## Math 9 Chapter 3 Lesson 1: The central angle and the measure of the arc

## 1. Summary of theory

### 1.1. Angle in the center

An angle whose vertex coincides with the center of the circle is called a central angle.

The two sides of the central angle intersect the circle at two points, thus dividing the circle into two arcs.

- For angles α ( 0 < α < 180°), the arc inside the angle is called the minor arc.
- The arc outside the angle is called the major arc.

### 1.2. Bow measurement

**DEFINE**

The measure of the arc of the minor arc is equal to the measure of the angle at the center of that arc.

The measure of the major arc is \(360^0\) minus the measure of the minor arc

The measure of a semicircle is \(180^0\)

**Sign:** the measure of arc AB is sđ\(\stackrel\frown{AB}\).

**ATTENTION:**

– Minor arc has measure less than \(180^0\)

– Major arc has measure greater than \(180^0\)

– The arc has the same starting and ending points with the measure \(0^0\).

– An arc that includes a circle has a measure of \(360^0\)

### 1.3. Compare two signs

In the same circle or two equal circles:

– Two arcs are said to be equal if they have the same measure.

– Of the two arcs, the one with the greater measure is called the greater arc.

### 1.4. When is sđ\(\stackrel\frown{AB}\) = sđ\(\stackrel\frown{AC}\) + sd\(\stackrel\frown{CB}\)?

**THEOREM: **If C is a point on arc AB then: sđ\(\stackrel\frown{AB}\) = sđ\(\stackrel\frown{AC}\) + sđ\(\stackrel\frown{CB}\)

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1: **Draw a circle and then draw two equal arcs

Draw AC and BD as any two diameters of the circle (O).

We have: \(\widehat {AOB} = \widehat {COD}\) (two opposite angles)

So AmB = inferred CnD arc \( \overparen{AmB}=\overparen {CnD}\)

**Solution guide**

By definition we have sđ\(\stackrel\frown{BC}=30^{0}\), sđ\(\stackrel\frown{AC}=45^{0}\)

Since point C is on arc AB, so sđ\(\stackrel\frown{AB}\)=sđ\(\stackrel\frown{AC}\)+sđ\(\stackrel\frown{BC}\)\(=45 ^0+30^0=75^0\)

**Verse 2:** Based on the following figure, calculate the measure of minor arc BC:

**Solution guide**

\(\bigtriangleup OAC\) has \(OA=OC\) so \(\bigtriangleup OAC\) is bounded at \(O\), so \(\widehat{AOC}=180^0-2\widehat{OAC }=180^0-2.30^0=120^0\).

Then, the measure of major arc BC is: sđ\(\stackrel\frown{BC}\)large \(=110^0+120^0=230^0\).

Infer sđ\(\stackrel\frown{BC}\)small\(=360^0-\)sđ\(\stackrel\frown{BC}\)large \(=360^0-230^0=130^ 0\)

So the measure of minor arc BC is \(130^0\).

**Question 3: **Given a circle with center O and diameter BC. A is a point on the circle such that \(\widehat{AOC}=75^0\). Compare the measures of the two minor arcs AC and AB:

**Solution guide**

We have \(\widehat{AOC}+\widehat{AOB}=180^0\Rightarrow \widehat{AOB}=180^0-\widehat{AOC}=180^0-75^0=105^0\) .

From this it follows that sđ\(\stackrel\frown{AC}=75^0\), sđ\(\stackrel\frown{AB}=105^0\) so sđ\(\stackrel\frown{AB}\) >sđ\(\stackrel\frown{AC}\).

### 2.2. Advanced exercises

**Question 1: **Given the following figure:

Calculate the measure of minor arc AB, \(\widehat{ADB}\) and then compare the two sides AC and AD.

**Solution guide**

\(\bigtriangleup ACO\) square at A has \(\widehat{ACO}=20^0\) so \(\widehat{AOC}=90^0-20^0=70^0\Rightarrow\)sđ\ (\stackrel\frown{AB}=70^0\)

\(\widehat{AOB}\) is the exterior angle of the isosceles triangle AOD. So \(\widehat{ADB}=\frac{1}{2}\widehat{AOB}=\frac{1}{2}.70^0=35^0\).

Considering \(\bigtriangleup ACD\) has \(\widehat{ACD}<\widehat{ADC}(20^0<35^0)\) so \(AC>AD\).

**Verse 2:** Based on the figure below, calculate the measure of minor arc AB, knowing that B is the midpoint of OC.

**Solution guide**

Triangle ABC is right-angled at A and B is the midpoint of OC, so BO=BC=BA.

Which OB=OA infers OB=OA=AB, from which \(\bigtriangleup OAB\) is regular. Then \(\widehat{AOB}=60^0\) so the measure of minor arc AB is \(60^0\).

## 3. Practice

### 3.1. Essay exercises

**Question 1: **

a. From 1 o’clock to 3 o’clock, by how many degrees does the hour hand rotate in the center?

b. From 3 o’clock to 6 o’clock, by how many degrees does the hour hand rotate in the center?

**Verse 2: **Fold a piece of paper to cut into an even five-pointed star. If you want to cut the paper with just one stroke of scissors, you must fold the paper into a shape with a central angle of how many degrees?

**Question 3:** Two tangents at A,B of the circle (O ;R) intersect at M. Knowing OM = 2R. Calculate the measure of the angle at the center of AOB?

**Question 4: **Given two circles (O; R) and (O’ ; R’) intersect at A ,B . Compare R and R’ in the following cases:

a. The measure of minor arc AB of (O ;R) is greater than the measure of minor arc AB of (O’ ;R’)

b. The measure of minor arc AB of (O ;R) is less than the measure of minor arc AB of (O’ ;R’)

c. Measures of two equal minor arcs

### 3.2. Multiple choice exercises

**Question 1: **Calculate the measure of minor arc BD and compare the measures of two minor arcs CD and AB:

A. sd\(\stackrel\frown{BD}=80^0\) and sd\(\stackrel\frown{AB}\) > sd\(\stackrel\frown{CD}\)

B. sd\(\stackrel\frown{BD}=80^0\) and sd\(\stackrel\frown{AB}\) < sd\(\stackrel\frown{CD}\)

C. sd\(\stackrel\frown{BD}=85^0\) and sd\(\stackrel\frown{AB}\) > sd\(\stackrel\frown{CD}\)

D. sd\(\stackrel\frown{BD}=85^0\) and sd\(\stackrel\frown{AB}\) < sd\(\stackrel\frown{CD}\)

**Verse 2: **Which of the following assertion is true:

A. If C is a point on arc AB, then sđ\(\stackrel\frown{BC}\) = sđ\(\stackrel\frown{AC}\) – sđ\(\stackrel\frown{AB}\)

B. Major arc measuring less than 1800

C. An angle whose vertex coincides with the center is called a central angle

D. Minor arc with measure greater than 1800

**Question 3: **Which of the following statements are incorrect?

a) Of two arcs on a circle, the one with the smaller measure is the smaller one.

b) In two arcs on the same circle or two equal circles, two equal arcs have the same measure.

c) Of the two arcs, the one with the greater measure is greater.

d) The measure of a semicircle is \(90^0\).

A. 4

B. 2

C. 1

D. 3

**Question 4: **Find the measure of minor arc AB and minor arc CD in the figure:

A. sđ\(\stackrel\frown{AB}=120^0\),sđ\(\stackrel\frown{CD}=80^0\)

B. sđ\(\stackrel\frown{AB}=130^0\),sđ\(\stackrel\frown{CD}=100^0\)

C. sđ\(\stackrel\frown{AB}=115^0\),sđ\(\stackrel\frown{CD}=80^0\)

D. sđ\(\stackrel\frown{AB}=120^0\),sđ\(\stackrel\frown{CD}=100^0\)

**Question 5:** Knowing the measure of minor arc AC is 1100, calculate \(\widehat{BOC}\)?

A. \(120^0\)

B. \(100^0\)

C. \(130^0\)

D. \(90^0\)

## 4. Conclusion

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

- Recognizing the central angle, can indicate 2 corresponding arcs, including the intercepted arc.
- Compare 2 arcs on a circle based on their measure (degrees).
- The first step is to use the theorem to add arcs.

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