4. The formula for the radius of a circle, which is described about a rectangle through the diagonal of a square:

5. The formula for the radius of a circle, which is described near a rectangle through the diameter of a circle (circumscribed):

6. The formula for the radius of a circle, which is described near a rectangle through the sine of the angle that is adjacent to the diagonal, and the length of the side opposite this angle:

7. The formula for the radius of a circle, which is described about a rectangle in terms of the cosine of the angle that is adjacent to the diagonal, and the length of the side at this angle:

8. The formula for the radius of a circle, which is described near a rectangle through the sine of an acute angle between the diagonals and the area of ​​the rectangle:

Angle between a side and a diagonal of a rectangle.

Formulas for determining the angle between the side and the diagonal of a rectangle:

1. The formula for determining the angle between the side and the diagonal of a rectangle through the diagonal and the side:

2. The formula for determining the angle between the side and the diagonal of a rectangle through the angle between the diagonals:

The angle between the diagonals of the rectangle.

Formulas for determining the angle between the diagonals of a rectangle:

1. The formula for determining the angle between the diagonals of a rectangle through the angle between the side and the diagonal:

β = 2α

2. The formula for determining the angle between the diagonals of a rectangle through the area and the diagonal.

Rectangle is a quadrilateral in which every corner is a right angle.

Proof

The property is explained by the action of feature 3 of the parallelogram (i.e. \angle A = \angle C , \angle B = \angle D )

2. Opposite sides are equal.

AB = CD,\enspace BC = AD

3. Opposite sides are parallel.

AB \parallel CD,\enspace BC \parallel AD

4. Adjacent sides are perpendicular to each other.

AB \perp BC,\enspace BC \perp CD,\enspace CD \perp AD,\enspace AD ​​\perp AB

5. The diagonals of the rectangle are equal.

AC=BD

Proof

According to property 1 the rectangle is a parallelogram, which means AB = CD.

Therefore, \triangle ABD = \triangle DCA along two legs (AB = CD and AD - joint).

If both figures - ABC and DCA are identical, then their hypotenuses BD and AC are also identical.

So AC = BD .

Only a rectangle of all figures (only from parallelograms!) Has equal diagonals.

Let's prove this too.

ABCD is a parallelogram \Rightarrow AB = CD , AC = BD by condition. \Rightarrow \triangle ABD = \triangle DCA already on three sides.

It turns out that \angle A = \angle D (like the corners of a parallelogram). And \angle A = \angle C , \angle B = \angle D .

We deduce that \angle A = \angle B = \angle C = \angle D. They are all 90^(\circ) . The total is 360^(\circ) .

Proven!

6. The square of the diagonal is equal to the sum of the squares of its two adjacent sides.

This property is valid by virtue of the Pythagorean theorem.

AC^2=AD^2+CD^2

7. The diagonal divides the rectangle into two identical right triangles.

\triangle ABC = \triangle ACD, \enspace \triangle ABD = \triangle BCD

8. The intersection point of the diagonals bisects them.

AO=BO=CO=DO

9. The point of intersection of the diagonals is the center of the rectangle and the circumscribed circle.

10. The sum of all angles is 360 degrees.

\angle ABC + \angle BCD + \angle CDA + \angle DAB = 360^(\circ)

11. All corners of the rectangle are right.

\angle ABC = \angle BCD = \angle CDA = \angle DAB = 90^(\circ)

12. The diameter of the circumscribed circle around the rectangle is equal to the diagonal of the rectangle.

13. A circle can always be described around a rectangle.

This property is valid due to the fact that the sum of the opposite corners of a rectangle is 180^(\circ)

\angle ABC = \angle CDA = 180^(\circ),\enspace \angle BCD = \angle DAB = 180^(\circ)

14. A rectangle can contain an inscribed circle and only one if it has the same side lengths (it is a square).

Rectangle. Since the rectangle has two axes of symmetry, its center of gravity is located at the intersection of the axes of symmetry, i.e. at the point of intersection of the diagonals of the rectangle.

Triangle. The center of gravity lies at the point of intersection of its medians. It is known from geometry that the medians of a triangle intersect at one point and divide in a ratio of 1:2 from the base.

A circle. Since the circle has two axes of symmetry, its center of gravity is at the intersection of the axes of symmetry.

Semicircle. The semicircle has one axis of symmetry, then the center of gravity lies on this axis. Another coordinate of the center of gravity is calculated by the formula: .

Many structural elements are made from standard rolled products - angles, I-beams, channels and others. All dimensions, as well as the geometric characteristics of rolled profiles, are tabular data that can be found in the reference literature in standard assortment tables (GOST 8239-89, GOST 8240-89).

Example 1 Determine the position of the center of gravity of the figure shown in the figure.

Decision:

We select the coordinate axes so that the Ox axis passes along the extreme lower overall dimension, and the Oy axis - along the extreme left overall dimension.

We break a complex figure into the minimum number of simple figures:

rectangle 20x10;

triangle 15x10;

circle R=3 cm.

We calculate the area of ​​each simple figure, its coordinates of the center of gravity. The results of the calculations are entered in the table

Answer: C(14.5; 4.5)

Example 2 . Determine the coordinates of the center of gravity of a composite section consisting of a sheet and rolled profiles.

Decision.

We select the coordinate axes, as shown in the figure.

We denote the figures by numbers and write out the necessary data from the table:

We calculate the coordinates of the center of gravity of the figure using the formulas:

Answer: C(0; 10)

Laboratory work No. 1 "Determining the center of gravity of composite flat figures"

Target: Determine the center of gravity of a given flat complex figure by experimental and analytical methods and compare their results.

Work order

Draw in notebooks your flat figure in size, indicating the coordinate axes.

Determine the center of gravity analytically.

1. Break the figure into the minimum number of figures, the centers of gravity of which, we know how to determine.

   Indicate the numbers of areas and the coordinates of the center of gravity of each figure.

   Calculate the coordinates of the center of gravity of each figure.

   Calculate the area of ​​each figure.

   Calculate the coordinates of the center of gravity of the entire figure using the formulas (put the position of the center of gravity on the drawing of the figure):

Installation for experimental determination of the coordinates of the center of gravity by suspension consists of a vertical rack 1 (see fig.) to which the needle is attached 2 . flat figure 3 Made of cardboard, which is easy to pierce a hole. holes BUT and AT pierced at randomly located points (preferably at the most distant distance from each other). A flat figure is hung on a needle, first at a point BUT , and then at the point AT . With the help of a plumb 4 , fixed on the same needle, a vertical line is drawn on the figure with a pencil corresponding to the plumb line. Center of gravity With figure will be located at the intersection of the vertical lines drawn when hanging the figure at points BUT and AT .

Often, a home craftsman needs to find the center of a circle or a round part. I already wrote about one of the ways to solve this problem in the article how to find the center of a circle. But it has one significant drawback - it is necessary to accurately find the middle of the chord and accurately build a perpendicular from it.

Fortunately, there is another method for accurately finding the center of a circle that does not require any precise measurements. It is based on the simple principle that if a right triangle is inscribed in a circle, then its hypotenuse (the longest side) will be the diameter of this circle or circle.

This is confirmed by the fact that the sum of the angles of a triangle is 180 degrees. And the whole circle is 360 degrees. And any rectangle whose hypotenuse is equal to the diameter of the circle will be rectangular. And vice versa - any right triangle with its hypotenuse represents the diameter of the circle.

And what will give us the center of the circle more precisely, if not the intersection of the two diameters of the circle?

As a "source" of a right angle, it is easiest to take a sheet of writing paper. In paper mills, they are cut with very high precision. You can use the page of any magazine, etc.

We put a sheet of paper on the round part so that one of its corners is on the circle or the edge of the circle. And mark the points where the sheet touches the other edges of the circle. We mark these points.

We draw a straight line between the marked points. The distance between them is the diameter of this circle. We cut off the excess paper and draw a straight line on the part - the diameter.

It is enough to move our triangle to another position and draw another diameter of the circle, and right there at the point of intersection of the diameters we will get the desired center of the circle ...

Thus, without taking absolutely no measurements, we can find the center of any circle.