Formulas and properties of a rectangle. Geometric figures

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).

is a parallelogram in which all angles are 90° and opposite sides are pairwise parallel and equal.

The rectangle has several irrefutable properties that are used in solving many problems, in the formulas for the area of ​​\u200b\u200bthe rectangle and its perimeter. Here they are:

The length of the unknown side or diagonal of the rectangle is calculated by or by the Pythagorean theorem. The area of ​​a rectangle can be found in two ways - by the product of its sides or by the formula for the area of ​​\u200b\u200ba rectangle through the diagonal. The first and simplest formula looks like this:

An example of calculating the area of ​​a rectangle using this formula is very simple. Knowing the two sides, for example a = 3 cm, b = 5 cm, we can easily calculate the area of ​​the rectangle:
We get that in such a rectangle the area will be equal to 15 square meters. cm.

Area of ​​a rectangle in terms of diagonals

Sometimes you need to apply the formula for the area of ​​a rectangle in terms of diagonals. For it, you will need not only to know the length of the diagonals, but also the angle between them:

Consider an example of calculating the area of ​​a rectangle using diagonals. Let a rectangle with diagonal d = 6 cm and angle = 30° be given. We substitute the data in the already known formula:

So, the example of calculating the area of ​​a rectangle through the diagonal showed us that finding the area in this way, given the angle, is quite simple.
Consider another interesting puzzle that will help us stretch our brains a little.

Task: Given a square. Its area is 36 sq. cm. Find the perimeter of a rectangle whose length of one of its sides is 9 cm, and the area is the same as that of the square given above.
So we have a few conditions. For clarity, we write them down to see all known and unknown parameters:
The sides of the figure are pairwise parallel and equal. Therefore, the perimeter of the figure is equal to twice the sum of the lengths of the sides:
From the formula for the area of ​​a rectangle, which is equal to the product of the two sides of the figure, we find the length of the side b
From here:
We substitute the known data and find the length of the side b:
Calculate the perimeter of the figure:
So, knowing a few easy formulas, you can calculate the perimeter of a rectangle, knowing its area.

Definition.

Rectangle It is a quadrilateral with two opposite sides equal and all four angles equal.

Rectangles differ from each other only in the ratio of the long side to the short side, but all four of them are right, that is, 90 degrees each.

The long side of a rectangle is called rectangle length, and the short rectangle width.

The sides of a rectangle are also its heights.

Basic properties of a rectangle

A rectangle can be a parallelogram, a square or a rhombus.

1. Opposite sides of a rectangle have the same length, that is, they are equal:

AB=CD, BC=AD

2. Opposite sides of the rectangle are parallel:

3. Adjacent sides of a rectangle are always perpendicular:

AB ┴ BC, BC ┴ CD, CD ┴ AD, AD ┴ AB

4. All four corners of the rectangle are straight:

∠ABC = ∠BCD = ∠CDA = ∠DAB = 90°

5. The sum of the angles of a rectangle is 360 degrees:

∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°

6. The diagonals of a rectangle have the same length:

7. The sum of the squares of the diagonal of a rectangle is equal to the sum of the squares of the sides:

2d2 = 2a2 + 2b2

8. Each diagonal of a rectangle divides the rectangle into two identical figures, namely right triangles.

9. The diagonals of the rectangle intersect and are divided in half at the point of intersection:

		AO=BO=CO=DO=	d
			2

10. The intersection point of the diagonals is called the center of the rectangle and is also the center of the circumscribed circle

11. The diagonal of a rectangle is the diameter of the circumscribed circle

12. A circle can always be described around a rectangle, since the sum of opposite angles is 180 degrees:

∠ABC = ∠CDA = 180° ∠BCD = ∠DAB = 180°

13. A circle cannot be inscribed in a rectangle whose length is not equal to its width, since the sums of opposite sides are not equal to each other (a circle can only be inscribed in a special case of a rectangle - a square).

Sides of a rectangle

Definition.

Rectangle length call the length of the longer pair of its sides. Rectangle width name the length of the shorter pair of its sides.

Formulas for determining the lengths of the sides of a rectangle

1. The formula for the side of a rectangle (the length and width of the rectangle) in terms of the diagonal and the other side:

a = √ d 2 - b 2

b = √ d 2 - a 2

2. The formula for the side of a rectangle (the length and width of the rectangle) in terms of the area and the other side:

b = dcos	β
	2

Rectangle Diagonal

Definition.

Diagonal Rectangle Any segment connecting two vertices of opposite corners of a rectangle is called.

Formulas for determining the length of the diagonal of a rectangle

1. The formula for the diagonal of a rectangle in terms of two sides of the rectangle (via the Pythagorean theorem):

d = √ a 2 + b 2

2. The formula for the diagonal of a rectangle in terms of area and any side:

4. The formula for the diagonal of a rectangle in terms of the radius of the circumscribed circle:

d=2R

5. The formula for the diagonal of a rectangle in terms of the diameter of the circumscribed circle:

d = D o

6. The formula of the diagonal of a rectangle in terms of the sine of the angle adjacent to the diagonal and the length of the side opposite to this angle:

8. The formula of the diagonal of a rectangle in terms of the sine of an acute angle between the diagonals and the area of ​​the rectangle

d = √2S: sinβ

Perimeter of a rectangle

Definition.

The perimeter of a rectangle is the sum of the lengths of all sides of the rectangle.

Formulas for determining the length of the perimeter of a rectangle

1. The formula for the perimeter of a rectangle in terms of two sides of the rectangle:

P = 2a + 2b

P = 2(a+b)

2. The formula for the perimeter of a rectangle in terms of area and any side:

P=	2S + 2a 2	=	2S + 2b 2
	a		b

3. Formula for the perimeter of a rectangle in terms of the diagonal and any side:

P = 2(a + √ d 2 - a 2) = 2(b + √ d 2 - b 2)

4. The formula for the perimeter of a rectangle in terms of the radius of the circumscribed circle and any side:

P = 2(a + √4R 2 - a 2) = 2(b + √4R 2 - b 2)

5. The formula for the perimeter of a rectangle in terms of the diameter of the circumscribed circle and any side:

P = 2(a + √D o 2 - a 2) = 2(b + √D o 2 - b 2)

Rectangle area

Definition.

Rectangle area called the space bounded by the sides of the rectangle, that is, within the perimeter of the rectangle.

Formulas for determining the area of ​​a rectangle

1. The formula for the area of ​​a rectangle in terms of two sides:

S = a b

2. The formula for the area of ​​a rectangle through the perimeter and any side:

5. The formula for the area of ​​a rectangle in terms of the radius of the circumscribed circle and any side:

S = a √4R 2 - a 2= b √4R 2 - b 2

6. The formula for the area of ​​a rectangle in terms of the diameter of the circumscribed circle and any side:

S \u003d a √ D o 2 - a 2= b √ D o 2 - b 2

Circle circumscribed around a rectangle

Definition.

A circle circumscribed around a rectangle A circle is called a circle passing through four vertices of a rectangle, the center of which lies at the intersection of the diagonals of the rectangle.

Formulas for determining the radius of a circle circumscribed around a rectangle

1. The formula for the radius of a circle circumscribed around a rectangle through two sides:

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.

Content:

A diagonal is a line segment that connects two opposite vertices of a rectangle. A rectangle has two equal diagonals. If the sides of the rectangle are known, the diagonal can be found using the Pythagorean theorem, because the diagonal divides the rectangle into two right triangles. If the sides are not given, but other quantities are known, for example, the area and perimeter or the ratio of the sides, you can find the sides of the rectangle, and then calculate the diagonal using the Pythagorean theorem.

Steps

1 Side by side

1. 1 Write down the Pythagorean theorem. Formula: a 2 + b 2 = c 2
2. 2 Plug the sides into the formula. They are given in the problem or they need to be measured. Side values ​​are substituted for a 3
   • In our example:
     4 2 + 3 2 = c 2 4

     2 By area and perimeter

     1. 1 Formula: S \u003d l w (In the figure, the symbol A is used instead of S.)
     2. 2 This value is substituted for S 3 Rewrite the formula so as to isolate w 4 Write down the formula for calculating the perimeter of a rectangle. Formula: P = 2 (w + l)
     3. 5 Substitute the value of the perimeter of the rectangle into the formula. This value is substituted for P 6 Divide both sides of the equation by 2. You will get the sum of the sides of the rectangle, namely w + l 7 In the formula, substitute the expression to calculate w 8 Get rid of fractions. To do this, multiply both parts of the equation by l 9 Set the equation to 0. To do this, subtract the term with the first-order variable from both sides of the equation.
        • In our example:
          12 l \u003d 35 + l 2 10 Order the terms of the equation. The first member will be the second variable member, then the first variable member, and then the free member. At the same time, do not forget about the signs (“plus” and “minus”) that are in front of the members. Note that the equation will be written as a quadratic equation.
          • In our example, 0 = 35 + l 2 − 12 l 11
            • In our example, the equation 0 = l 2 − 12 l + 35 12 Find l 13 Write down the Pythagorean theorem. Formula: a 2 + b 2 = c 2
              • Use the Pythagorean theorem, because each diagonal of a rectangle divides it into two equal right triangles. Moreover, the sides of the rectangle are the legs of the triangle, and the diagonal of the rectangle is the hypotenuse of the triangle.
            • 14 These values ​​are substituted for a 15 Square the length and width, and then add the results. Remember that when squaring a number, it is multiplied by itself.
              • In our example:
                5 2 + 7 2 = c 2 16 Take the square root of both sides of the equation. Use a calculator to quickly find the square root. You can also use the online calculator. you will find c

                3 By area and aspect ratio

                1. 1 Write down an equation characterizing the ratio of the sides. Isolate l 2 Write down the formula for calculating the area of ​​a rectangle. Formula: S = l w (Notation A is used instead of S in the figure.)
                   • This method is also applicable when the value of the perimeter of the rectangle is known, but then you need to use the formula to calculate the perimeter, not the area. Formula for calculating the perimeter of a rectangle: P = 2 (w + l)
                2. 3 Plug in the area of ​​the rectangle into the formula. This value is substituted for S 4 Substitute the expression characterizing the ratio of the sides into the formula. In the case of a rectangle, you can substitute an expression to calculate l 5 Write down a quadratic equation. To do this, open the brackets and equate the equation to zero.
                   • In our example:
                     35 = w (w + 2) 6 Factorize the quadratic equation. Read on for detailed instructions.
                     • In our example, the equation 0 = w 2 − 12 w + 35 7 Find w 8 Substitute the value of the width (or length) found in the equation characterizing the ratio of the sides. So you can find the other side of the rectangle.
                       • For example, if you calculated that the width of a rectangle is 5 cm and the aspect ratio is given by the equation l = w + 2 9 Write down the Pythagorean theorem. Formula: a 2 + b 2 = c 2
                         • Use the Pythagorean theorem, because each diagonal of a rectangle divides it into two equal right triangles. Moreover, the sides of the rectangle are the legs of the triangle, and the diagonal of the rectangle is the hypotenuse of the triangle.
                       • 10 Plug in the length and width values ​​into the formula. These values ​​are substituted for a 11 Square the length and width, and then add the results. Remember that when squaring a number, it is multiplied by itself.
                         • In our example:
                           5 2 + 7 2 = c 2 12 Take the square root of both sides of the equation. Use a calculator to quickly find the square root. You can also use the online calculator. You'll find c (displaystyle c) , which is the hypotenuse of the triangle, and hence the diagonal of the rectangle.
                           • In our example:
                             74 = c 2 (displaystyle 74=c^(2))
                             74 = c 2 (displaystyle (sqrt (74))=(sqrt (c^(2))))
                             8, 6024 = c (displaystyle 8,6024=c)
                             Thus, the diagonal of a rectangle whose length is 2 cm more than its width and whose area is 35 cm 2 is approximately 8.6 cm.