NSW Syllabuses

# Mathematics K–10 - Stage 4 - Measurement and Geometry Right-Angled Triangles (Pythagoras)

## Outcomes

#### A student:

• MA4-1WM

communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

• MA4-2WM

applies appropriate mathematical techniques to solve problems

• MA4-16MG

applies Pythagoras’ theorem to calculate side lengths in right-angled triangles, and solves related problems

## Content

• identify the hypotenuse as the longest side in any right-angled triangle and also as the side opposite the right angle
• establish the relationship between the lengths of the sides of a right-angled triangle in practical ways, including with the use of digital technologies
• describe the relationship between the sides of a right-angled triangle (Communicating)
• use Pythagoras' theorem to find the length of an unknown side in a right-angled triangle
• explain why the negative solution of the relevant quadratic equation is not feasible when solving problems involving Pythagoras' theorem (Communicating, Reasoning)
• write answers to a specified or sensible level of accuracy, using an 'approximately equals' sign, ie $$\, \doteqdot$$ or $$\, \approx \,$$
• solve a variety of practical problems involving Pythagoras' theorem, approximating the answer as a decimal
• apply Pythagoras' theorem to solve problems involving the perimeters and areas of plane shapes (Problem Solving)
• identify a Pythagorean triad as a set of three numbers such that the sum of the squares of the first two equals the square of the third
• use the converse of Pythagoras' theorem to establish whether a triangle has a right angle
• use technology to explore decimal approximations of surds
• recognise that surds can be represented by decimals that are neither terminating nor have a repeating pattern (Communicating)
• solve a variety of practical problems involving Pythagoras' theorem, giving exact answers (ie as surds where appropriate), eg $$\sqrt{5}$$

### Background Information

Students should gain an understanding of Pythagoras' theorem, rather than just being able to recite the formula. By dissecting squares and rearranging the dissected parts, they will appreciate that the theorem is a statement of a relationship among the areas of squares.

In Stage 5, Pythagoras' theorem becomes the formula for the circle on the Cartesian plane. These links can be developed later in the context of circle geometry and the trigonometry of the general angle.

Pythagoras' theorem is named for the Greek philosopher and mathematician Pythagoras (c580–c500 BC), who is credited with its discovery. However, it is probable that the theorem was known to the Babylonians 1000 years earlier.

In the 1990s, the British mathematician Andrew Wiles (b 1953) finally proved a famous conjecture made by the French lawyer and mathematician Pierre de Fermat (1601–1665), known as 'Fermat's last theorem', which states that if $$n$$ is an integer greater than 2, then $$\, a^n + b^n = c^n$$ has no positive-integer solutions.

#### Purpose/Relevance of Substrand

Pythagoras' theorem is of great importance in the mathematics learned in secondary school and is also important in many areas of further mathematics and science study. The theorem is part of the foundations of trigonometry, an important area of mathematics introduced in Stage 5, which allows the user to determine unknown sides and angles in both right-angled and non-right-angled triangles. Pythagoras' theorem has many everyday applications, such as building construction (to ensure that buildings, rooms, additions, etc are square) and determining the length of any diagonal (eg the screen size of a television, the shortest distance between two points).

### Language

Students need to understand the difference between an 'exact' answer and an 'approximate' answer.

They may find some of the terminology encountered in word problems involving Pythagoras' theorem difficult to interpret, eg 'foot of a ladder', 'inclined', 'guy wire', 'wire stay', 'vertical', 'horizontal'. Teachers should provide students with a variety of word problems and explain such terms explicitly.