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Chapter 11: Pythagoras’ Theorem Form 3
Chapter 11: Pythagoras’ Theorem (Edexcel Chapter 19 pg 295, exercises 19A, 19B) Pythagoras was a famous mathematician in Ancient Greece. The theorem which is named after him is an important result about right – angled triangles. The angle B is the right angle. The side, AB, opposite the right angle is called the hypotenuse. It is the longest side in the triangle.
A
B C
Chapter 11: Pythagoras’ Theorem Form 3
The area of the square on the side of length 3 units is 9 units2. The area of the square on the side of length 4 units is 16 units2. The area of the square on the side of length 5 units (the hypotenuse) is 25 units2. We can notice that 25 units2 = 9 units2 + 16 units2 Thus, 52 = 32 + 42
Pythagoras’s theorem states: In a right-angled triangle, the area of the square on the hypotenuse is equal to the sum of areas of the square on the other two sides. Pythagoras’ theorem can be used to find the length of the third side of a right-angled triangle when the lengths of the other two sides are known. For this, the theorem is usually stated in terms of the lengths of the sides of the triangle.
c2 = a2 + b2 Apply Pythagoras’ theorem to this right-angled triangle: CB2 = CA2 + BA2 11.1 Finding Lengths Aim: 19A pg 297 Example 1: Work out the length of the hypotenuse in this triangle.
A B
C
Chapter 11: Pythagoras’ Theorem Form 3
No matter the position of the right angled-triangle, Pythagoras’ Theorem still applies.
Example 2: In triangle XYZ, angle X = 90°, XY = 8.6cm and XZ = 13.9 cm. Work out the length of YZ. Give your answer correct to 3 significant figures.
Pythagoras’ theorem can also be used to work out the length of one of the shorter sides in a right- angled triangle when the lengths of the other two sides are known.
Example 3: Find the missing length. Give you answer to 3 s.f
Chapter 11: Pythagoras’ Theorem Form 3
Example 4: In triangle ABC, angle A = 90°, BC = 17.4cm and AC = 5.8cm. Work out the length of AB. Give your answer correct to 3 significant figures.
11.2 Applying Pythagoras’s Theorem Aim: 19B pg 300 Example1: Below is the plan of a rectangular field. There is a footpath across the field from A to C. How much shorter is it to use the footpath than to walk from A to B and then to C?
Chapter 11: Pythagoras’ Theorem Form 3
Example 2: A helicopter flies 26km north from a heliport, then 19 km west. How far is it from the heliport now?
H
N 19 Km
26 km
Chapter 11: Pythagoras’ Theorem Form 3
Example 3: The diagram shows an Isosceles triangle ABC. The midpoint of BC is the point M. In the triangle, AB = AC = 8 cm and BC = 6cm.
a) Work out the height, AM, of the triangle. Give your answer to 3 s.f.
