## Triangles¶

### Right triangle¶

In a right triangle, one angle is the same , the other two angles and result together .

The following applies to the area and the circumference of a right triangle: The Pythagorean theorem

Right-angled triangles have a special feature: if you square the lengths of the sides of the triangle, the square number corresponds the longest side of the triangle (the "hypotenuse") exactly the sum of the square numbers and the shorter sides of the triangle (the "cathetus").

This law, known as the “Pythagorean theorem”, can be graphically illustrated by running along the hypotenuse and the two catheters and Draws squares with the corresponding side lengths and compares the areas with each other: The areas of the two smaller squares and are with the big square surface area.

In practice, the Pythagorean theorem proves to be useful to work around two boards, poles, etc. of known lengths and to be arranged at right angles to each other. Solve equation (3) for the length of the connecting line on, so it turns out Are the corner points and exactly around apart, the angle is between and exactly . The length ratio is particularly suitable because here applies; the length of the base unit can be freely selected.

Elevation and cathetus set

In addition, two other relationships apply in the right triangle:

• Height rate:

The product of the two parts of the hypotenuse and the right and left of the height is equal to the square of the height: • Cathetus set: The product of a cathetus is equal to the product of the hypotenuse and the adjacent hypotenuse: These two laws were already discovered by Euclid. They are based on the triangles and the two by the height resulting triangles and are similar to each other: All contain a right angle and each have one side of the triangle in common; in addition, all triangles have the angle because of equation (1) together.

Due to the similarity, the proportions of the side lengths are the same, it applies, for example, to the triangles and the aspect ratio , which also turns out to be can be written and thus corresponds to the height rate. The two sets of cathets also follow from the length ratios of triangles and as of triangles and .

Other properties

Further relationships in triangles are discussed in more detail in the Trigonometry section.

Remarks: