1. 1. What measure is needed to find the circumference
1. What measure is needed to find the circumference or area of a circle?
ANSWER
radius or diameter
2. Find the radius of a circle with diameter 8 centimeters.
ANSWER
4 cm
3. A right triangle has legs with lengths 5 inches and 12 inches. Find the length of the hypotenuse.
ANSWER
13 in.
4. Solve 6x + 15 = 33.
ANSWER 3
5. Solve (x + 18)2 = x
ANSWER 7

2. Use properties of segments that intersect circles.
Target
Use properties of segments that intersect circles.
You will…
Use properties of a tangent to a circle.

3. Vocabulary
circle = set of all points equidistant from a given point
chord
radius
center
tangent
diameter
point of tangency
secant

4. Vocabulary
circle – the set of all points in a plane equidistant from a given point called the center of the circle
radius – a segment whose endpoints are the center and any point on the circle
chord – a segment whose endpoints are on the circle
diameter – a chord that contains the center
secant – a line that intersects a circle in two points
tangent – a line, segment, or ray in the plane of a circle that intersects the circle in exactly one point called the point of tangency

5. Vocabulary
circle = set of all points equidistant from a given point
chord
radius
center
tangent
diameter
point of tangency
secant

6. Vocabulary
Properties of Tangents
Theorem 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 (point of tangency) Q T P
Theorem 10.2 – Tangent segments from a common external point are congruent R S

7. EXAMPLE 1
Identify special segments and lines
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. AC a.
SOLUTION
is a radius because C is the center and A is a point on the circle. AC a.

8. EXAMPLE 1
Identify special segments and lines
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. b. AB
SOLUTION b. AB
is a diameter because it is a chord that contains the center C.

9. EXAMPLE 1
Identify special segments and lines
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. DE c.
SOLUTION c. DE
is a tangent ray because it is contained in a line that intersects the circle at only one point.

10. EXAMPLE 1
Identify special segments and lines
Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. AE d.
SOLUTION d. AE
is a secant because it is a line that intersects the circle in two points.

11. GUIDED PRACTICE
for Example 1
1. In Example 1, what word best describes
AG ? CB ?
SOLUTION
Is a chord because it is a segment whose endpoints are on the circle. AG CB
is a radius because C is the center and B is a point on the circle.

12. GUIDED PRACTICE
for Example 1
2. In Example 1, name a tangent and a tangent segment.
SOLUTION
A tangent is DE
A tangent segment is DB

13. EXAMPLE 2
Find lengths in circles in a coordinate plane
Use the diagram to find the given lengths. a.
Radius of A b.
Diameter of A
Radius of B c.
Diameter of B d.
SOLUTION a.
The radius of A is 3 units. b.
The diameter of A is 6 units. c.
The radius of B is 2 units. d.
The diameter of B is 4 units.

14. GUIDED PRACTICE
for Example 2
3. Use the diagram to find the radius and diameter of C and D.
SOLUTION a.
The radius of C is 3 units. b.
The diameter of C is 6 units. c.
The radius of D is 2 units. d.
The diameter of D is 4 units.

15. EXAMPLE 3
Draw common tangents
Tell how many common tangents the circles have and draw them. a. b. c.
SOLUTION a. 4 3 b. c. 2

16. GUIDED PRACTICE
for Example 3
Tell how many common tangents the circles have and draw them. 4.
SOLUTION
2 common tangents

17. GUIDED PRACTICE
for Example 3
Tell how many common tangents the circles have and draw them. 5.
SOLUTION
1 common tangent

18. GUIDED PRACTICE
for Example 3
Tell how many common tangents the circles have and draw them. 6.
SOLUTION
No common tangents

19. EXAMPLE 4
Verify a tangent to a circle
In the diagram, PT is a radius of P. Is ST tangent to P ?
SOLUTION
Use the Converse of the Pythagorean Theorem. Because = 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.

20. Find the radius of a circle
EXAMPLE 5
Find the radius of a circle
In the diagram, B is a point of tangency. Find the radius r of C.
SOLUTION
You know from Theorem 10.1 that AB BC , so ABC is a right triangle. You can use the Pythagorean Theorem.
AC2 = BC2 + AB2
Pythagorean Theorem
(r + 50)2 = r
Substitute.
r r = r
Multiply.
100r = 3900
Subtract from each side.
r = 39 ft .
Divide each side by 100.

21. Find the radius of a circle
EXAMPLE 6
Find the radius of a circle
RS is tangent to C at S and RT is tangent to C at T. Find the value of x.
SOLUTION
RS = RT
Tangent segments from the same point are
28 = 3x + 4
Substitute.
8 = x
Solve for x.

22. GUIDED PRACTICE
for Examples 4, 5 and 6
Is DE tangent to C?
Explain your reasoning.
ANSWER
Since the radius of the circle is 3 units, CE = = 5. Using the converse of Pythagorean’s Theorem, since CD2 + DE2 = CE2 , a Pythagorean Triple, angle D is right.
Therefore DE CD and tangent to C.

23. GUIDED PRACTICE
for Examples 4, 5 and 6
8. ST is tangent to Q. Find the value of r.
9. Find the value(s) of x.
+3 = x
ANSWER
ANSWER
r = 7