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LESSON 10–7

LESSON 10–7. Special Segments in a Circle. 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

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LESSON 10–7

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  1. 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. A. 70 B. 75 C. 80 D. 85 5-Minute Check 1

  4. Find x. Assume that any segment that appears to be tangent is tangent. A. 110 B. 115 C. 125 D. 130 5-Minute Check 2

  5. Find x. Assume that any segment that appears to be tangent is tangent. A. 100 B. 110 C. 115 D. 120 5-Minute Check 3

  6. Find x. Assume that any segment that appears to be tangent is tangent. A. 40 B. 38 C. 35 D. 31 5-Minute Check 4

  7. What is the measure of XYZ if is tangent to the circle? A. 55 B. 110 C. 125 D. 250 5-Minute Check 5

  8. 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. 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. chord segment • secant segment • external secant segment • tangent segment Vocabulary

  11. Concept

  12. Use the Intersection of Two Chords A. Find x. AE• EC=BE• EDTheorem 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. Use the Intersection of Two Chords B. Find x. PT• TR=QT• TSTheorem 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. 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. Formulate Draw a model using a circle. Let xrepresent 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. 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. Example 2

  19. Find Measures of Segments in Circles Determine The measure of EB = 2.00 – 0.25 or 1.75 mm. HB● BF = EB ● BG Segment products x ● x = 1.75 ● 0.25 Substitution x2 = 0.4375 Simplify. x ≈ 0.66 Take the square root of each side. Answer: The length of the organism is 0.66 millimeters. Example 2

  20. ? 12≈ (0.75)2 + (0.66)2 ? 1≈ 0.56 + 0.44 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.  1≈ 1 Example 2

  21. 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. 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. 28.125 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 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. A. 22.36 B. 25 C. 28 D. 30 Example 4

  30. LESSON 10–7 Special Segments in a Circle

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