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Circles: Chords and Segments, Geometry Concepts Explained

Explore chord intersections, segment measurements, and circle theorems in geometry. Learn how to calculate lengths and angles in circle diagrams with clear examples.

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Circles: Chords and Segments, Geometry Concepts Explained

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  1. A B C D Find x. Assume that any segment that appears to be tangent is tangent. 70 5-Minute Check 1

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

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

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

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

  6. 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

  7. Concept

  8. 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

  9. 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

  10. 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

  11. A B C D A. Find x. 16 Example 1

  12. A B C D B. Find x. 4 Example 1

  13. 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

  14. Plan 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 Understand 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

  15. Find Measures of Segments in Circles Solve The measure of ED = 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

  16. ? 12≈ (0.75)2 + (0.66)2 ? 1≈ 0.56 + 0.44 Find Measures of Segments in Circles Check 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

  17. A B C D 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? 10 ft Example 2

  18. Concept

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

  20. 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

  21. A B C D Find x. 26 Example 3

  22. Concept

  23. 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

  24. A B C D Find x. Assume that segments that appear to be tangent are tangent. 30 Example 4

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