Properties of the diagonals of a parallelogram. Complete lessons - Knowledge Hypermarket. Parallelogram in problems

Task 4.

One of the diagonals of a parallelogram with a length of 4√6 makes an angle of 60° with the base, and the second diagonal makes an angle of 45° with the same base. Find the second diagonal.

Solution.

1. AO = 2√6.

2. We apply the sine theorem to triangle AOD.

AO/sin D = OD/sin A.

2√6/sin 45 o = OD/sin 60 o.

ОD = (2√6sin 60 о) / sin 45 о = (2√6 √3/2) / (√2/2) = 2√18/√2 = 6.

Answer: 12.

Task 5.

For a parallelogram with sides 5√2 and 7√2, the smaller angle between the diagonals is equal to the smaller angle of the parallelogram. Find the sum of the lengths of the diagonals.

Solution.

Let d 1, d 2 be the diagonals of the parallelogram, and the angle between the diagonals and the smaller angle of the parallelogram is equal to φ.

1. Let's count two different
ways its area.

S ABCD = AB AD sin A = 5√2 7√2 sin f,

S ABCD = 1/2 AC ВD sin AOB = 1/2 d 1 d 2 sin f.

We obtain the equality 5√2 · 7√2 · sin f = 1/2d 1 d 2 sin f or

2 · 5√2 · 7√2 = d 1 d 2 ;

2. Using the relationship between the sides and diagonals of the parallelogram, we write the equality

(AB 2 + AD 2) 2 = AC 2 + BD 2.

((5√2) 2 + (7√2) 2) 2 = d 1 2 + d 2 2.

d 1 2 + d 2 2 = 296.

3. Let's create a system:

(d 1 2 + d 2 2 = 296,
(d 1 + d 2 = 140.

Let's multiply the second equation of the system by 2 and add it to the first.

We get (d 1 + d 2) 2 = 576. Hence Id 1 + d 2 I = 24.

Since d 1, d 2 are the lengths of the diagonals of the parallelogram, then d 1 + d 2 = 24.

Answer: 24.

Task 6.

The sides of the parallelogram are 4 and 6. The acute angle between the diagonals is 45 degrees. Find the area of ​​the parallelogram.

Solution.

1. From triangle AOB, using the cosine theorem, we write the relationship between the side of the parallelogram and the diagonals.

AB 2 = AO 2 + VO 2 2 · AO · VO · cos AOB.

4 2 = (d 1 /2) 2 + (d 2 /2) 2 – 2 · (d 1/2) · (d 2 /2)cos 45 o;

d 1 2 /4 + d 2 2 /4 – 2 · (d 1/2) · (d 2 /2)√2/2 = 16.

d 1 2 + d 2 2 – d 1 · d 2 √2 = 64.

2. Similarly, we write the relation for the triangle AOD.

Let's take into account that<АОD = 135 о и cos 135 о = -cos 45 о = -√2/2.

We get the equation d 1 2 + d 2 2 + d 1 · d 2 √2 = 144.

3. We have a system
(d 1 2 + d 2 2 – d 1 · d 2 √2 = 64,
(d 1 2 + d 2 2 + d 1 · d 2 √2 = 144.

Subtracting the first from the second equation, we get 2d 1 · d 2 √2 = 80 or

d 1 d 2 = 80/(2√2) = 20√2

4. S ABCD = 1/2 AC ВD sin AOB = 1/2 d 1 d 2 sin α = 1/2 20√2 √2/2 = 10.

Note: In this and the previous problem there is no need to solve the system completely, anticipating that in this problem we need the product of diagonals to calculate the area.

Answer: 10.

Task 7.

The area of ​​the parallelogram is 96 and its sides are 8 and 15. Find the square of the shorter diagonal.

Solution.

1. S ABCD = AB · AD · sin ВAD. Let's make a substitution in the formula.

We get 96 = 8 · 15 · sin ВAD. Hence sin ВAD = 4/5.

2. Let's find cos VAD. sin 2 VAD + cos 2 VAD = 1.

(4 / 5) 2 + cos 2 VAD = 1. cos 2 VAD = 9 / 25.

According to the conditions of the problem, we find the length of the smaller diagonal. The diagonal ВD will be smaller if the angle ВАD is acute. Then cos VAD = 3 / 5.

3. From the triangle ABD, using the cosine theorem, we find the square of the diagonal BD.

ВD 2 = АВ 2 + АD 2 – 2 · АВ · ВD · cos ВAD.

ВD 2 = 8 2 + 15 2 – 2 8 15 3 / 5 = 145.

Answer: 145.

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1. Parallelogram

Compound word "parallelogram"? And behind it lies a very simple figure.

Well, that is, we took two parallel lines:

Crossed by two more:

And inside there is a parallelogram!

What properties does a parallelogram have?

Properties of a parallelogram.

That is, what can you use if the problem is given a parallelogram?

The following theorem answers this question:

Let's draw everything in detail.

What does it mean first point of the theorem? And the fact is that if you HAVE a parallelogram, then you will certainly

The second point means that if there IS a parallelogram, then, again, certainly:

Well, and finally, the third point means that if you HAVE a parallelogram, then be sure to:

Do you see what a wealth of choice there is? What to use in the problem? Try to focus on the question of the task, or just try everything one by one - some “key” will do.

Now let’s ask ourselves another question: how can we recognize a parallelogram “by sight”? What must happen to a quadrilateral for us to have the right to give it the “title” of a parallelogram?

Several signs of a parallelogram answer this question.

Signs of a parallelogram.

Attention! Let's begin.

Parallelogram.

Please note: if you found at least one sign in your problem, then you definitely have a parallelogram, and you can use all the properties of a parallelogram.

2. Rectangle

I think that it will not be news to you at all that

First question: is a rectangle a parallelogram?

Of course it is! After all, he has - remember, our sign 3?

And from here, of course, it follows that in a rectangle, like in any parallelogram, the diagonals are divided in half by the point of intersection.

But the rectangle also has one distinctive property.

Rectangle property

Why is this property distinctive? Because no other parallelogram has equal diagonals. Let's formulate it more clearly.

Please note: in order to become a rectangle, a quadrilateral must first become a parallelogram, and then show the equality of the diagonals.

3. Diamond

And again the question: is a rhombus a parallelogram or not?

With full right - a parallelogram, because it has and (remember our feature 2).

And again, since a rhombus is a parallelogram, then it must have all the properties of a parallelogram. This means that in a rhombus, opposite angles are equal, opposite sides are parallel, and the diagonals bisect at the point of intersection.

Properties of a rhombus

Look at the picture:

As in the case of a rectangle, these properties are distinctive, that is, for each of these properties we can conclude that this is not just a parallelogram, but a rhombus.

Signs of a diamond

And again, pay attention: there must be not just a quadrilateral whose diagonals are perpendicular, but a parallelogram. Make sure:

No, of course, although its diagonals are perpendicular, and the diagonal is the bisector of the angles and. But... diagonals are not divided in half by the point of intersection, therefore - NOT a parallelogram, and therefore NOT a rhombus.

That is, a square is a rectangle and a rhombus at the same time. Let's see what happens.

Is it clear why? - rhombus is the bisector of angle A, which is equal to. This means it divides (and also) into two angles along.

Well, it's quite clear: the diagonals of a rectangle are equal; The diagonals of a rhombus are perpendicular, and in general, a parallelogram of diagonals is divided in half by the point of intersection.

MIDDLE LEVEL

Properties of quadrilaterals. Parallelogram

Properties of a parallelogram

Attention! Words " properties of a parallelogram"mean that if in your task There is parallelogram, then all of the following can be used.

Theorem on the properties of a parallelogram.

In any parallelogram:

Let's understand why this is all true, in other words WE'LL PROVE theorem.

So why is 1) true?

If it is a parallelogram, then:

• lying criss-cross
• lying like crosses.

This means (according to criterion II: and - general.)

Well, that’s it, that’s it! - proved.

But by the way! We also proved 2)!

Why? But (look at the picture), that is, precisely because.

Only 3 left).

To do this, you still have to draw a second diagonal.

And now we see that - according to the II characteristic (angles and the side “between” them).

Properties proven! Let's move on to the signs.

Signs of a parallelogram

Recall that the parallelogram sign answers the question “how do you know?” that a figure is a parallelogram.

In icons it's like this:

Why? It would be nice to understand why - that's enough. But look:

Well, we figured out why sign 1 is true.

Well, it's even easier! Let's draw a diagonal again.

Which means:

AND It's also easy. But...different!

Means, . Wow! But also - internal one-sided with a secant!

Therefore the fact that means that.

And if you look from the other side, then - internal one-sided with a secant! And that's why.

Do you see how great it is?!

And again simple:

Exactly the same, and.

Please note: if you found at least one sign of a parallelogram in your problem, then you have exactly parallelogram and you can use everyone properties of a parallelogram.

For complete clarity, look at the diagram:

Properties of quadrilaterals. Rectangle.

Rectangle properties:

Point 1) is quite obvious - after all, sign 3 () is simply fulfilled

And point 2) - very important. So, let's prove that

This means on two sides (and - general).

Well, since the triangles are equal, then their hypotenuses are also equal.

Proved that!

And imagine, equality of diagonals is a distinctive property of a rectangle among all parallelograms. That is, this statement is true^

Let's understand why?

This means (meaning the angles of a parallelogram). But let us remember once again that it is a parallelogram, and therefore.

Means, . Well, of course, it follows that each of them! After all, they have to give in total!

So they proved that if parallelogram suddenly (!) the diagonals turn out to be equal, then this exactly a rectangle.

But! Pay attention! We're talking about parallelograms! Not just anyone a quadrilateral with equal diagonals is a rectangle, and only parallelogram!

Properties of quadrilaterals. Rhombus

And again the question: is a rhombus a parallelogram or not?

With full right - a parallelogram, because it has (Remember our feature 2).

And again, since a rhombus is a parallelogram, it must have all the properties of a parallelogram. This means that in a rhombus, opposite angles are equal, opposite sides are parallel, and the diagonals bisect at the point of intersection.

But there are also special properties. Let's formulate it.

Properties of a rhombus

Why? Well, since a rhombus is a parallelogram, then its diagonals are divided in half.

Why? Yes, that's why!

In other words, the diagonals turned out to be bisectors of the corners of the rhombus.

As in the case of a rectangle, these properties are distinctive, each of them is also a sign of a rhombus.

Signs of a diamond.

Why is this? And look,

That means both These triangles are isosceles.

To be a rhombus, a quadrilateral must first “become” a parallelogram, and then exhibit feature 1 or feature 2.

Properties of quadrilaterals. Square

That is, a square is a rectangle and a rhombus at the same time. Let's see what happens.

Is it clear why? A square - a rhombus - is the bisector of an angle that is equal to. This means it divides (and also) into two angles along.

Well, it's quite clear: the diagonals of a rectangle are equal; The diagonals of a rhombus are perpendicular, and in general, a parallelogram of diagonals is divided in half by the point of intersection.

Why? Well, let's just apply the Pythagorean theorem to...

SUMMARY AND BASIC FORMULAS

Properties of a parallelogram:

1. Opposite sides are equal: , .
2. Opposite angles are equal: , .
3. The angles on one side add up to: , .
4. The diagonals are divided in half by the point of intersection: .

Rectangle properties:

1. The diagonals of the rectangle are equal: .
2. A rectangle is a parallelogram (for a rectangle all the properties of a parallelogram are fulfilled).

Properties of a rhombus:

1. The diagonals of a rhombus are perpendicular: .
2. The diagonals of a rhombus are the bisectors of its angles: ; ; ; .
3. A rhombus is a parallelogram (for a rhombus all the properties of a parallelogram are fulfilled).

Properties of a square:

A square is a rhombus and a rectangle at the same time, therefore, for a square all the properties of a rectangle and a rhombus are fulfilled. And also:

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Signs pa-ral-le-lo-gram-ma

1. Definition and basic properties of a parallelogram

Let's start by recalling the definition of par-ral-le-lo-gram-ma.

Definition. Parallelogram- what-you-re-gon-nick, which has every two pro-ti-false sides that are parallel (see Fig. .1).

Rice. 1. Pa-ral-le-lo-gram

Let's remember basic properties of pa-ral-le-lo-gram-ma:

In order to be able to use all these properties, you need to be sure that the fi-gu-ra, about someone -roy we are talking about, - par-ral-le-lo-gram. To do this, it is necessary to know such facts as signs of pa-ral-le-lo-gram-ma. We are looking at the first two of them now.

2. The first sign of a parallelogram

Theorem. The first sign of pa-ral-le-lo-gram-ma. If in a four-coal the two opposite sides are equal and parallel, then this four-coal nickname - parallelogram. .

Rice. 2. The first sign of pa-ral-le-lo-gram-ma

Proof. We put the dia-go-nal into the four-reh-coal-ni-ke (see Fig. 2), she divided it into two tri-coal-ni-ka. Let's write down what we know about these triangles:

according to the first sign of the equality of triangles.

From the equality of the indicated triangles it follows that, by the sign of the parallelism of straight lines when crossing, ch-nii their s-ku-shchi. We have that:

Do-ka-za-but.

3. Second sign of a parallelogram

Theorem. The second sign is pa-ral-le-lo-gram-ma. If in a four-corner every two pro-ti-false sides are equal, then this four-corner is parallelogram. .

Rice. 3. The second sign of pa-ral-le-lo-gram-ma

Proof. We put the dia-go-nal in the four-corner (see Fig. 3), she splits it into two triangles. Let's write down what we know about these triangles, based on the theory's form:

according to the third sign of the equality of triangles.

From the equality of triangles it follows that, by the sign of parallel lines, when intersecting them s-ku-shchey. Let's eat:

par-ral-le-lo-gram by definition. Q.E.D.

Do-ka-za-but.

4. An example of using the first parallelogram feature

Let's look at an example of the use of signs of pa-ral-le-lo-gram.

Example 1. In the bulge there are no coals Find: a) the corners of the coals; b) hundred-ro-well.

Solution. Illustration Fig. 4.

pa-ral-le-lo-gram according to the first sign of pa-ral-le-lo-gram-ma.

A. by the property of a par-ral-le-lo-gram about pro-ti-false angles, by the property of a par-ral-le-lo-gram about the sum of angles, when lying to one side.

B. by the nature of equality of pro-false sides.

re-tiy sign pa-ral-le-lo-gram-ma

5. Review: Definition and Properties of a Parallelogram

Let's remember that parallelogram- this is a four-square-corner, which has pro-ti-false sides in pairs. That is, if - par-ral-le-lo-gram, then (see Fig. 1).

The parallel-le-lo-gram has a number of properties: pro-ti-false angles are equal (), pro-ti-false angles -we are equal ( ). In addition, the dia-go-na-li pa-ral-le-lo-gram-ma at the point of re-se-che-niya is divided according to the sum of the angles, at-le- pressing to any side pa-ral-le-lo-gram-ma, equal, etc.

But in order to take advantage of all these properties, it is necessary to be absolutely sure that the ri-va-e-my th-you-rekh-coal-nick - pa-ral-le-lo-gram. For this purpose, there are signs of par-ral-le-lo-gram: that is, those facts from which one can draw a single-valued conclusion , that what-you-rekh-coal-nick is a par-ral-le-lo-gram-mom. In the previous lesson, we already looked at two signs. Now we're looking at the third time.

6. The third sign of a parallelogram and its proof

If in a four-coal there is a dia-go-on at the point of re-se-che-niya they do-by-lams, then the given four-you Roh-coal-nick is a pa-ral-le-lo-gram-mom.

Given:

What-you-re-coal-nick; ; .

Prove:

Parallelogram.

Proof:

In order to prove this fact, it is necessary to show the parallelism of the parties to the par-le-lo-gram. And the parallelism of straight lines most often appears through the equality of internal cross-lying angles at these right angles. Thus, here is the next method to obtain the third sign of par-ral -le-lo-gram-ma: through the equality of triangles .

Let's see how these triangles are equal. Indeed, from the condition it follows: . In addition, since the angles are vertical, they are equal. That is:

(first sign of equalitytri-coal-ni-cov- along two sides and the corner between them).

From the equality of triangles: (since the internal crosswise angles at these straight lines and separators are equal). In addition, from the equality of triangles it follows that . This means that we understand that in four-coal two hundred are equal and parallel. According to the first sign, pa-ral-le-lo-gram-ma: - pa-ral-le-lo-gram.

Do-ka-za-but.

7. Example of a problem on the third sign of a parallelogram and generalization

Let's look at the example of using the third sign of pa-ral-le-lo-gram.

Example 1

Given:

- parallelogram; . - se-re-di-na, - se-re-di-na, - se-re-di-na, - se-re-di-na (see Fig. 2).

Prove:- pa-ral-le-lo-gram.

Proof:

This means that in the four-coal-no-dia-go-on-whether at the point of re-se-che-niya they do-by-lam. By the third sign of pa-ral-le-lo-gram, it follows from this that - pa-ral-le-lo-gram.

Do-ka-za-but.

If you analyze the third sign of pa-ral-le-lo-gram, then you can notice that this sign is with-vet- has the property of a par-ral-le-lo-gram. That is, the fact that the dia-go-na-li de-la-xia is not just a property of the par-le-lo-gram, and its distinctive, kha-rak-te-ri-sti-che-property, by which it can be distinguished from the set what-you-rekh-coal-ni-cov.

SOURCE

http://interneturok.ru/ru/school/geometry/8-klass/chyotyrehugolniki/priznaki-parallelogramma

http://interneturok.ru/ru/school/geometry/8-klass/chyotyrehugolniki/tretiy-priznak-parallelogramma

http://www.uchportfolio.ru/users_content/675f9820626f5bc0afb47b57890b466e/images/46TThxQ8j4Y.jpg

http://cs10002.vk.me/u31195134/116260458/x_56d40dd3.jpg

http://wwww.tepka.ru/geometriya/16.1.gif