Pythagoras’ Theorem and Right-Angled Triangles

Calculating the Hypotenuse in a right-angled triangle

Lengths in Right-angled triangles

Finding any length of a Right-Angled Triangle

Pythagoras’ Theorem – Solving Complex Problems

Prerequisite Knowledge

• Draw and measure line segments and angles in geometric figures, including interpreting scale drawings
• Apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles
• Derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon

Key Concepts

• For a right-angled triangle, Pythagoras’ Theorem states that a² +b² = c² where c is the hypotenuse.

• A Pythagorean triple is a set of three integers that exactly fits the Pythagoras relationship.
• If the lengths of the three sides of a triangle obey Pythagoras’ Theorem the triangle is right-angled.
• Students should look for right-angled triangles in shapes with problem solving with Pythagoras’ Theorem.

Working mathematically

Develop fluency

• Use language and properties precisely to analyse 2-D and 3-D shapes.
• Use algebra to generalise the structure of arithmetic, including to formulate mathematical relationships
• Select and use appropriate calculation strategies to solve increasingly complex problems

Reason mathematically

• Make and test conjectures about patterns and relationships; look for proofs or counter-examples
• Begin to reason deductively in geometry, number and algebra, including using geometrical constructions

Solve problems

• Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems
• Develop their use of formal mathematical knowledge to interpret and solve problems
• Begin to model situations mathematically and express the results using a range of formal mathematical representations
• Select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems

Subject Content

Geometry and measures

• Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs
• Use Pythagoras’ Theorem in similar triangles to solve problems involving right-angled triangles
• Interpret mathematical relationships both algebraically and geometrically.