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Special Segment with Circles

Explore the construction of tangent lines and the measurement of segments in and around circles. Practice problem-solving skills and discover circle vocabulary and theorems. Examples illustrate the application of formulas and the geometric properties of circles.

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Special Segment with Circles

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  1. Special Segment with Circles Concept 60

  2. Content Standards Reinforcement of G.C.4 Construct a tangent line from a point outside a given circle to the circle. Mathematical Practices 1 Make sense of problems and persevere in solving them. 7 Look for and make use of structure. CCSS

  3. 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

  4. chord segment – two pieces of a chord, when they intersect inside the circle. • Examples: AF, BF, CF, DF B A F D C Vocabulary

  5. Concept

  6. 1. 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

  7. 2. Find x. PT• TR=QT• TS x• (x + 10) = (x + 2) • (x + 4) x2 + 10x = x2 + 4x + 2x + 8 x2 + 10x = x2 + 6x + 8 10x = 6x + 8 4x = 8 x = 2 Answer:x = 2 Example 1

  8. 3. Find x. UE• ES=RE• ET 10• 8 = x • 5 80 = 5x 16 = x Answer:x = 16 Example 1

  9. 4. Find x. GJ• JI=FJ• JH x• (x + 20) = (x + 8) • (x + 4) x2 + 20x = x2 + 4x + 8x + 32 x2 + 20x = x2 + 12x + 32 20x = 12x + 32 8x = 32 x = 4 Example 1

  10. 5. 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. E HB● BF = EB ● BG x ● x = 1.75 ● 0.25 x2 = 0.4375 x ≈ 0.66 1.75 2 mm H x B x F G Example 2

  11. 6. 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? 6 ● 6 = 2 ● x 36= 2x 18 = x 6 6 X represents what? The missing piece of the diameter. x Find the radius. 18 + 2 = 20 20/2 = 10 ft

  12. Special Segment with Circles Concept 61

  13. secant segment – a segment of a secant line that has exactly one endpoint on the far side of the circle and the other is a point of intersection with another line. • Example: AB and AC • external secant segment – a secant segment that lies in the exterior of the circle. • Example: AF and AD B F A D C

  14. tangent segment – a piece of a tangent line that goes from the point of tangency to a point of intersection with another line. • Example A B C

  15. Concept

  16. 7. Find x. EH• EI=EF• EG 8• (8 + x) = 10 • (10 + 24) 64 + 8x = 10(34) 64 + 8x = 340 8x = 276 x = 34.5 Example 3

  17. 8. Find x. MK• MI=MO• KI x• (x + 24) = 25 • (25 + 27) x2 + 24x = 25(52) x2 + 24x = 1300 x2 + 24x -1300 = 0 (x – 26)(x + 50) = 0 So x – 26 = 0 x + 50 = 0 x = 26 x = -50 Example 3

  18. Concept

  19. 9. Find x. Assume that segments that appear to be tangent are tangent. MN2= MO ● MP x2= 20(20 + 25) x2 = 20(45) x2 = 900 x = 30 Example 4

  20. 10. LM is tangent to the circle. Find x. Round to the nearest tenth. LM2= LK ● LJ 122= x(x + x + 2) 144= x(2x + 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

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