How to find the area of \u200b\u200ba rectangular triangle in an unusual way. How to find the area of \u200b\u200ba rectangular triangle with an unusual way area of \u200b\u200bhypotenuse and cathetu

In the lessons of geometry in high school, we all told about the triangle. However, as part of the school program, we only receive the most necessary knowledge and learning the most common and standard methods for computing. Are there unusual ways of finding this magnitude?

As an introduction, we remember which triangle is considered rectangular, as well as we denote the concept of area.

The rectangular triangle is called a closed geometric shape, one of the angles of which is 90 0. Inalienable concepts in the definition are cathets and hypotenuses. Under the categories imply two sides, which at the point of the connection form a straight angle. The hypotenuse is the side opposite to the straight corner. A straight triangle can be equally feasible (two sides will have the same value), but will never be equilateral (all sides of the same length). The determination of height, medians, vectors and other mathematical terms will not disassemble in detail. They are easy to find in reference books.

The area of \u200b\u200bthe straight triangle. Unlike rectangles, rule about

the work of the parties in the definition does not work. If we speak dry language terms, then under the triangle area understand the property of this figure to occupy a part of the plane, expressed by the number. Quite difficult to perceive, agree. We will not try to get deep deep into the definition, our goal is not in this. Let's go to the main thing - how to find the area of \u200b\u200ba rectangular triangle? We will not produce the calculations, we indicate only formulas. To do this, we will define with the notation: a, b, c - the sides of the triangle, kartettes - AB, BC. ACB angle - straight. S is the triangle area, 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 \u200b\u200ba rectangular triangle if the magnitude of its cathets is known

METHOD 2. Find an area of \u200b\u200ban equilibried rectangular triangle

Method 3. Calculating area through a rectangle

Rectangular triangle to the square (if a triangle

equal) or rectangle. We obtain a simple quadrangle, composed of 2 identical rectangular triangles. In this case, the significance of one of them will be equal to half the area of \u200b\u200bthe figure of the figure. S Rectangle Calculate the product of the parties. Denote this value M. The desired area of \u200b\u200bthe area will be equal to half M.

Method 4. "Pythagora pants." The famous Pythagora theorem

We all remember its wording: "The sum of the squares of the cathets ...". But not everyone can

to say, there is some "pants". The fact is that initially Pyfagor studied the relationship built on the sides of the straight triangle. Having reveaning patterns in the ratio of the sides of the squares, he was able to withdraw and the formula known to all of us. It can be used in the case when the value of one of the parties is unknown.

Method 5. How to find the area of \u200b\u200ba rectangular triangle according to the formula of Geron

Also a fairly simple calculation method. The formula involves the expression of the triangle area through the numerical values \u200b\u200bof its parties. For calculations, it is necessary to know the magnitude of all sides of the triangle.

S \u003d (P-AC) * (P-BC), where p \u003d (AB + BC + AC) * 0.5

In addition to the above, there are many other ways to find the value of such a mysterious figure like a triangle. Among them: the calculation of the method inscribed or described circle, calculation using the coordinates of the vertices, the use of vectors, absolute value, sinuses, tangents.

Triangle is a flat geometric figure with one angle equal to 90 °. At the same time, in geometry, it is very often necessary to calculate the area of \u200b\u200bsuch a figure. How to do it, we will tell further.

The simplest formula for determining the area of \u200b\u200bthe rectangular triangle

Initial data, where: A and B - the sides of the triangle leaving the direct angle.

That is, the area is equal to half the work of both sides, which come out of the direct angle. Of course, there is a Geron formula used to calculate the area of \u200b\u200ban ordinary triangle, but to determine the value you need to know the length of three sides. Accordingly, you will have to calculate the hypotenuse, and this is too much time.

Find the area of \u200b\u200ba rectangular triangle through the Geron formula

This is a well-known and source formula, but for this you will have to calculate the hypotenuse in two categories using the Pythagore's theorem.

In this formula: A, B, C - the sides of the triangle, and P is a half-meter.

Find the area of \u200b\u200bthe rectangular triangle on hypotenuse and corner

If none of the cathets are known in your task, then you can not use the easiest way. To determine the magnitude, you need to calculate the length of cathets. It is simply made on hypotenuse and cosine of the adjacent angle.

b \u003d C × COS (α)

Having learned the length of one of the cathets, on the Pythagore Theorem, you can calculate the second side, which emerges from the straight corner.

b 2 \u003d C 2 -A 2

In this formula C and A - hypotenuse and catat, respectively. Now you can calculate the area on the first formula. Similarly, one of the cathets can be calculated, having a second and angle. In this case, one of the sides will be equal to the product of the category of the corner. There are other ways to calculate the area, but knowing the main theorems and rules, you can easily find a desired value.

If you do not have any side of the triangle, but there is only a median and one of the corners, then you can calculate the length of the parties. To do this, use the median properties to share a rectangular triangle for two. Accordingly, it can act as hypotenuse if it comes out of an acute angle. Use the Pythagore theorem and determine the length of the side of the triangle coming out of the direct angle.

As you can see, knowing the main formulas and the theorem of Pythagora, you can calculate the area of \u200b\u200bthe rectangular triangle, having only one of the corners and the length of one of the sides.

A rectangular triangle is called a triangle, in which one of the corners is 90 °. Its area can be found if two categories are known. You can, of course, go and long - to find the hypotenuse and calculate the area of \u200b\u200bthe software, but in most cases it will only take over time. That is why the formula of the area of \u200b\u200bthe rectangular triangle looks like this:

The area of \u200b\u200bthe rectangular triangle is equal to half the work of cathets.

An example of calculating the area of \u200b\u200ba rectangular triangle.
Dan is a rectangular triangle with customs a. \u003d 8 cm, b. \u003d 6 cm.
Calculate the area:
Square is: 24 cm 2

Also in a rectangular triangle uses the Pythagora theorem. - The sum of the squares of two cathets is equal to the square of the hypotenuse.
The formula of an equifiable rectangular triangle is calculated as well as a conventional rectangular triangle.

An example of calculating the area of \u200b\u200ban equilibried rectangular triangle:
Dan triangle with customs a. \u003d 4 cm, b. \u003d 4 cm. Calculate the area:
Calculate the area: \u003d 8 cm 2

The formula of the area of \u200b\u200bthe rectangular triangle on hypotenuse can be used if one rolls is given. From the Pythagora theorem we find the length of the unknown category. For example, given hypotenuse c. and cathet a., cathe b. will be equal to:
Next, calculate the area along the usual formula. An example of calculating the formula for the area of \u200b\u200bthe rectangular triangle on hypotenuse is identical to the above described above.

Consider an interesting task that will help consolidate the knowledge of the formulas for solving a triangle.
Task: The area of \u200b\u200bthe rectangular triangle is 180 square meters. See Find a smaller triangle catat if it is less than 31 cm.
Decision: Denote by kartets a. and b.. Now we will substitute the data in the Square formula: we also know that one roll is less than another a. – b. \u003d 31 cm
From the first condition we get that
We substitute this condition in the second equation: