Week 9 · Pythagoras' theorem and its converse: finding sides and classifying triangles · 3 lessons
1
Pythagoras' theorem: finding a missing side of a right-angled triangle
Mathematics Prep 2 Term 1, Unit 3, Lesson 2
GoalUse c² = a² + b² to find a hypotenuse or a leg, and solve simple problems with two right-angled triangles or a ladder.
New words
hypotenuse · the side opposite the right angle; it is the longest sidelegs · the two sides that make the right anglePythagoras’ theorem · c² = a² + b², where c is the hypotenusesquare root · undoes squaring: √25 = 5
Explanation
In a right-angled triangle the square on the hypotenuse equals the sum of the squares on the other two sides: c² = a² + b². The picture shows it for the 3-4-5 triangle: the squares on the legs have areas 9 and 16, and 9 + 16 = 25 is the square on the hypotenuse. To find the hypotenuse, add the squares of the legs and take the square root: legs 6 and 8 give c² = 36 + 64 = 100, so c = 10. To find a leg, subtract: b² = c² − a². If the answer is not a whole number, leave it as a root (legs 4 and 5 give c = √41) and use a calculator only when the problem asks for a decimal. Be careful: c² = 100 is not yet the length; the length is c = 10. The hypotenuse is always the longest side, so c is larger than both legs.
Worked examples
Find the hypotenuse when the legs are 6 cm and 8 cm. c² = 6² + 8² = 36 + 64 = 100, so c = 10 cm. Now find a leg: the hypotenuse is 13 cm and one leg is 5 cm. b² = 13² − 5² = 169 − 25 = 144, so b = 12 cm.
A 5 m ladder leans against a wall with its foot 3 m from the wall. How high does it reach? The ladder is the hypotenuse: h² = 5² − 3² = 25 − 9 = 16, so h = 4 m.
Two right-angled triangles share a side. In Δ ABC the right angle is at B, AB = 3 and BC = 4, so AC = 5. In Δ ACD the right angle is at C and CD = 12. Then AD² = 5² + 12² = 169, so AD = 13. The area of a right-angled triangle with hypotenuse 17 and a leg of 8 needs the other leg: √(289 − 64) = 15, so the area is ½ × 8 × 15 = 60.
Class activity
3-4-5 rope. Mark a rope with knots at 3, 4 and 5 equal spaces (for example 30 cm each) and join the ends to make a triangle. Check the corner opposite the longest side with a book corner. Is it a right angle?
Practice
1
Find the hypotenuse of a right-angled triangle with legs 9 and 12.
81 + 144 = 225, so c = 15.
2
The hypotenuse is 25 and one leg is 7. Find the other leg.
625 − 49 = 576, so the leg is 24.
3
Find the hypotenuse for legs 4 and 5. Leave the answer as a root.
16 + 25 = 41, so c = √41.
4
A 2.5 m ladder has its foot 1.5 m from a wall. How high does it reach?
h² = 6.25 − 2.25 = 4, so h = 2 m.
5
Mariam says the hypotenuse for legs 6 and 8 is 100. What did she forget?
