Easy
In a right-angled triangle, the hypotenuse is 10 cm and the angle θ is 30^°. Calculate the length of the side opposite to θ.
🔍 Worked example — Standard
A ladder 5 m long leans against a vertical wall. The foot of the ladder is 3 m from the base of the wall. Calculate the angle θ that the ladder makes with the ground.
🔍 Worked example — Exam-style
An observer at point A looks up at the top of a tower at point B. The angle of elevation is 25^°. The perpendicular distance from A to the base of the tower is 40 m. Calculate the height of the tower to 1 decimal place.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Elevation, depression and bearings
🧑🏫 Teacher's explanation
Imagine you are standing on a cliff looking out at a boat. If you look straight ahead (horizontally) and then tilt your head up to see a bird, that upward tilt is your angle of elevation. If you tilt your head down to see the boat, that downward tilt is your angle of depression.
📖 Definitions
🔍 Worked example — Easy
A person stands 50 m from the base of a vertical tower. The angle of elevation to the top of the tower is 35^°. Calculate the height of the tower.
🔍 Worked example — Standard
A ship sails from Port A on a bearing of 070^° for 12 km to reach Point B. How far East has the ship traveled?
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Shortest distance
The shortest distance from a point to a line is the perpendicular distance, so it makes a right angle with the line.
🧑🏫 Teacher's explanation
Imagine you are standing at the edge of a straight road and you want to walk to the road as quickly as possible. You wouldn't walk at an angle; you would walk directly towards it so that your path hits the road at a 90^° angle. This 'straight-on' path is the shortest possible way to reach the line.
📖 Definitions
🔍 Worked example — Easy
A point A is located 5 cm from a straight line L. Calculate the shortest distance from A to L.
🔍 Worked example — Standard
A ship is 12 km due North of a lighthouse. The ship moves due East until it is directly North of a buoy. If the shortest distance from the lighthouse to the ship's path is 12 km, find the distance from the lighthouse to the buoy if the buoy is 5 km East of the lighthouse's longitude.
🔍 Worked example — Exam-style
A vertical pole of height 8 m stands on horizontal ground. An observer is 15 m away from the base of the pole. Calculate the shortest distance from the top of the pole to the observer.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
A 6 m ladder leans against a wall with its foot 1.8 m from the wall. Height reached = √(6^2 - 1.8^2) = 5.72 m. The angle with the ground is cos^-1 (1.8/6 ) = 72.5^°.
🔍 Worked example — Easy
A 6 m ladder leans against a wall. The foot of the ladder is 1.8 m from the base of the wall. Calculate the height the ladder reaches up the wall and the angle it makes with the ground.
🔍 Worked example — Standard
From a point 30 m away from the base of a tree, the angle of elevation to the top of the tree is 35^°. Calculate the height of the tree.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 15 cm. (2) tan^-15/12 = 22.6^°. (3) 15sin 28^° = 7.04 m. (4) √(144 + 64) = 14.4 km.