# Pythagoras' Theorem

Students are required to understand that Pythagoras’ Theorem only applies to right angled triangles and need to use Pythagoras’ Theorem appropriately to solve problems. Students are also required to use side ratios in similar triangles to solve problems in right-angled triangles.

Visit the secondary mathematics webpage to access all lists.

### Teaching Pythagoras

This video can be used as inspiration for teachers when lesson planning. Students are presented with a starter activity, with half the class using one approach and the rest taking a different approach. The starter activity is followed up by students being presented with a problem related to playing a hole of golf. The starter activity is then revisited as a means to solving the golf problem. The lesson then moves on by asking the students to solve a murder mystery problem by using Pythagoras’s Theorem to solve problems. The whole lesson is designed such that students discover Pythagoras’s Theorem rather than it being taught to them.

### Pythagoras' Theorem

The text, **Pythagoras’ Theorem**, begins by explaining what is meant by the hypotenuse of a right-angled triangle and explains how Pythagoras’s Theorem shows a connection between the areas of the squares drawn on each side of the right-angled triangle. There follows simple, straightforward examples and exercises to consolidate understanding of the underlying principles of Pythagoras’s Theorem. The next section contains explanations, examples and exercises finding the length of the hypotenuse in a variety of situations before moving on to require the students to find the length of one of the other sides of the triangle.

More complex examples covered include finding the length of the sides of an isosceles right-angled triangle when the length of the hypotenuse is given. Students then solve problems in a variety of different contexts including problems where they are required to draw an appropriate diagram. The final section explains how Pythagoras’s Theorem can be used to identify whether a triangle is right angled, acute angled or contains an obtuse angle.

The activities sheet contains an activity in which students work through Bronowski’s proof of Pythagoras’s Theorem, explore the areas of squares on the sides of a right-angled triangle, investigate the length of the perpendicular height of an equilateral triangle and investigate Pythagorean triples.

### Task Maths 4

**Tilted squares, Chapter 10** is a good introduction to Pythagoras’s Theorem and affords the opportunity to revise finding areas. The area of the tilted square can be found by boxing in the square, finding the area of the larger square and taking away the areas of the triangles formed in the corners. Students can then investigate the connection with the ‘tilt’ - for example, a 3 unit across and 4 unit up produces a square with an area of 25 square units. The resource contains a number of extension activities such as asking whether the areas of the two semi-circles drawn on the two shorter sides of a right angled triangle have a total area equal to the area of the semi-circle drawn on the hypotenuse.

### Pythagoras' Theorem

This resource consists of 23 instant maths ideas, ideal for use as starter questions, extension question or probing questions to assess understanding. Tasks include asking students to investigate the area of tilted squares, explore whether Pythagoras’s Theorem works for all triangles, find a proof for Pythagoras’s Theorem, find families of Pythagorean triples, discuss Fermat’s last theorem and solve a variety of problems including investigating whether a square peg fits into a round hole better than a round peg fits into a square hole.

### Circles with Whole Number Points

This resource provides teacher inspiration for investigating equations of a circle and Pythagoras’ theorem. Produced originally for post 16 students it could be used as an investigation to challenge more able students. The student task sheet refers to the use of Geogebra. The task consists of drawing circle, centre (0,0), radius 5 and finding all the points with integer coordinates through which the circle passes. The activity can be extended by altering the radius of the circle or moving the centre of the circle from the origin.

### Quadthagoras

A rectangle is presented. The length of a diagonal is marked. The sides of the rectangle are given in terms of expressions in x. The challenge is to determine the value of x. Solving the problem involves the use of Pythagoras’ theorem which then gives rise to a quadratic with quadratic coefficient greater than one. The quadratic can be solved by factorisation.

The ‘something in common’ is that in each example given x turns out to be 8. A subsequent challenge is to work out how all of the answers are the same.