Equilateral triangle: all rules

This article describes all the properties, rules and determination of an equilateral triangle.

Mathematics is a favorite subject of many schoolchildren, especially those who work to solve problems. Geometry is also an interesting science, but not all children can understand the new material in the lesson. Therefore, they have to modify and finish at home. Let's repeat the rules of an equilateral triangle. Read below.

All the rules of an equilateral triangle: properties

In the very word “equilateral”, the definition of this figure is hidden.

Determination of an equilateral triangle:This is a triangle in which all sides are equal to each other.

Due to the fact that an equilateral triangle is in some way an isosceles triangle, it has signs of the latter. For example, in these triangles, the angle bisector is still median and height.

Recall: The bisector is a beam dividing the corner in half, the median is a beam released from the top, dividing the opposite side in half, and the height is a perpendicular coming from the top.

The second sign of an equilateral triangle It is that all its corners are equal to each other and each of them has a degree measure of 60 degrees. The conclusion about this can be made from the general rule on the sum of the angles of the triangle equal to 180 degrees. Therefore, 180: 3 \u003d 60.

The next property: The center of an equilateral triangle, as well as the circuits described in it and described near it and described near it, is the intersection point of all its median (bisector).

The fourth property: The radius of the circle described near the equilateral triangle exceeds the radius of the inscribed circle into this figure. You can verify this by looking at the drawing. OS is a radius of a circle described near the triangle, and OV1 is inscribed by the radius. The point O is the intersection of the median, which means it shares it as 2: 1. From this we conclude that OS \u003d 2s1.

The fifth property It is that in this geometric figure it is easy to calculate the components of the elements if the length of one side is indicated in the condition. In this case, the Pythagoras theorem is most often used.

The sixth property: The area of \u200b\u200bsuch a triangle is calculated by the formula S \u003d (a^2*3) /4.
Seventh property: radii of the circle described near the triangle, and the circle inscribed in the triangle, respectively
R \u003d (A3) /3 and R \u003d (A3) /6.

Consider examples of tasks:

Example 1:

Task: The radius of a circle inscribed in an equilateral triangle is 7 cm. Find the height of the triangle.

Solution:

The radius of the inscribed circle is associated with the last formula, therefore, OM \u003d (BC3) /6.
BC \u003d (6 * OM) /3 \u003d (6 * 7) /3 \u003d 143.
Am \u003d (BC3) /2; Am \u003d (143*3) /2 \u003d 21.
Answer: 21 cm.

This problem can be solved differently:

Based on the fourth property, we can conclude that OM \u003d 1/2 AM.
Therefore, if OM is 7, then the AO is 14, and am equal to 21.

Example 2:

Task: The radius of the circle described near the triangle is 8. Find the height of the triangle.

Solution:

Let ABC be an equilateral triangle.
As in the previous example, you can go in two ways: a simpler one - AO \u003d 8 \u003d ›Ohm \u003d 4. Then AM \u003d 12.
And longer - to find AM through the formula. AM \u003d (AS3) /2 \u003d (83*3) /2 \u003d 12.
Answer: 12.

As you can see, knowing the properties and definition of an equilateral triangle, you can solve any problem on geometry on this topic.