Pythagoras theorem and its use

Pythagoras theorem and its use

You have no doubt already met Pythagoras theorem and found it particularly useful in solving particular right–angled triangles. I remind you again of this important theorem.

Pythagoras theorem states that in a right–angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

Pictorially, the theorem is illustrated in Figure 4.1.

For the triangle shown, c² = a² + b² or a² = c² – b² or b² = c² – a2. We can use this theorem to find the height of equilateral and isosceles triangles.

An equilateral triangle: has all its sides of equal length.

An isosceles triangle: has two sides of equal length.

Example 4.6

Find the height of an isosceles triangle that has equal sides of length 13 cm and a base length of 10 cm. The triangle is illustrated in the Figure 4.2.

Then applying Pythagoras theorem and noting that the hypotenuse is opposite the right angle then:

h2 = 13² – 52 and so h2 = 169 – 25 = 144, then square rooting both sides, h = 12cm.

Note that the height h is always at right angles to the base and meets at the apex. A perpendicular can be constructed from any side of the triangle, providing it cuts the base at right–angles and extended to the apex of the triangle.

Pythagoras has helped us to partially solve right–angled triangles. Another important technique is concerned with the ratio of the sides in similar triangles.

Similar triangles have the same shape, one is simply an enlargement or reduction of the other. These triangles have two important properties; these are illustrated in Figure 4.3.

From Figure 4.3 you should note that their corresponding angles are equal and their corresponding sides are proportional. Thus in both triangles:

Taking A (angle A), as the reference, then side a is the opposite, side b is the adjacent and side c is the hypotenuse and the ratio of these sides are equal, that is for both similar triangles:

and in general the ratios of the corresponding sides are constant, that is the ratios a/b, b/c and a/b are the same for all similar right–angles triangles.

Also note that for right–angled triangles to be similar, all corresponding angles must be the same. That is, for the triangles illustrated: A =D, B = E and of course they both have a right angle (90^°). Non–right–angled triangles are similar if the ratios of their sides are the same and each of the corresponding angles is the same but no angle is a right angle. Remember that the only stipulation about the angles in a triangle is that their sum has to add up to 180°.