## Math 9 Chapter 2 Lesson 5: Signs to recognize the tangent of a circle

## 1. Summary of theory

### 1.1. Signs to identify the tangent of a circle

**Theorem**

If a line passes through a point of a circle and is perpendicular to the radius passing through that point, then the line is a tangent to the circle.

### 1.2. Apply

**Problem: **Through the point A outside the circle (O), construct a tangent to the circle

**How to build:**

Let M be the midpoint of AO. Construct a circle with center M and radius MO intersecting (O) at B, C. Draw AB, AC as tangents to (O)

## 2. Illustrated exercise

### 2.1. Basic exercises

**Question 1: **Let M and (O). Draw a tangent to (O) passing through M in the cases

a) M lies outside the circle

b) M lies on the circle

**Solution guide**

a) Make K the midpoint of OM. Then draw a circle with center K and radius KM. (K;KM) intersects (O) at A, B. Then MA, MB are tangent to the circle

b) Connect the radius OM. Draw a line d perpendicular to OM at M. d is tangent to (O).

**Verse 2: **Let (O;12) M distance from O 20. Draw tangent MA (A is the point of contact)

1) Calculate MA

2) Draw string AB perpendicular to OM. Prove that MB is tangent

**Solution guide**

1) Apply the Pythagorean Theorem: \(MA=\sqrt{MO^2-OA^2}=\sqrt{20^2-12^2}=16\)

2) Let H be the intersection of AB with OM

Consider 2 triangles OAH and OBH are 2 right triangles at H; OA=OB=R; General OH should \(\Delta OAH=\Delta OBH\Rightarrow HA=HB\)

Triangle MAB has both altitude and median MH, so MAB is isosceles at M \(\Rightarrow \widehat{MAH}=\widehat{MBH}\)

we have an isosceles triangle OAB, so: \(\widehat{OAB}=\widehat{OBA}\). Then: \(\widehat{MBO}=\widehat{MBH}+\widehat{OBA}=\widehat{MAH}+\widehat{OAB}=90^{\circ}\)

So MB is tangent

**Question 3: **Given a circle (O) with diameter AB. C is a point on the circle such that \(\widehat{CAB}=30^{\circ}\). M is the point of symmetry with O through B. Prove that MC is a tangent to (O)

**Solution guide**

Triangle ABC is right angled at C. \(\widehat{CAB}=30^{\circ}\Rightarrow \widehat{CBA}=60^{\circ}\) where \(CO=OB\) should be equilateral triangle COB deduce CB=OB

Triangle COM has medians CB and CB=OB=BM, so triangle COM is right-angled at C, so MC is tangent to (O)

### 2.2. Advanced exercises

**Question 1:** Let ABC be a right triangle at A, altitude AH. Let M, N be the points of symmetry with H through AB and AC, respectively. E,D are projections of H onto AB, AC

Prove that: MN is a tangent to the circle with diameter BC .

**Solution guide**

We have: \(\widehat{BMA}=\widehat{BME}+\widehat{AME}=\widehat{BHE}+\widehat{AHE}=90^{\circ}\). Similarly \(\widehat{ANC}=90^o\)

\(\widehat{MAN}=\widehat{MAB}+\widehat{BAC}+\widehat{CAN}=2.\widehat{BAC}=180^o\Rightarrow\) M, A, N align

Let K be the midpoint of BC. Considering the quadrilateral MBCN has \(MB\parallel CN\) so MBCN is a trapezoid.

KA is the median of the trapezoid, so \(KA\perp MN\) is at A. So MN is tangent to (K;KA) (circle with diameter BC)

**Verse 2: **Let ABC be an acute triangle with CE and BD the altitudes. H is the intersection of CE and BD.

a) Prove that A,E,H,D are on the same circle set as (O)

b) Let M be the midpoint of BC. Prove that ME, MD are tangents to (O)

**Solution guide**

a) The triangles AEH and ADH are right triangles at E and D respectively, with AH being the common hypotenuse. Let O be the midpoint of AH then

(O;OA) will pass through the points A, E, H,

b) Considering triangle AOE with OA=OE, triangle AOE is isosceles at O, deduce \(\widehat{OEA}=\widehat{OAE}\) (1)

Let F be the intersection point AH with BC. Since H is orthocenter, \(AF\perp BC\) is at F.

Again we have: \(\widehat{OAE}=\widehat{MCE}\) (because it is the same as \(\widehat{MBE}\)). where \(\widehat{MCE}=\widehat{MEC}\) (2)

From (1) and (2) deduce: \(\widehat{MEC}=\widehat{OEA}\)so: \(\widehat{MEO}=\widehat{MEC}+\widehat{CEO}=\widehat {OEA}+\widehat{CEO}=90^{\circ}\). So ME is tangent

Same for MD

## 3. Practice

### 3.1. Essay exercises

**Question 1: **Given a circle (O), point A lies outside the circle. Using a ruler and compass, construct points B and C on the circle (O) such that AB and AC are tangents to the circle (O).

**Verse 2:** Let point A lie on line d and point B lie on line d. Construct a circle (O) passing through A and B, taking the line d as a tangent

**Question 3: **Let ABC be a right triangle at A. Draw a circle (B; BA) and a circle (C; CA), they intersect at point D (other than A). Prove that CD is a tangent to the circle (B).

Question 4: Given triangle ABC is isosceles at A, altitudes AD and BE intersect at H. Draw a circle (O) with diameter AH. Prove that:

a. Point E lies on the circle (O).

b. DE is the tangent to the circle (O).

### 3.2. Multiple choice exercises

**Question 1: **Which of the following statements is correct?

A. The line d is said to be tangent to (O) when they have a common point

B. The line d is said to be tangent when d is perpendicular to the radii OA and OA

C. The line d is said to be tangent to (O) when d is perpendicular to the radius OA and A belongs to the circle.

D. The line d is said to be tangent to (O) when d is perpendicular to OA at A and OA>R

**Verse 2: **Given a circle (O). A, B, C are 3 points on the circle such that triangle ABC is isosceles at A. Which of the following statements is true?

The tangent to the circle at A is

A. Pass through A and perpendicular to AB

B. Pass through A and parallel AC

C. Passes through A and parallels BC

D. Pass through A and perpendicular to BC

**Question 3:** Let ABC be a triangle with AH being the altitude (H belongs to BC). How will the circle (A;AH) be located with the sides of triangle ABC

A. (A;AH) tangent to AB,AC and intersects BC

B. (A;AH) touches BC, AC and does not cut AB

C. (A;AH) cuts AB, AC and touches BC

D. (A;AH) cuts AB and touches BC, AC

**Question 4: **Given a circle (O;15), the chord AB does not pass through the center. Through O draw a line perpendicular to AB that intersects the tangent at A at C and cuts AB at D. Know AB=24. The OC length is:

A. 12

B. 19

C. 25

D. 32

**Question 5:** Let ABC be a triangle with AB=3, AC=4, BC=5. Then:

A. AB is a tangent to (C;3)

B. AC is tangent to (B;3)

C. AB is a tangent to (B;4)

D. AC is tangent to (C;4)

## 4. Conclusion

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

- Understand the telltale signs of a tangent to a circle.
- Draw a tangent at a point of the circle, draw a tangent passing through a point outside the circle.
- Apply theory to solve related problems.

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