Positions of the center of gravity of some figures. Geometric figures. Rectangle. Formulas: All angles of a rectangle are right angles

Rectangle is a quadrilateral in which each angle is right.

Proof

The property is explained by the action of feature 3 of the parallelogram (that is, \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 on 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.

Of all the figures (only of parallelograms!), only the rectangle 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 angles of a parallelogram). And \angle A = \angle C , \angle B = \angle D .

We conclude that \angle A = \angle B = \angle C = \angle D. They are all 90^(\circ) . In total - 360^(\circ) .

Proven!

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

This property is true due to 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 point of intersection of the diagonals divides them in half.

AO = BO = CO = DO

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

10. The sum of all angles is 360 degrees.

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

11. All angles of a rectangle are right.

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

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

13. You can always describe a circle around a rectangle.

This property is true due to the fact that the sum of the opposite angles 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 equal side lengths (it is a square).

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4. Formula for the radius of a circle, which is described around a rectangle through the diagonal of a square:

5. Formula for the radius of a circle, which is described around a rectangle through the diameter of the circle (described):

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

7. Formula for the radius of a circle, which is described around a rectangle through the cosine of the angle that is adjacent to the diagonal, and the length of the side of this angle:

8. Formula for the radius of a circle, which is described around a rectangle through the sine of the acute angle between the diagonals and the area of ​​the rectangle:

The angle between the side and the diagonal of a rectangle.

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

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

2. 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 a rectangle.

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

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

β = 2α

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

Often a home craftsman needs to find the center of a circle or round part. I have already written 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 midpoint of the chord and accurately construct 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 you inscribe in a circle right triangle, then its hypotenuse (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 a circle.

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

The easiest way to use a right angle as a “source” is to take a sheet of writing paper. At paper mills they are cut with very high precision. You can use a page from a magazine, etc.

We place a sheet of paper on the round part so that one of its corners is on the circumference or edge of the circle. And we mark the points where the sheet comes into contact with the other edges of the circle. We mark these points.

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 then at the point of intersection of the diameters we will get the desired center of the circle...

Thus, without making absolutely any measurements, we can find the center of any circle.