## Math 9 Chapter 2 Lesson 2: Diameter and chord of a circle

## 1. Summary of theory

### 1.1. Compare the length of the diameter

**Theorem 1: **Of the strings of a circle, the largest wire is the diameter

### 1.2. Perpendicular relationship between diameter and string

**Theorem 2**

In a circle, a diameter perpendicular to a chord passes through the midpoint of that chord.

In a circle, the diameter passing through the midpoint of a string but not passing through the center is perpendicular to the chord.

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1: **Given figure 67. Calculate the length of string AB, knowing OA = 13 cm, AM = MB, OM = 5 cm.

**Solution guide**

Consider (O) where OM is the part of the diameter passing through the midpoint M of the string AB

\( \Rightarrow OM \bot AB\) (theorem)

Consider triangle OAM, right angled at M, with:

\(\eqalign{& O{A^2} = A{M^2} + O{M^2} \cr & \Rightarrow AM = \sqrt {O{A^2} – O{M^2}} = \sqrt {{{13}^2} – {5^2}} = 12 \cr & \Rightarrow AB = 2AM = 24\,\,\left( {cm} \right) \cr} \)

**Verse 2:** Given a circle with center O and diameter AB, the chord CD does not intersect AB. Let H and K be the perpendicular projections of A and B onto CD, respectively. CM: CH=DK

**Solution guide**

Construct OE perpendicular to CD (E belongs to CD) according to theorem 2, then E is the midpoint of CD. (first)

Consider trapezoid ABKH with O as midpoint AB and \(OE\parallel AH\parallel BK\) so E is mid point HK. (2)

From (1) and (2), we have CH=DK

### 2.2. Advanced exercises

**Question 1: **Let (O;R) diameter AB, H is the midpoint of OB. Draw the string CD perpendicular to AB to H, K is the midpoint of AC and I is the symmetrical midpoint of A through H

a) CMR: 4 points C, H, O, K on the same circle

b) CM ADIC is a rhombus. Calculate area in

__Solution guide__

a) Draw OK, because K is the midpoint of AC, so OK is perpendicular to AC, then 4 points K, O, H, C will belong to the circle with diameter OC.

b) Consider quadrilateral ADIC with 2 diagonals intersecting at the midpoint of each line, so ADIC is a parallelogram.

Consider triangle ADC with AH being the altitude and the median ( OH is perpendicular to CD, then it passes through the midpoint of CD), so triangle ACD is isosceles at A, so AC = AD

Then ADIC is a rhombus.

\(S_{ADIC}=S_{\Delta ADC}+S_{\Delta DIC}=2.S_{\Delta ADC}=AH.CD\)

Which \(AH=\frac{3R}{2}\) ; \(CD=2.CH=2.\sqrt{OC^2-OH^2}=2\sqrt{R^2-\frac{R^2}{4}}=R\sqrt{3}\)

\(\Rightarrow S_{ADIC}=\frac{3R}{2}.R\sqrt{3}=\frac{3R^2\sqrt{3}}{2}\)

**Verse 2: **Let ABC be an acute triangle (AB

a) Prove that quadrilateral BHCD is a parallelogram

b) Prove \(OM=\frac{1}{2}.AH\)

**Solution guide**

a) Triangle ABD has OA=OB=OD with O being the midpoint of AD so ABD is square at B \(\Rightarrow BD\perp AB\Rightarrow BD\parallel CH\)

same for triangle ADC right angled at C \(\Rightarrow CD\perp AC\Rightarrow BH\parallel CD\)

Quadrilateral BHCD has pairs of opposite sides parallel, so BHCD is a parallelogram

b) we have OM perpendicular to BC so M is the midpoint of BC. Since BHCD is a parallelogram, the diagonal HD passing through the midpoint of BC is M .

Consider triangle AHD where O is the midpoint of AD, M is the midpoint of HD, so OM is the median of triangle AHD \(\Rightarrow OM=\frac{1}{2}.AH\)

## 3. Practice

**Question 1: **Given triangle ABC, altitudes BH and CK. Prove:

a. Four points B, C, H, K are on the same circle

b. HK < BC

**Verse 2:** Quadrilateral ABCD has

a. Prove that the four points A, B, C, D are on the same circle

b. Compare the lengths AC and BD. If AC = BD then what is the quadrilateral ABCD?

**Question 3: **Given a semicircle with center O, diameter AB, and a string EF that does not intersect the diameter. Let I and K be the feet of the perpendiculars drawn from A and B to EF, respectively. Prove that IE = KF.

**Question 4: **Given a circle (O) with radius OA = 3cm. The chord BC of the circle is perpendicular to OA at the midpoint of OA. Calculate the length BC.

### 3.2. Multiple choice exercises

**Question 1: **Given a circle (O;R) of diameter AB. M is a point between A and B. Through M draw a chord CD perpendicular to AB. Knowing AM=4, R=6.5. What is the value of the area of triangle BCD?

A. 50

B. 52

C. 54

D. 56

**Verse 2: **Given a circle (O;R) and a string CD. From O draw a ray perpendicular to CD at M, cut (O) at H. Knowing CD=16, MH=4. R=?

A. 8

B. 9

C. 10

D. 11

**Question 3: **Let the circle (O;R) and the two chords AB and CD are equal and perpendicular to each other at I. Assume IA=2, IB=4. Distance from center O to AB is d and to CD is d’

Values of d and d’

A. \(d=2;d’=1\)

B. \(d=d’=1\)

C. \(d=d’=2\)

D. \(d=1;d’=2\)

**Question 4: **Given a circle (O;12) of diameter CD. The series MN passes through the midpoint I of OC such that the angle NID is 30 degrees. MN=?

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

B. \(2\sqrt{3}\)

C. \(3\)

D. \(2\)

**Question 5: **Given (O;25), string AB=40. Draw string CD parallel to AB and whose distance to AB is 22. What is the length of cord CD?

A. 42

B. 44

C. 46

D. 48

## 4. Conclusion

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

- Knowing the diameter is the largest of the strings of the circle, grasping the two theorems about the diameter perpendicular to the string and the diameter passing through the midpoint of the string not passing through the center.
- Know how to apply theorems to prove that the diameter passing through the midpoint of a string, the diameter is perpendicular to the string.

.

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

## Leave a Reply