## Math 9 Chapter 2 Lesson 1: Determination of circles and symmetry of circles

## 1. Summary of theory

### 1.1. Reminds me of the circle

A circle with center O and radius R (R>0) is a figure consisting of points a distance from O equal to R denoted by (O;R).

- M lies on the circle (0;R) If and only if OM=R
- M lies outside the circle if and only if OM>R
- M lies inside the circle if and only if OM

### 1.2. How to define a circle

We already know: A circle is determined when the center and radius of that circle is known, or when a line segment is known as the diameter of that circle.

Through 3 noncollinear points, we can draw one and only one circle.

**Attention: **

No circle can be drawn through three collinear points

The circle passing through the three vertices of a triangle is called the circumcircle of the triangle

A circle is a shape with a center of symmetry. The center of a circle is the center of opposites of the circle

### 1.4. Axis of symmetry

A circle is a shape with an axis of symmetry, Any diameter is an axis of symmetry of the circle

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1: **Prove that the circumcircle of a right triangle has its center at the midpoint of the hypotenuse.

**Solution guide**

Consider triangle ABC, right angled at A

Let M be the midpoint of BC, so MB=MC. It is easy to prove that MA=MB=MC, thereby deducing the circumcircle ABC whose center is the midpoint of the hypotenuse

(For example, let N be the midpoint of AB. According to the properties of the mean line \(MN\parallel AC\Rightarrow MN\perp AB\) MN is both the altitude and the median of the triangle ABM, so ABM is isosceles at M)

**Verse 2: **Given Rectangle ABCD with AB=10, BC=8. Prove that A,B,C,D are on the same circle and find the radius of that circle

**Solution guide**

Applying the proof as in lesson 1, we have: The circumcircle of triangle ABC has center E and radius EA. Similarly for triangle ADC also circumcircle at E radius is EA

then we have A, B, C, D on the same circle with center E and radius AE.

\(AE=\frac{1}{2}.AC=\frac{1}{2}.\sqrt{AB^2+BC^2}=\frac{1}{2}.\sqrt{10^ 2+8^2}=\sqrt{41}\)

**Question 3:** Given a rhombus ABCD with angle A 60 degrees, Let E, F, G, H be the midpoints of AB, BC, CD, DA respectively. Prove that E, F, G, H, B, D are on the same circle

**Solution guide**

ABCD is a rhombus, so O is the midpoint of AC and BD. Since OE is the median of triangle ABD, \(OE\parallel AD\Rightarrow \widehat{OEB}=\widehat{DAB}=60^{\circ}\)

An isosceles triangle ABD has \(\widehat{A}=60^{\circ}\) so ABD should be \(\widehat{ABD}=60^{\circ}\Rightarrow \Delta EOB\) equilateral. Similar for HOD, DOG, FOB . Triangle

Combining OB = OD infers the top 6 points lying on a circle with center O and radius OB.

### 2.2. Advanced exercises

**Question 1:** Given a circle with center (O) and diameter AB. Draw a circle with center (I) diameter OA. radius OC of (O) intersects circle (I) at D. draw CH perpendicular to AB.

Prove that quadrilateral ACHD is an isosceles trapezoid

**Solution guide**

It is easy to prove that triangle ADO is right-angled at D. Consider two right triangles ADO and CHO with AO=OC; \(\widehat{AOD}=\widehat{COH}\)

\(\Rightarrow \Delta ADO=\Delta FOR\Rightarrow OD=OH; AD=CH\)

\(\Delta DOH\sim \Delta COA\Rightarrow \widehat{OHD}=\widehat{HIGH}\Rightarrow DH\parallel AC\) \(\Rightarrow ADHC\) is an isosceles trapezoid

**Verse 2: **Given a circle with center (O) and diameter AB. H and K are the perpendicular projections of A and B onto CD, respectively. CM: CH=DK

**Solution guide**

Through O draw a line perpendicular to CD intersecting CD at F, HB at G. Consider triangle ABH with O as midpoint AB, \(OG\parallel AH\) \(\Rightarrow G\) as midpoint BH

Consider triangle HKB with \(FG\parallel KB\) and G is mid point BH so F is mid point HK.

and F is the midpoint of CD, so we have something to prove

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Let ABCD be a rectangle with AD = 12cm, CD = 16cm. Prove that four points ABCD are on the same circle. Calculate the radius of that circle.

**Verse 2:** Given acute angle xOy and two points D and E on ray Oy. Construct a circle with center M passing through D and E such that the center M lies on the ray Ox.

**Question 3:** Which of the following sentences is true and which is false?

a. Two distinct circles can have two points in common

b. Two distinct circles can have three distinct points in common

c. The center of the circumcircle of a triangle is always inside that triangle.

**Question 4:** Given square ABCD, O is the intersection of two diagonals, OA = √2 cm. Draw a circle with center A and radius 2cm. Of the five points A, B, C, D, O, which point lies on the circle? Which point is inside the circle? Which point lies outside the circle?

### 3.2. Multiple choice exercises

**Question 1:** Given trapezoid ABCD \((AB\parallel CD), \widehat{C}=\widehat{D}=60^{\circ}, CD=2AD=8\) Then A, B, C, D always belong Which circle?

A. \((I;R=4\sqrt{2})\) I is the midpoint of CD

B. \((O=AC\cap BD;R=4\sqrt{2})\)

C. \((O=AC\cap BD;R=4)\)

D. \((I;R=4)\) I is the midpoint of CD

**Verse 2: **Given a circle with center A and diameter BC. Let D be the midpoint of AB. String EF is perpendicular to AB at D. What is the quadrilateral EBFA?

A. Rectangle

B. Square

C. Rhombus

D. Not enough data to conclude

**Question 3: **Let two lines xy and x’y’ perpendicular to each other intersect at O. A line segment AB=8 moves such that A always lies on xy and B always lies on x’y’. Then on what path does the midpoint M of AB move?

A. A straight line parallel to xy is 4 . distance from xy

B. The line parallel to x’y’ is 4 . distance from x’y’

C. Circle with center O and radius 4

D. Circle with center O and radius 8

**Question 4: **Let ABC be a triangle with BH and CE being altitudes. Let M be the intersection point BH and CE. I is the midpoint of BC. Then B, C, E, H belong to which circle?

A. \((I;R=IA)\)

B. \((I;R=IB)\)

C. \((M;R=MB)\)

D. \((M;R=MA)\)

**Question 5: **Let ABC be an acute triangle. The circle with diameter BC intersects AB at N, AC at M. Let H be the intersection of CN and BM. Then A, N, H, M lie on which circle?

A. \((I;IM)\), I is the midpoint of MN

B. \((I;IH)\), I is the midpoint of MN

C. \((F;FA)\), F is the intersection of the circle with AH

D. \((E;EA)\), E is the midpoint of AH

## 4. Conclusion

Through this lesson, you will understand some of the main topics as follows:

- Understand the definition of a circle, ways to determine a circle, circumcircle of a triangle, triangle inscribed in a circle. Students know that a circle is a shape with a center of symmetry and an axis of symmetry.
- Know how to construct a circle passing through a non-collinear point.Know how to prove that a point lies on, inside, and outside the circle. Students know how to apply their knowledge in practice.

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