How to find the area of a rectangular triangle in an unusual way

At the geometry lessons in high school, we were all told how to find the area of a right-angled triangle. However, within the framework of the school curriculum, we receive only the most necessary knowledge and learn the most common and standard methods of calculation. Are there unusual ways of finding this value?

As an introduction, let us recall which triangle is considered to be rectangular, and also denote the concept of area.

A rectangular triangle is a closed geometric figure, one of whose angles is 90 ⁰ . Inherent concepts in the definition of a right triangle are the legs and the hypotenuse. By the legs are meant two sides, which at the junction point form a right angle. Hypotenuse is the opposite of the right angle. A right triangle can be isosceles (its two sides will have the same value), but it will never be equilateral (all sides of the same length). Definitions of height, medians, vectors and other mathematical terms will not be discussed in detail. They are easy to find in the reference books.

Square of a right triangle. Unlike rectangles, the rule of The work of the parties in determining the area of the triangle does not work. If we speak the dry language of terms, then by the area of the triangle we mean the property of this figure to occupy part of the plane expressed in number. It is quite difficult to perceive, you will agree. We will not try to penetrate deeply into the definition, our goal is not this. Let's move on to the main thing - how to find the area of a right-angled triangle? We will not make the calculations ourselves, we shall only indicate the formulas. For this we define the notation: A, B, C - the sides of the triangle, the legs - AB, BC. The angle of the ACB is straight. S is the area of the triangle, h _{n n} is the height of the triangle, where nn is the side to which it is omitted.

Method 1. How to find the area of a right triangle, if the size of its legs is known

S = 0.5 * a * b

Method 2. Find the area of an isosceles right triangle

S = 0.5 * h _BC * BC

Method 3: Calculate the area through a rectangle

We finish the rectangular triangle to the square (if the triangle Isosceles) or a rectangle. We obtain a simple quadrilateral composed of 2 identical rectangular triangles. In this case, the value of the area of one of them will be equal to half the area of the figure obtained. S of the rectangle is calculated by the product of the sides. Let's designate this value M. The required value of the area will be equal to half M.

S = 0.5 * M

Method 4. "Pythagorean Pants". The famous theorem of Pythagoras

We all remember her wording: "the sum of the squares of the legs ...". But not everyone can Say, and here some "pants". The fact is that from the very beginning Pythagoras studied the interrelation of the areas of squares constructed on the sides of a straight triangle. Having discovered the regularities in the ratio of the sides of the squares, he was able to derive the formula known to all of us. It can be used in cases where the value of one of the parties is unknown.

Method 5. How to find the area of a right-angled triangle by Heron's formula

It is also an easy way to calculate. The formula assumes the expression of the area of the triangle through the numerical values of its sides. For calculations it is necessary to know the values of all sides of the triangle.

S = (p-AC) * (p-BC), where p = (AB + BC + AC) * 0.5

In addition to the above, there are many other ways to find the value of such a mysterious figure as a triangle. Among them: calculation by the method of inscribed or circumscribed circle, calculation by means of coordinates of vertices, use of vectors, absolute value, sines, tangents.