Lesson 25 Triangle ratios

# Episode 1

Reasoning Resources: Set of similar right-angles triangles (see below), Rulers, Blu-tack
Developing an understanding of the properties of similar triangles, in particular that corresponding sides have a constant ratio between them. Prepare a set of similar right-angled triangles, each with a 30° angle: one with adjacent side 10 cm, one with side 20 cm and the third with 30 cm, using different coloured paper. If you prefer to display the triangles on OHP use triangles with 3,6 and 9 cm sides. Whole class work
Mathematical similarity of shapes is precise: the size is different but the elements are either the same (e.g. angle) or are related in the same way (lengths of corresponding sides).

By keeping the angle the same the dominant ratio is the enlargement. The shift of focus to the ratio of the sides within the triangle in the second episode can be prefigured: now let’s look at within the triangle itself. What happens when we change the angle?
For this episode the class works as a whole, but you can ask them to consult in pairs on ideas for some questions. Suppose I want to draw some right-angled triangles with one angle of 30°. How many could I draw? Show the three triangles in different orientations by attaching them to the board. Could there be others?

• What can you say about the triangles?
• What can you say about the angles?
• How can we be sure we are talking about the same side?

Pupils should explain to each other the notions of similarity and corresponding sides, moving to the need for good labels. Ask for suggestions for labels for the sides, moving towards opposite, adjacent (to the 30° angle and the right angle) and hypotenuse (opposite to the right angle).

Superimpose the smaller triangle on the 20 cm triangle with the 30° angle overlapping. The adjacent side of the large triangle is double the size of the adjacent side of the middle triangle. What can you say about the opposite sides? The hypotenuses? Check by measuring and record the measurements for each triangle in a table on the board: Repeat for the 10 cm and 30 cm triangles (factor of 3). Link to enlargement and scale factor, from small to large and vice versa, pointing out that ratio of small to large is a fraction/decimal less than 1.