Math 9 Chapter 3 Lesson 8: The circumcircle and the incircle

1.1. Define

– The circle that passes through all the vertices of a polygon is called the circumcircle of the polygon and this polygon is called the incircle.

The circle tangent to all sides of a polygon is called the incircle of the polygon and the polygon is called the circumcircle.

1.2. Theorem

Every regular polygon has an circumcircle and an incircle. The centers of these two circles coincide and are called the centers of the regular polygon

– An equilateral triangle ABC has the same circumcenter and incircle

– The square XYZT has the same center of the circumcircle and inscribed circle

1.3. Formula for calculating radius of circumcircle and incircle of regular polygon.

Regular polygon \(n\) side length \(a, R\) is the radius of the circumcircle and \(r\) is the radius of the incircle of the polygon.

We have: \( R\) = \(\dfrac{a}{2sin\dfrac{180^{\circ}}{n}}\); \(r\) = \(\dfrac{a}{2tan\dfrac{180^{\circ}}{n}}\).

2. Illustrated exercise

2.1. Basic exercises

Question 1:

a) Draw a circle with center O and radius R = 2cm.

b) Draw a regular hexagon ABCDEF with all vertices lying on the circle (O).

c) Why is the center O equidistant from the sides of a regular hexagon? Call this distance r.

d) Draw a circle (O; r).

Solution guide

a)

b) How to draw a regular hexagon with all vertices lying on the circle (O)

Draw chords AB = BC = CD = DE = EF = FA = R = 2 cm

c) Since the chords AB = BC = CD = DE = EF = FA are equal, the distances from O to the chords are equal.

Verse 2: Let XYZT square with center I. Find the circumference of the circumcircle of the square given that the circumference of the incircle of square XYZT is \(20\pi\)(cm)

Solution guide

Let \(R,r (cm)\) be the radii of the circumcircle and incircle of square XYZT, respectively.

According to the problem, the circumference of the incircle of square XYZT is \(20\pi\)(cm) so \(2r.\pi=20\Rightarrow r=10 cm\)

Draw \(ID\perp XY (D\in XY)\)

Then triangle IXD is right-angled at D, applying Pythagorean theorem we have \(R^2=2r^2\Rightarrow R=\sqrt{2.10^2}=10\sqrt{2} cm\)

Perimeter of the circumcircle of the square is: \(2\pi R=20\sqrt{2} \pi (cm)\)

Question 3: Let MNPQ square with side 4cm. Calculate the area of ​​the square, the area of ​​the circle inscribed and circumscribed in the square MNPQ.

Solution guide

Area of ​​square MNPQ is: \(S_{MNPQ}=4^2=16(cm^2)\)

Draw \(OS\perp PQ (S\in PQ)\) then \(SQ=SP=2cm\)

It is easy to prove that triangle OSQ is right-angled at S

Applying Pythagorean theorem to right triangle OSQ we have \(OQ=\sqrt{2.OS^2}=2\sqrt{2}(cm)\)

The area of ​​the circle inscribed in the square is: \(S_{1}=OS^2.\pi=4\pi (cm^2)\)

The area of ​​the circle circumscribed about the square is: \(S_{2}=OQ^2.\pi=(2\sqrt{2})^2\pi=8\pi (cm^2)\)

2.2. Advanced exercises

Question 1: Prove that: In a square, the radius of the circumcircle is always greater than the radius of the incircle of that square.

Solution guide

Consider a square ABCD with center O, draw \(OM\perp CD (M\in CD)\)

Then OD is the radius of the circumcircle, OM is the radius of the incircle of square ABCD .

\(\bigtriangleup OMD\) is square at M so \(OD\geq OM\) (1)

Suppose \(OD= OM\) then the incircle and the circumcircle are two circles that have the same center O and the lengths of the two radii are equal, so they coincide.

Then there is no square that has both a vertex on the circle (O) and a side tangent to the circle (O).

Therefore \(OD\neq OM\) combined with (1) we have \(OD > OM\) (dpcm)

Verse 2: Let ABCDEF be a regular hexagon with center O. Let R,r be the radii of the circumcircle and incircle of the hexagon, respectively. Write an expression for the relationship between R and r.

Solution guide

The regular hexagon ABCDEF should divide the circumcircle (O) into 6 equal arcs, deducing \(\widehat{AOF}=\frac{360^0}{6}=60^0\)

The triangle AOF is isosceles at O ​​with \(\widehat{AOF}=60^0\) so \(\bigtriangleup AOF\) is regular.

Draw the altitude AH of \(\bigtriangleup AOF\) then \(OH=r\) and \(AH=\frac{R}{2}\)

\(\bigtriangleup AOH\) is square at H so \(AO^2=OH^2+AH^2\Rightarrow R^2=r^2+(\frac{R}{2})^2\Rightarrow r^ 2=\frac{3R^2}{4}\Rightarrow r=\frac{R\sqrt{3}}{2}\)

3. Practice

3.1. Essay exercises

Question 1: Draw a square \(ABCD\) with center \(O\) then draw an equilateral triangle with a vertex \(A\) and take \(O\) as the center. Show how to draw.

Verse 2: Draw a circle with center \(O\) and radius \(R = 2cm\) and then draw a regular octagon inscribed in circle \((O; 2 cm).\) Describe how to draw.

Question 3:

\(a)\) Draw a regular hexagon \(ABCDEG\) inscribed in a circle of radius \(2cm\) and then draw \(12\) with equilateral side \(AIBJCKDLEMGN\) inscribed in the circle. Show how to draw.

\(b)\) Calculate the side length \(AI.\)

\(c)\) Calculate the radius \(r\) of the circle inscribed in \(AIBJCKDLEMGN.\)

Tutorial. Apply the formulas in lesson \(46.\)

Question 4:

\(a)\) Calculate the side of a regular pentagon inscribed in a circle of radius \(3cm.\)

\(b)\) Calculates the side of a regular pentagon circumscribed about a circle of radius \(3cm.\)

3.2. Multiple choice exercises

Question 1: An equilateral triangle ABC has center (O), radius of the circumcircle of the triangle is 12cm. Then, the perimeter of the triangle is:

A. \(12\sqrt{3}(cm)\)

B. \(24\sqrt{3}(cm)\)

C. \(36\sqrt{3}(cm)\)

D. \(48\sqrt{3}(cm)\)

Verse 2: Which of the following statements is false:

A. A square can always be inscribed in a circle.

B. A triangle can always be inscribed in a circle

C. Regular pentagons always have inscribed and circumscribed circles

D. In the square, let R,r be the radii of the circumcircle and incircle of the square, respectively. Then R=2r

Question 3: Which of the following assertion is true:

A. Any polygon has an inscribed circle and an circumscribed circle.

B. The radius of the inscribed circle of a square is always greater than the radius of the circumcircle of that square.

C. Any polygon with the center of the circumcircle and the incircle coincide.

D. A triangle always has an incircle and an circumcircle.

Question 4: Let ABCD be a square with center O. Let R,r be the radii of the circumcircle and inscribed circle of square ABCD. Know \(R+r=3\sqrt{2}(cm)\). Find the circumference of the circle inscribed in square ABCD.

A. \((12-6\sqrt{2})\pi (cm)\)

B. \((18-6\sqrt{2}) \pi (cm)\)

C. \(8 (cm)\)

D. \(12-6\sqrt{2} (cm)\)

Question 5: Given a regular hexagon ABCDEF whose circumcircle radius is \(2\sqrt{3} (cm)\). The radius of the circle inscribed in this hexagon is:

A. \(\frac{3\sqrt{2}}{2} (cm)\)

B. \(4 (cm)\)

C. \(3\sqrt{2}(cm)\)

D. \(3(cm)\)

4. Conclusion

Through this lesson, you will learn some key topics as follows:

• State defined, properties of circumcircle, incircle of a polygon.
• Recognize that any regular polygon has an circumscribed circle and an inscribed circle.
• Draw the center of a regular polygon, thereby drawing the circumcircle and incircle of a given regular polygon.
• Calculate side a in terms of R and vice versa, calculate R in terms of side a of equilateral triangles, squares, and hexagons.