Properties of a trapezoid circumscribed about a circle: formulas and theorems

A trapezoid is a geometric figure with four angles. When constructing a trapezoid, it is important to take into account that two opposite sides are parallel, and the other two, on the contrary, are not parallel relative to each other. This word came to modern times from Ancient Greece and sounded like “trapezion”, which meant “table”, “dining table”.

This article talks about the properties of a trapezoid circumscribed about a circle. We will also look at the types and elements of this figure.

Properties of a trapezoid

Properties of a trapezoid... What are they and what should you know about them?

Why is that? Well, of course, simply because the bases are parallel and the side is a secant. So it turns out that and are internal one-sided angles with parallel and and transversal. That's why . And in exactly the same way and are interior one-sided angles with the same parallel and , but the secant is now - .

You see: the main thing that plays a role is the parallelism of the bases. Let's look at some more properties of the trapezoid.

Like any quadrilateral, a trapezoid has diagonals. There are two of them - look at the pictures:

Well, now let's talk about angles again.

And now - 2 diagonals and 4 corners at once:

What can follow from this? A very important fact: triangles and are similar at two angles. Their similarity coefficient is equal to the ratio of the bases: .

Rectangle

The next special quadrilateral is the rectangle (sometimes abbreviated as rectangle). Its distinctive feature is that all its angles are right.

Let's show some rectangles:

It is obvious that the opposite sides of a rectangle are parallel, because they are perpendicular to the same straight line. Consequently, every rectangle is at the same time a parallelogram and has all its properties. It is worth especially noting that the converse statement is not true - not every parallelogram is a rectangle. In other words, a rectangle is a special case of a parallelogram, which differs in that its angles are 90°.

This implies two properties of a rectangle:

• its opposite sides are equal;
• the point of intersection of its diagonals is the middle of these same diagonals.

However, there is one more property that is NOT characteristic of other parallelograms.

It's very easy to prove. Let there be a rectangle ABCD:

Let's compare ∆АВD and ∆АСD. They are rectangular, they have a common leg AD, and the other two legs, AB and CD, are equal as opposite sides of the rectangle. It turns out that the triangles under consideration are equal, and therefore their hypotenuses, which are precisely the diagonals of the rectangle, are also equal.

It turns out that the converse theorem is also true, which is called the rectangle test:

Indeed, let there be some parallelogram ABCD whose diagonals AC and BD are the same.

Opposite sides in one parallelogram are the same:

As a result, all angles ABCD turn out to be right angles, and this figure, by definition, turns out to be a rectangle.

Exercise. in rectangle ABCD, which divides side C D into segments SK and K D of length 27 and 45 cm, respectively. Find the perimeter of AB CD.

Solution: To find the perimeter, you need to find the lengths of all sides.

If AK is a bisector, then

∆CAD is rectangular and we have just found one of its acute angles. Then you can find the 2nd angle:

It turns out that in ∆AKD two angles are equal to 45°, which means it is isosceles, and

We have found two adjacent sides of the rectangle, AD and CD. The other two sides will be equal to them:

Midline of trapezoid

To begin with, what is the midline of the trapezoid?

It turns out that the length of this midline can be expressed in terms of the lengths of the bases of the trapezoid. Namely, the following formula holds:

And also:

Trapezoid inscribed in a circle.

Even if you have not yet studied the topic “Circle. Inscribed angle" and "Inscribed quadrilateral", you will find it useful (and hopefully interesting) to know the following amazing fact:

We won’t prove this (here, anyway), but it would be good to remember – it will come in handy!

Let's summarize - it's short. The most important thing about a trapezoid is that it has two parallel sides and the BCE properties of a trapezoid are determined by this.

So, if you have a trapezoid in your problem, use parallelism and everything will work out!

Shrugs with a barbell behind your back

Technique: Take a barbell or dumbbells in your hands. Stand up straight and straighten your elbows completely. Raise your arms using only the trapezius muscles. Then gently lower it back to feel the muscle. Here you are allowed to take heavy weights, but keep in mind that the amplitude must be ideal. If you took it too big, just reduce it.

In female version with photo.

Trapezoid. Basic concepts and definitions

The parallel sides are called bases , and the non-parallel sides are called sides .

Properties of a trapezoid

Properties of a trapezoid... What are they and what should you know about them? Let's look at the basic properties of a trapezoid.

First property of a trapezoid

Why? and are parallel, and and are secant, therefore:

Second property of a trapezoid

The similarity coefficient of triangles is equal to the ratio of the bases:

Third property of a trapezoid

First, let's formulate the basic definition that you need to know to understand this property of a trapezoid:

And now the formula:

And here is the third property of the trapezoid:

Why is this? That one is a little more complicated - you will need to draw as much as one extra line!

So, let's do it. Then the quadrilateral is a parallelogram. Let's take the middle of the side and the middle of the side. Both: and are again parallelograms ( and ; and ). Well, that means, yes.

Let's move on.

What follows from all this?

That's what they proved!

The fourth property of a trapezoid

Why? For more details, see the topic “Inscribed quadrilateral”, and here in two lines: (it’s a trapezoid!) (inscribed quadrilateral). Well, the same.

Fifth property of a trapezoid

We won’t prove this theorem - don’t be alarmed.

Let us only note that the REVERSE is also TRUE:

The sixth property of a trapezoid

Seventh property of a trapezoid

Here we will see once again how useful it is to draw a line in a trapezoid, parallel to either the side or diagonal - a new look immediately appears. We have already done this once - in the paragraph about the middle line. And now you have learned a new fact that is relatively common in problems.

Let's prove it! This is already a whole task that may well come up right in the exam!

Well, now try to solve problems about the trapezoid with the help of new knowledge and methods - they are usually not too difficult. The main thing is to firmly remember all the properties of a trapezoid and not to forget about the parallelism of the bases, and sometimes (in more complex problems) it is useful to draw something parallel or connect the sides.

Let's carry out and.

Let's denote ; .

Then:

1. – rectangular

This means (the median drawn to the hypotenuse is equal to half of it). That is . But after all (since it’s a parallelogram).

Close-grip barbell row to the chin

Grasp the barbell with a close overhand grip. Stand up straight. The narrower you take the barbell, the more the trapezius is loaded. The arms must be straight at the elbows. Start moving vertically up to your chin. Slowly return to the starting position. Important: elbows should be directed to the sides.

In female version with photo.

TRAPEZE. SUMMARY AND BASIC FORMULAS

A trapezoid is a quadrilateral in which two sides are parallel (they are called the bases), and the other two are not (these are the sides).

• The middle line is parallel to the bases: .
• The length of the midline of a trapezoid is equal to half the sum of the lengths of the bases: .

• Triangles formed by the bases of a trapezoid and the diagonal segments ( and ) are similar at two angles with a similarity coefficient equal to the ratio of the bases: .
• The areas of the triangles formed by the lateral sides and segments of the diagonals of the trapezoid are equal to: .

Properties of an isosceles trapezoid:

• diagonals are equal: ;
• the angles at the base are equal: ;
• the sum of opposite angles is equal to: .

The sides and diagonal of an isosceles trapezoid are related by the relation: .

The area of ​​a trapezoid is equal to half the sum of the bases multiplied by the height: .

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