Tangents

Key Concepts

• Draw tangents to a circle
• Identify special segments and lines.
• Find lengths in a coordinate plane.
• Draw common tangents.
• Verify a tangent to a circle.
• Find the radius of a circle.
• Tangents from a common external point are congruent.

Tangents

Question: 

How are the lengths of tangent segments related? 

A line can intersect a circle at 0, 1, or 2 points. If a line is in the plane of a circle and intersects the circle at 1 point, the line is a tangent. 

Step 1: 

Draw tangents to a circle 

Draw a circle. Use a compass to draw a circle. Label the center P. 

Draw tangents  

Draw lines and BC so that they intersect  ⊙  P only at A and C, respectively. These lines are called tangents. 

Measure segments 

AB↔  and BC↔ are called tangent segments. Measure and compare the lengths of the tangent segments. 

A circle is the set of all points in a plane that are equidistant from a given point called the center of the circle.  

A circle with center P is called “circle P” and can be written ⊙  P.  

A segment whose endpoints are the center and any point on the circle is a radius. 

A chord is a segment whose endpoints are on a circle.  

A diameter is a chord that contains the center of the circle. 

A secant is a line that intersects a circle in two points.  

A tangent is a line in the plane of a circle that intersects the circle in exactly one point, the point of tangency.  

The tangent ray 

AB and the tangent segment AB are also called tangents. 

Example 1:

Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of ⊙C. 

a. AC

b. AB

c.  DE

d.  AE

Solution  

1. AC is a radius because C is the center and A is a point on the circle.  
2. AB is a diameter because it is a chord that contains the center C.  
3. DE  is a tangent ray because it is contained in a line that intersects the circle at only one point.  
4. AE is a secant because it is a line that intersects the circle in two points. 

Guided Practice 

Example 1: 

1. In Example 1, what word best describes AG and CB ?  
2. In Example 1, name a tangent and a tangent segment. 

Solution: 

1. AG = Chord/Diameter, CB =radius, 
2. DE = Tangent and DE

Read Vocabulary  

The plural of radius is radii. All radii of a circle are congruent. 

Radius and Diameter

The words radius and diameter are used for lengths as well as segments.  

For a given circle, think of a radius and a diameter as segments and the radius and the diameter as lengths. 

Find lengths in circles in a coordinate plane   

Use the diagram to find the given lengths.  

1. Radius of A  
2. Diameter of A  
3. Radius of B  
4. Diameter of B 

Solution: Count squares  

1. The radius of ⨀A is 3 units.  
2. The diameter of ⨀A is 6 units.  
3. The radius of ⨀B is 2 units.  
4. The diameter of ⨀B is 4 units. 

Guided Practice 

Example 2:  

Use the diagram in Example 2 to find the radius and diameter of ⨀C and ⨀D. 

Solution: 

The radius of ⨀C is 3 units. 

The diameter of ⨀C is 6 units 

The radius of ⨀D is 2 units. 

The radius of ⨀D is 4 units. 

Coplanar Circles  

Two circles can intersect in two points, one point, or no points.  

Coplanar circles that intersect in one point are called tangent circles.  

Coplanar circles that have a common center are called concentric circles. 

Common Tangents

A line, ray, or segment that is tangent to two coplanar circles is called a common tangent. 

Draw common tangents  

Guided Practice 

Example 3: 

Tell how many common tangents the circles have and draw them. 

Theorem 10.1 

In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle. 

GIVEN: Line m is tangent to ⨀Q at P. 

 PROVE c: m ⊥ QP

1. Assume m is not perpendicular to QP

Then the perpendicular segment from Q to m intersects m at some other point R.  

Because m is a tangent, R cannot be inside ⨀Q. Compare the length QR to QP. 

1. Because QR is the perpendicular segment from Q to m. QR is the shortest segment from Q to m. Now compare QR to QP.  

2. Use your results from parts (a) and (b) to complete the indirect proof.  

Verify a tangent to a circle   

EXAMPLE 4: 

In the diagram

PT is a radius of ⊙P. Is ST tangent to ⨀P? 

Solution: 

 Use the Converse of the Pythagorean Theorem. 

Because 12² + 35²= 37² △PST is a right triangle and ST ⊥ PT

 So, ST is perpendicular to a radius of ⨀P at its endpoint on ⨀P. 

 By Theorem 10.1,

ST is tangent to ⨀P. 

EXAMPLE 5: 

In the diagram, B is a point of tangency. Find the radius r of ⨀C. 

You know from Theorem 10.1 that

AB ⊥ BC , so △ABC is a right triangle.  

You can use the Pythagorean Theorem.  

AC² =BC ² +AB ²                         (Pythagorean Theorem)  

  (r + 50)² = r ² + 80²                            (Substitute) 

 r ² + 100r + 2500 = r ² + 6400               (Multiply) 

  100r = 3900                                    (Subtract from each side) 

    ∴   r =39 ft                                   (Divide each side by 100)

Theorem 10.2  

Tangent segments from a common external point are congruent. 

Proving Theorem10.2 

Write proof that tangent segments from a common external point are congruent. 

GIVEN: 

SR and ST are tangent to ⊙P. 

PROVE: SR > ST

Plan for Proof  

Use the hypotenuse–leg congruence 

Theorem to show that

△ SRP △STP 

Join P, A; PB and PS 

∠PRS= ∠STP= 900 

Angle between radii and tangents 

Now, the right triangles  

∆PRS and ∆PTS  

PR=PT (radii of the same circle) 

PS=PS (Common) 

Therefore, by R.H.S congruency axiom 

∆PRS ≅ ∆PTS  

This gives SR ≅ ST 

Hence proved. 

Example 6: 

RS is tangent to ⨀C at S and RT is tangent to ⨀C at T. Find the value of x. 

Solution: 

RT = RS                  (Tangent segments from the same point are ≅) 

3x+4 =28               (Substitute) 

3x =28-4 

3x=24 

x=8                        (solve for x) 

Problem solving

GLOBAL POSITIONING SYSTEM (GPS)  

GPS satellites orbit about 11,000 miles above Earth. The mean radius of Earth is about 3959 miles. Because GPS signals cannot travel through Earth, a satellite can transmit signals only as far as points A and C from point B, as shown. Find BA and BC to the nearest mile. 

Solution: 

ED =3959 mi     is the radius of Earth  

DB=1100 mi 

AB=?     BC=?          AB, BC are tangents 

AB=BC (∵ Tangent segments from a common external point are congruent.) 

AE=ED     Radius of Earth 

BE=ED+DB=3959+11000=14959 mi 

AE ⊥ AB The tangent at any point of a circle is perpendicular to the radius through the point of contact. 

△EAB is right angled triangle. By Pythagoras theorem 

AE² +AB²= BE²

AB² = BE²– AE²

= 14959² – 3959²

= (14959+3959) (14959–3959) 

 =11000 x 18918 

AB= √18918 × 11000

=14425.60 mi 

Let’s check your knowledge

1. Copy the diagram. Tell how many common tangents the circles have and draw. 

2. Determine whether AB is to ⊙C. Explain. 

3. Find the values of the variable. 

4. Find the values of x and y. 

Answers

1.  

2. From △CAB 

BC² = AC²+ AB² (∵ Pythagoras theorem) 

5² = 3²+ 4²

25=25 

So△CAB is right angled triangle. 

∠CAB = 900

CA ⊥ AB     

The tangent at any point of a circle is perpendicular to the radius through the point of contact.  

∴ AB is a tangent 

3. Tangent segments from a common external point are congruent. 

So, AD=AB 

7x – 6 =3x+10 

7x-3x=10+6 

4x=16 

∴   x=4  

Step 1: 

Linear pair = 180⁰

(2x+3)⁰ + 137⁰= 180⁰

x⁰ + 140⁰= 180⁰

2x⁰ = 40⁰

x⁰ = 20⁰

(2x+3)⁰ =2 x 20⁰+3= 43⁰

Step 2:  Two lines are parallel to each other, then 

  Alternate exterior angles are equal 

(4y−7)^o = (2x+3)⁰

(4y−7)^o = 43⁰

4y⁰ = 50⁰

y⁰ = (252)⁰

Exercise

1. Find the value of x and y in the following figure

2. When will two lines tangent to the same circle not intersect?
3. Find the value of the variables in the following figures:

4. Determine whether AB tangent is tangent to circle C. Explain.

Concept map

What have we learned

• How to draw tangents to a circle.
• How to Identify special segments and lines.
• How to find lengths in a coordinate plane.
• How to draw common tangents.
• How to verify a tangent to a circle.
• How to find the radius of a circle.
• How tangents from a common external point are congruent