What are the alternative ways of finding the length of the hypotenuse of a right angle triangle without using Pythagoras theorem?

Answer 1

The least accurate is to draw the triangle and then measuring it.

Alternatively you can use trigonometric ratios:

tan = opposite/adjacent

sin = opposite/hypotenuse → hypotenuse = opposite/sin

cos = adjacent/hypotenuse → hypotenuse = adjacent/cos

Using the tangent ration one of the non-right angles of the triangle can be found. Then using either the sine or cosine ratio the hypotenuse can be found.

eg if the two "legs" are 1 cm and √2 cm, then:

The angle at the end of the √2 cm side is: arc tan(1/√2) = 30°

Then the hypotenuse is: 1 cm / sin (arc tan(1/√2)) = 1 cm / ½ = 2 cm.

or √2 / cos (arc tan(1/√2)) = √2 / (1/√2) = √2 × √2 = 2.

eg if the two "legs" are 3 cm and 4 cm, then:

The angle at the end of the 4 cm side is: arc tan ¾ ≈ 36.87°

The the hypotenuse is: 3 / sin(arc tan ¾) = 3/0.6 = 5

or 4 / cos(arc tan ¾) = 3/0.8 = 5