Tứ giác nội tiếp toán 9 – Chi tiết lý thuyết và bài tập

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As you all know, Mathematics is one of the important and main subjects in the curriculum. In particular, 9th grade math helps students practice thinking, analytical ability, calculation and application of knowledge to do exercises, and at the same time is the foundation for higher learning programs in the future. In today’s article, Kien Guru will introduce you to the inscribed quadrilateral.

What is an inscribed quadrilateral?

First, to better understand this part of knowledge, you need to understand what is an inscribed quadrilateral?

1. Define.

A quadrilateral with four vertices lying on the same circle is called a cyclic quadrilateral (referred to as inscribed quadrilateral).

Eg:

The quadrilateral ABCD in figure a) is called a quadrilateral inscribed in the circle with center O because all four vertices A, B, C and D lie on the circle with center O.

The quadrilateral MNPE in figure b) is not an inscribed quadrilateral because there is a vertex E that is not on the circle.

Thus, as long as at least one vertex of the quadrilateral is not on the circle, then the quadrilateral is not a quadrilateral inscribed in that circle.

Attention:

– The circle with center O is called the circumcircle of quadrilateral ABCD.

– Through three distinct points that are not collinear, there is always only one circle, so every triangle has a circumcircle but not every quadrilateral has a circumcircle.

1. Nature

Theorem: In a cyclic quadrilateral, the sum of two opposite angles is 180 degrees

Prove:

Consider quadrilateral ABCD with four vertices lying on the circle (O). I need to prove

Inversion theorem: If a quadrilateral has the sum of the measures of two opposite angles equal to 180 degrees, then the quadrilateral is inscribed in the circle.

Prove:

We already know: Through three distinct non-collinear points, only one circle can be determined.

Because A, B, C are three distinct points that are not collinear, we can draw a circle with center O, through three points A, B, C . Then the string AC divides the circle into two arcs AnC and AmC .

Hence point D lies on arc AmC

So quadrilateral ABCD has all four vertices lying on the circle with center O.

We have gone through the properties of the inscribed quadrilateral, now Ant will continue with the signs to recognize the inscribed quadrilateral.

Signs to recognize inscribed quadrilaterals

Next, let’s learn about the telltale signs of an inscribed quadrilateral.

Quadrilaterals with one of the following characteristics are inscribed quadrilaterals:

1. Isosceles trapezoid, rectangle or square

Since ABCD is a square, ABCD is a quadrilateral inscribed in the circle.

2. A quadrilateral with four vertices lying on the same circle (or a quadrilateral with four vertices equidistant from one point)

OA = OB = OC = OD

3. A quadrilateral whose sum of opposite angles is 180 degrees

4. A quadrilateral with two adjacent vertices looking down at the opposite side at equal angles

5. A quadrilateral with an exterior angle at one vertex equal to the interior angle of the opposite vertex.

To continue the lesson, Kien will introduce you to ways to prove the inscribed quadrilateral.

Ways to prove the inscribed quadrilateral.

To solve the exercise well, you need to know the ways to prove the inscribed quadrilateral.

1. Prove that the four vertices of the quadrilateral are equidistant from a certain point

Example: Given a fixed point O and quadrilateral ABCD.

If students prove that four points A, B, C, D are equidistant from point O with a distance equal to R, ie OA = OB = OC = OD = R, then point O is the center of the circle passing through four points A, B , C, D. In other words, quadrilateral ABCD is inscribed in a circle with center O and radius R.

2. Prove that a quadrilateral whose sum of opposite angles is 180°

Let the quadrilateral ABCD

This method is derived from the very definition of the inscribed quadrilateral. The content of this method is as follows: “If quadrilateral ABCD has the sum of two opposite angles equal to 180 degrees, then the quadrilateral is inscribed”

The consequences of this content are:

For quadrilateral ABCD:

3. A quadrilateral whose exterior angle at a vertex is equal to the interior angle at the opposite vertex of that vertex is inscribed in a circle

Example: Given a quadrilateral PQRS, if it is proved that the exterior angle at vertex R is equal to the interior angle at vertex P (that is, angle P of that quadrilateral) because angle P and angle R are opposite, then the quadrilateral PQRS is inscribed in the circle.

In this method, you must pay attention to the right picture at the right angle, otherwise you will be wrong, but the result is correct and affect the next sentences. Specifically, when the problem is for the quadrilateral PQRS and it is proved that the exterior angle at the vertex R is equal to the angle P of the quadrilateral (angle P and angle R are opposite), it can be concluded that the quadrilateral PQRS is an inscribed quadrilateral.

4. Prove that from two vertices that are adjacent to the same edge, two angles are congruent

Let the quadrilateral ABCD

This method applies when the problem is for quadrilateral ABCD and the data suggests that DAC = DBC = 90 degrees. From there, students can conclude that quadrilateral ABCD is inscribed in a circle.

5. If a quadrilateral has the sum of the measures of two opposite angles equal, then the quadrilateral is inscribed in a circle

This method is a special case of the second proof method.

6. Prove by the method of refutation

You can prove quadrilateral ABCD given the problem by this method, into one of the special shapes is isosceles trapezoid, square and rectangle. Then, based on the basic properties of these shapes, it is easy to deduce quadrilateral ABCD inscribed in a circle.

In summary, we prove quadrilateral ABCD is one of the special shapes: Quadrilateral ABCD is isosceles trapezoid, rectangle or square.

Some notes when doing the test to prove the inscribed quadrilateral

– Children should draw clear, easy-to-see pictures and avoid drawing pictures in some special cases.

– Symbols of equal angles and lines should be clearly marked.

– Stick to assumptions, learned knowledge to do the test effectively.

– The requirements of the problem can also be suggested directions to solve the problem.

Do not use things that need to be proven to prove them.

Above, Kien and students learned the theory of inscribed quadrilaterals, the signs to recognize inscribed quadrilaterals, and ways to prove inscribed quadrilaterals. We will do exercises to reinforce our knowledge.

Practical exercise

After fully grasping the theory of theory, Kien will introduce students to practical exercises and the most detailed solutions.

I. Exercise page 89 Textbook 9 volume 2

Lesson 53: Know ABCD is a cyclic quadrilateral. Please fill in the blanks in the following table (if possible).

We fill in the table as follows:

Calculation:

Solution guide:

Solution guide:

Lesson 56: See figure 47. Find the measures of the angles of quadrilateral ABCD.

Solution guide:

Lesson 57: Which of the following figures is inscribed in a circle?

Parallelogram, rectangle, square, trapezoid, square trapezoid, isosceles trapezoid? Why?

Solution guide:

The figures inscribed in a circle are:

Since a square is a rectangle ⇒ The square is also inscribed in a circle.

II. Exercises page 90 Textbook 9 volume 2

a) Prove that quadrilateral ABDC is a cyclic quadrilateral.

b) Determine the center of the circle passing through the four points A, B, D, C.

Solution guide:

a) Prove that quadrilateral ABDC is a cyclic quadrilateral.

Quadrilateral ABDC has:

Quadrilateral ABDC is a cyclic quadrilateral

b) Determine the center of the circle passing through the four points A, B, D, C.

⇒ AD is the diameter of the circumcircle of triangle ABD Where ABDC is the inscribed quadrilateral

⇒ AD is the diameter of the circumcircle of quadrilateral ABDC.

⇒ Center O is mid point AD.

So the center of the circle passing through the four points A, B, D, C is the mid point AD.

Lesson 59: Given parallelogram ABCD. The circle passing through three vertices A, B, C intersects the line CD at P other than C. Prove AP = AD.

Solution guide:

Lesson 60: Proving QR // ST

Above, Kien Guru introduced you to the knowledge of solving 9 inscribed quadrilaterals, hoping that you will firmly grasp the knowledge to successfully complete the high school exam. Stay tuned for the next articles to learn more. Get more useful materials and knowledge.

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