The problem of finding the diagonal of a rectangle can be formulated in three different ways. Let's take a closer look at each of them. The methods depend on known data, so how do you find the diagonal of a rectangle?
If two sides are known
In the case when two sides of the rectangle a and b are known, to find the diagonal it is necessary to use the Pythagorean theorem: a 2 + b 2 =c 2, here a and b are the legs of the right triangle, c is the hypotenuse of the right triangle. When a diagonal is drawn in a rectangle, it is divided into two right triangle. We know two sides of this right triangle (a and b). That is, to find the diagonal of a rectangle, the following formula is needed: c=√(a 2 +b 2), here c is the length of the diagonal of the rectangle.
By known side and angle, between side and diagonal
Let the side of the rectangle a and the angle it forms with the diagonal of the rectangle α be known. First, let's remember the cosine formula: cos α = a/c, here c is the diagonal of the rectangle. How to calculate the diagonal of a rectangle from this formula: c = a/cos α.
Along a known side, the angle between the adjacent side of the rectangle and the diagonal.
Since the diagonal of a rectangle divides the rectangle itself into two right triangles, it is logical to turn to the definition of sine. Sine is the ratio of the leg opposite this angle to the hypotenuse. sin α = b/c. From here we derive the formula for finding the diagonal of a rectangle, which is also the hypotenuse of a right triangle: c = b/sin α.
Now you are savvy in this matter. You can please your geometry teacher tomorrow!
Definition.
Rectangle is a quadrilateral in which two opposite sides are equal and all four angles are equal.The rectangles differ from each other only in the ratio of the long side to the short side, but all four corners are right, that is, 90 degrees.
The long side of a rectangle is called rectangle length, and the short one - 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 rectangles have the same length, that is, they are equal:
AB = CD, BC = AD
2. Opposite sides of the rectangle are parallel:
3. The 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:
2d 2 = 2a 2 + 2b 2
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 intersection point:
| AO=BO=CO=DO= | d | ||
| 2 |
10. The point of intersection of the diagonals is called the center of the rectangle and is also the center of the circumcircle
11. The diagonal of a rectangle is the diameter of the circumcircle
12. You can always describe a circle around a rectangle, since the sum of opposite angles is equal to 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 the opposite sides are not equal to each other (a circle can only be inscribed in special case rectangle - square).
Sides of a rectangle
Definition.
Rectangle length is the length of the longer pair of its sides. Rectangle width is the length of the shorter pair of its sides.Formulas for determining the lengths of the sides of a rectangle
1. Formula for the side of a rectangle (length and width of the rectangle) through the diagonal and the other side:
a = √ d 2 - b 2
b = √ d 2 - a 2
2. Formula for the side of a rectangle (length and width of the rectangle) through the area and the other side:
| b = dcos | β |
| 2 |
Diagonal of a rectangle
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. Formula for the diagonal of a rectangle using two sides of the rectangle (via the Pythagorean theorem):
d = √ a 2 + b 2
2. Formula for the diagonal of a rectangle using the area and any side:
4. Formula for the diagonal of a rectangle in terms of the radius of the circumscribed circle:
d = 2R
5. Formula for the diagonal of a rectangle in terms of the diameter of the circumscribed circle:
d = D o
6. Formula for the diagonal of a rectangle using the sine of the angle adjacent to the diagonal and the length of the side opposite to this angle:
8. Formula for the diagonal of a rectangle through sine acute angle between the diagonals and the area of the rectangle
d = √2S: sin β
Perimeter of a rectangle
Definition.
Perimeter of a rectangle is the sum of the lengths of all sides of a rectangle.Formulas for determining the length of the perimeter of a rectangle
1. Formula for the perimeter of a rectangle using two sides of the rectangle:
P = 2a + 2b
P = 2(a + b)
2. Formula for the perimeter of a rectangle using area and any side:
| P= | 2S + 2a 2 | = | 2S + 2b 2 |
| a | b |
3. Formula for the perimeter of a rectangle using the diagonal and any side:
P = 2(a + √ d 2 - a 2) = 2(b + √ d 2 - b 2)
4. Formula for the perimeter of a rectangle using the radius of the circumcircle and any side:
P = 2(a + √4R 2 - a 2) = 2(b + √4R 2 - b 2)
5. Formula for the perimeter of a rectangle using the diameter of the circumcircle and any side:
P = 2(a + √D o 2 - a 2) = 2(b + √D o 2 - b 2)
Area of a rectangle
Definition.
Area of a rectangle called the space limited by the sides of the rectangle, that is, within the perimeter of the rectangle.Formulas for determining the area of a rectangle
1. Formula for the area of a rectangle using two sides:
S = a b
2. Formula for the area of a rectangle using the perimeter and any side:
5. Formula for the area of a rectangle using the radius of the circumcircle and any side:
S = a √4R 2 - a 2= b √4R 2 - b 2
6. Formula for the area of a rectangle using the diameter of the circumscribed circle and any side:
S = a √D o 2 - a 2= b √D o 2 - b 2
Circle circumscribed around a rectangle
Definition.
A circle circumscribed around a rectangle is a circle passing through the 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. Formula for the radius of a circle circumscribed around a rectangle through two sides:
is a parallelogram in which all angles are equal to 90°, and opposite sides are parallel and equal in pairs.
A rectangle has several irrefutable properties that are used in solving many problems, in formulas for the area of a rectangle and its perimeter. Here they are:
The length of an unknown side or diagonal of a rectangle is calculated using or using 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 a 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 two sides, for example a = 3 cm, b = 5 cm, we can easily calculate the area of the rectangle:
We find that in such a rectangle the area will be equal to 15 square meters. cm.
Area of a rectangle through diagonals
Sometimes you need to apply the formula for the area of a rectangle through the diagonals. It requires not only finding out the length of the diagonals, but also the angle between them:
Let's look at 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 into 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, if an angle is given, is quite simple.
Let's look at another interesting problem that will help us stretch our brains a little.
Task: Given a square. Its area is 36 square meters. cm. Find the perimeter of a rectangle whose length of one side is 9 cm and whose area is the same as the square given above.
So we have several conditions. For clarity, let’s write them down to see all the known and unknown parameters: 
The sides of the figure are parallel and equal in pairs. 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 side b
From here:
We substitute the known data and find the length of side b:
Calculate the perimeter of the figure: 
This is how, knowing a few simple formulas, you can calculate the perimeter of a rectangle, knowing its area.
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 a 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, such as area and perimeter or aspect ratio, you can find the sides of the rectangle and then use the Pythagorean theorem to calculate the diagonal.
Steps
1 On the sides
- 1 Write down the Pythagorean theorem. Formula: a 2 + b 2 = c 2
- 2
Substitute the values of the sides into the formula. They are given in the problem or need to be measured. The side values are substituted for a 3
- In our example:
4 2 + 3 2 = c 2 42 By area and perimeter
- 1 Formula: S = l w (In the figure, instead of S, the designation A is used.)
- 2 This value is substituted for S 3 Rewrite the formula to isolate w 4 Write down the formula to calculate the perimeter of a rectangle. Formula: P = 2 (w + l)
- 5
Substitute 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 Substitute the expression to calculate w 8 into the formula Get rid of the fraction. To do this, multiply both sides of the equation by l 9 Set the equation equal to 0. To do this, subtract the first-order variable term from both sides of the equation.
- In our example:
12 l = 35 + l 2 10 Order the terms of the equation. The first term will be the second-order variable term, then the first-order variable term, and then the free term. At the same time, do not forget about the signs (“plus” and “minus”) that appear 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 is 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 you square a number, it multiplies by itself.
- In our example:
5 2 + 7 2 = c 2 16 Remove square root from both sides of the equation. Use a calculator to quickly find the square root. You can also use an online calculator. You will find c3 By area and aspect ratio
- 1
Write down an equation characterizing the ratio of the sides. Isolate l 2 Write down the formula to calculate the area of a rectangle. Formula: S = l w (In the figure, instead of S, the designation A is used.)
- This method is also applicable when 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)
- 3
Substitute the area of the rectangle into the formula. This value is substituted for S 4 In the formula, substitute an expression characterizing the relationship of the parties. In the case of a rectangle, you can substitute an expression to calculate l 5 Write it down quadratic equation.
To do this, open the brackets and set the equation equal to zero.
- In our example:
35 = w(w+2)6 Factor the quadratic equation. To get detailed instructions, read.- In our example, the equation is 0 = w 2 − 12 w + 35 7 Find w 8 Substitute the found width (or length) into the equation characterizing the aspect ratio. This way you can find the other side of the rectangle.
- For example, if you calculate 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
Substitute 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 you square a number, it multiplies 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 an online calculator. You will find c (displaystyle c), that is, the hypotenuse of the triangle, and therefore 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 greater than its width and whose area is 35 cm 2 is approximately 8.6 cm.
- In our example:
- In our example:
- For example, if you calculate 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
- In our example, the equation is 0 = w 2 − 12 w + 35 7 Find w 8 Substitute the found width (or length) into the equation characterizing the aspect ratio. This way you can find the other side of the rectangle.
- In our example:
- 1
Write down an equation characterizing the ratio of the sides. Isolate l 2 Write down the formula to calculate the area of a rectangle. Formula: S = l w (In the figure, instead of S, the designation A is used.)
- In our example:
- In our example, the equation is 0 = l 2 − 12 l + 35 12 Find l 13 Write down the Pythagorean theorem. Formula: a 2 + b 2 = c 2
- In our example 0 = 35 + l 2 − 12 l 11
- In our example:
- In our example:
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
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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).
