1. Special Segments in a Circle
LESSON 10–7
Special Segments in a Circle

2. Five-Minute Check (over Lesson 10–6) TEKaS Then/Now New Vocabulary
Theorem 10.15: Segments of Chords Theorem
Example 1: Use the Intersection of Two Chords
Example 2: Real-World Example: Find Measures of Segments in Circles
Theorem 10.16: Secant Segments Theorem
Example 3: Use the Intersection of Two Secants
Theorem 10.17
Example 4: Use the Intersection of a Secant and a Tangent
Lesson Menu

3. Find x. Assume that any segment that appears to be tangent is tangent.
C. 80
D. 85
5-Minute Check 1

4. Find x. Assume that any segment that appears to be tangent is tangent.
C. 125
D. 130
5-Minute Check 2

5. Find x. Assume that any segment that appears to be tangent is tangent.
C. 115
D. 120
5-Minute Check 3

6. Find x. Assume that any segment that appears to be tangent is tangent.
C. 35
D. 31
5-Minute Check 4

7. What is the measure of XYZ if is tangent to the circle?
B. 110
C. 125
D. 250
5-Minute Check 5

8. Mathematical Processes G.1(B), G.1(E)
Targeted TEKS
G.12(A) Apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve non-contextual problems.
Mathematical Processes
G.1(B), G.1(E)
TEKS

9. Find measures of segments that intersect in the interior of a circle.
You found measures of diagonals that intersect in the interior of a parallelogram.
Find measures of segments that intersect in the interior of a circle.
Find measures of segments that intersect in the exterior of a circle.
Then/Now

10. external secant segment tangent segment
chord segment
secant segment
external secant segment
tangent segment
Vocabulary

11. Concept

12. A. Find x. AE • EC = BE • ED Theorem 10.15 x • 8 = 9 • 12 Substitution
Use the Intersection of Two Chords
A. Find x.
AE • EC = BE • ED Theorem 10.15
x • 8 = 9 • 12 Substitution
8x = 108 Multiply.
x = 13.5 Divide each side by 8.
Answer: x = 13.5
Example 1

13. x • (x + 10) = (x + 2) • (x + 4) Substitution
Use the Intersection of Two Chords
B. Find x.
PT • TR = QT • TS Theorem 10.15
x • (x + 10) = (x + 2) • (x + 4) Substitution
x2 + 10x = x2 + 6x + 8 Multiply.
10x = 6x + 8 Subtract x2 from each side.
Example 1

14. 4x = 8 Subtract 6x from each side. x = 2 Divide each side by 4.
Use the Intersection of Two Chords
4x = 8 Subtract 6x from each side.
x = 2 Divide each side by 4.
Answer: x = 2
Example 1

15. A. Find x.
A. 12
B. 14
C. 16
D. 18
Example 1

16. B. Find x.
A. 2
B. 4
C. 6
D. 8
Example 1

17. Find Measures of Segments in Circles
BIOLOGY Biologists often examine organisms under microscopes. The circle represents the field of view under the microscope with a diameter of 2 mm. Determine the length of the organism if it is located 0.25 mm from the bottom of the field of view. Round to the nearest hundredth.
Example 2

18. Find Measures of Segments in Circles
Analyze Two cords of a circle are shown. You know that the diameter is 2 mm and that the organism is 0.25 mm from the bottom.
Formulate Draw a model using a circle. Let x represent the unknown measure of the equal lengths of the chord which is the length of the organism. Use the products of the lengths of the intersecting chords to find the length of the organism.
Example 2

19. Determine The measure of EB = 2.00 – 0.25 or 1.75 mm.
Find Measures of Segments in Circles
Determine The measure of EB = 2.00 – 0.25 or mm.
HB ● BF = EB ● BG Segment products
x ● x = 1.75 ● 0.25 Substitution
x2 = Simplify.
x ≈ 0.66 Take the square root of each side.
Answer: The length of the organism is millimeters.
Example 2

20. Find Measures of Segments in Circles
Justify Use the Pythagorean Theorem to check the triangle in the circle formed by the radius, the chord, and part of the diameter.
12 ≈ (0.75)2 + (0.66)2 ?
1 ≈ ?
1 ≈ 1 
Example 2

21. label the unknown measure.
Find Measures of Segments in Circles
Evaluate When solving word problems involving circles, it is helpful to make a drawing and label all parts of the circle that are known. Use a variable to
label the unknown measure.
Example 2

22. ARCHITECTURE Phil is installing a new window in an addition for a client’s home. The window is a rectangle with an arched top called an eyebrow. The diagram below shows the dimensions of the window. What is the radius of the circle containing the arc if the eyebrow portion of the window is not a semicircle?
A. 10 ft
B. 20 ft
C. 36 ft
D. 18 ft
Example 2

23. Concept

24. Use the Intersection of Two Secants
Find x.
Example 3

25. Distributive Property
Use the Intersection of Two Secants
Theorem 10.16
Substitution
Distributive Property
Subtract 64 from each side.
Divide each side by 8.
Answer: 34.5
Example 3

26. Find x. A
B. 50
C. 26
D. 28
Example 3

27. Concept

28. LM is tangent to the circle. Find x. Round to the nearest tenth.
Use the Intersection of a Secant and a Tangent
LM is tangent to the circle. Find x. Round to the nearest tenth.
LM2 = LK ● LJ
122 = x(x + x + 2)
144 = 2x2 + 2x
72 = x2 + x
0 = x2 + x – 72
0 = (x – 8)(x + 9)
x = 8 or x = –9
Answer: Since lengths cannot be negative, the value of x is 8.
Example 4

29. Find x. Assume that segments that appear to be tangent are tangent.
C. 28
D. 30
Example 4

30. Special Segments in a Circle
LESSON 10–7
Special Segments in a Circle