## Math 7 Chapter 3 Lesson 8: Properties of the three perpendicular bisectors of a triangle

## 1. Summary of theory

### 1.1. The perpendicular bisector of the triangle

In a triangle, the perpendicular bisector of one side is called a perpendicular bisector of that triangle.

– Each triangle has three perpendicular bisectors.

**Comment: **In an isosceles triangle, the perpendicular bisector of the base side is also the median to this side.

### 1.2. Properties of the three perpendicular bisectors of a triangle

**Theorem: **The three perpendicular bisectors of a triangle pass through the same point. This point is equidistant from the three vertices of the triangle.

**Eg: **The three perpendicular bisectors of triangle ABC go through the same point O. infer OA = OB = OC

**Attention:**

- Since the intersection O of the three perpendicular bisectors of a triangle ABC is equidistant from the three vertices of the triangle, there is a circle with center O passing through the three vertices A, B, and C.
- We call that circle the circumcircle of triangle ABC.

## 2. Illustrated exercise

**Question 1:** On the three sides AB, BC and CA of an equilateral triangle ABC. Take the points in the order M, N, P such that AM=BN=CP. Let O be the intersection of the three perpendicular bisectors of triangle ABC. Prove that O is also the intersection of the three perpendicular bisectors of the triangle MNP.

**Solution guide**

Under the assumption that O is the intersection of the three perpendicular bisectors of triangle ABC, we have:

OA = OB = OC

\( \Rightarrow \) The triangles AOM, BON and COP have:

AM = BN = CP (hypothetical)

\(\widehat {{A_1}} = \widehat {{B_1}} = \widehat {{C_1}} = {30^0}\) (Since ABC is an equilateral triangle, the perpendicular bisector is also the bisector) and OA = OB = OC

\(\begin{array}{l} \Rightarrow \Delta AOM = \Delta BON = \Delta COP\,\,\,(cgc)\\ \Rightarrow \,\,OM = ON = OP\end{array} \)

This proves that O is the intersection of the three perpendicular bisectors of the triangle MNP .

**Verse 2: **Let ABC be a triangle. Find a point O equidistant from three points A, B, C.

**Solution guide**

Point O is equidistant from two points A and B, so point O lies on the bisector of the line segment AB.

Point O is equidistant from two points B and C, so O lies on the perpendicular bisector of the line segment BC.

Point O is equidistant from three points A, B, and C, so O is the intersection of the perpendicular bisectors of triangle ABC.

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Triangle ABC has \(\widehat A\) as an obtuse angle. The perpendicular bisectors of AB and AC intersect at O and intersect BC at P and E respectively. The circle with center O and radius OA passes through which points in the figure.

**Verse 2:** Given triangle ABC and bisector AK of angle A. Know that the intersection point of the bisector of triangle ABK coincides with the intersection of the three perpendicular bisectors of triangle ABC. Find the measures of the angles of triangle ABC.

**Question 3: **Determine the form of a triangle whose intersection of the bisectors coincides with the intersection of the perpendicular bisectors.

### 3.2. Multiple choice exercises

**Question 1:** Given ΔABC, two altitudes BD and CE. Let M be the mid point of BC. Choose the wrong sentence:

A. BM = MC

B. ME = MD

C. DM = MB

D. M is not on the perpendicular bisector of DE

**Verse 2: **Let ΔABC have AC > AB. On the side AC take a point E such that CE = AB. The perpendicular bisectors of BE and AC intersect at O. Choose the correct answer

A. ABO = COE

B. BOA = COE

C. AOB = COE

D. ABO = CEO

**Question 3:** Let ΔABC have AC > AB. On the side AC take a point E such that CE = AB. The perpendicular bisectors of BE and AC intersect at O. Choose the correct answer

A. AO is the median of triangle ABC

B. AO is the perpendicular bisector of triangle ABC

C. AO BC

D. AO is the bisector of angle A

**Question 4:** Let ΔABC where ∠A = 100°. The perpendicular bisectors of AB and AC intersect side BC at E and F respectively. Calculate ∠EAF .

A. 20°

B. 30°

C. 40°

D. 50°

**Question 5: **Given ΔABC is square at A, draw altitude AH. On the side AC take a point K such that AK = AH. Draw KD ⊥ AC (D ∈ BC). Select the correct answers

A. AHD = AKD

B. AD is the perpendicular bisector of the line segment HK

C. AD is the bisector of angle HAK

D. Both A, B, C are correct

## 4. Conclusion

Through this lesson, you should know the following:

- Define the perpendicular bisector of a triangle.
- Understand the property of the three perpendicular bisectors of a triangle.
- Apply to solve related problems.

.

=============