
In ancient Egypt people used the Pythagorean theorem to measure out land. They constructed right angles through the use of Pythagorean triplets. On the other hand, they were also capable of calculating unknown distances inside a right triangle if they knew the two other distances. Even though the Pythagorean Theorem can be successfully used to measure out land on a small scale, it is pretty much useless for making maps of big areas. To find the distance from one place to another, when there is no way to measure it directly, trigonometry can help. Trigonometry is the study of the relation between angles and sides within
triangles. Using trigonometry you can find the length of an unknown side
inside a right triangle if you know the length of one side and one
angle. You can also find the size of an unknown angle if you know the
lengths of two of the sides of the right triangle. To do this, trigonometry
uses right triangles which have equal angles and studies the relation between
the length of the sides.
Take a look at the grid on the left. Move your mouse over it. Do you see the right triangle which has one corner following your mouse? Now move your mouse so that you have a triangle with a base of 2 and a height of 1. (The base and height are displayed on the top of the grid.) If you move your mouse 2 more squares to the right and 1 more square up, you get a triangle with a base of 4 and a height of 2. Take a look at the hypotenuse (red side) of the triangle. It passes through the point where you had your mouse before. Since the hypotenuse passes through the same two points, it is at the same angle. At the bottom you see the ratio 2:1. These two numbers show the smallest possible triangle you can get on this grid which has the same angles as the one currently displayed. The triangle that you have displayed now has a base of 4 and a height of 2. When you compare the numbers at the top and the numbers at the bottom, you see that if you divide them through each other, you get the same answer; the first of the bottom numbers divided by the second (2/1) is 2. If you do the same for the top two numbers (the base and heigt of the triangle) you also get 2. This means these two pairs of numbers have the same ratio. This is the case for any two triangles which have the same angles. Two triangles with equal ratio of their sides have equal angles. They are called similar triangles.



