CIE IGCSE Mathematics (0580) Pythagoras’ theorem Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Pythagoras’ theorem Study Notes
LEARNING OBJECTIVE
- When to Use Pythagoras’ Theorem
Key Concepts:
- Pythagoras’ Theorem
Pythagoras’ Theorem
Pythagoras’ Theorem
Pythagoras’ Theorem relates the lengths of the sides in a right-angled triangle. It applies only to right-angled triangles.
If a triangle has a right angle (90°), then:
\( \text{(Hypotenuse)}^2 = \text{(Opposite side)}^2 + \text{(Adjacent side)}^2 \)
Or, more formally: \( c^2 = a^2 + b^2 \) where \( c \) is the hypotenuse (the side opposite the right angle).
When to Use:
- To find the length of a missing side in a right-angled triangle.
- To check if a triangle is a right triangle by verifying if \( c^2 = a^2 + b^2 \).
Important:
- Always identify the hypotenuse—it is the longest side, opposite the right angle.
- Use square roots to find side lengths when solving.
Example:
A triangle has two sides of length 6 cm and 8 cm. Find the length of the hypotenuse.
▶️ Answer/Explanation
\( c^2 = a^2 + b^2 = 6^2 + 8^2 = 36 + 64 = 100 \)
\( c = \sqrt{100} = 10 \)
Answer: 10 cm
Example:
Find the missing side of a right triangle if the hypotenuse is 13 cm and one side is 5 cm.
▶️ Answer/Explanation
\( a^2 + b^2 = c^2 \)
\( 5^2 + b^2 = 13^2 \Rightarrow 25 + b^2 = 169 \Rightarrow b^2 = 144 \)
\( b = \sqrt{144} = 12 \)
Answer: 12 cm
Example:
A ladder is leaning against a wall. The foot of the ladder is 2.5 m away from the base of the wall, and the top of the ladder reaches 6 m up the wall. How long is the ladder? Give your answer correct to 2 decimal places.
▶️ Answer/Explanation
This forms a right-angled triangle, where:
- Vertical side (height on wall): \( 6 \, \text{m} \)
- Horizontal side (distance from wall): \( 2.5 \, \text{m} \)
- Hypotenuse (length of ladder): \( c \)
\( c^2 = a^2 + b^2 = 6^2 + 2.5^2 = 36 + 6.25 = 42.25 \)
\( c = \sqrt{42.25} = 6.5 \)
Answer: The ladder is 6.50 m long.
Example:
\(ABCD\) is a trapezium.
Given:
- \( AD = 10 \, \text{cm} \)
- \( AB = 9 \, \text{cm} \)
- \( DC = 3 \, \text{cm} \)
- \( \angle ABC = \angle BCD = 90^\circ \)
Calculate the length of \( AC \). Give your answer correct to 3 significant figures.
▶️ Answer/Explanation
From the diagram, draw verticals:
- \( AB = 9 \, \text{cm} \)
- \( DC = 3 \, \text{cm} \)
So vertical difference between A and M is:
\( 9 – 3 = 6 \, \text{cm} \)
Apply Pythagoras in triangle \( \triangle ADM \) to find base \( BC =DM\):
Let horizontal distance from B to C be \( x \)
\( AD^2 = x^2 + 6^2 \Rightarrow 10^2 = x^2 + 36 \)
\( 100 = x^2 + 36 \Rightarrow x^2 = 64 \Rightarrow x = 8 \, \text{cm} \)
Now apply Pythagoras in triangle \( \triangle ABC \):
- Vertical side = \( AB = 9 \, \text{cm} \)
- Horizontal side = \( BC = 8 \, \text{cm} \)
\( AC^2 = 9^2 + 8^2 = 81 + 64 = 145 \)
\( AC = \sqrt{145} \approx 12.04159 \)