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Understanding Tangent, Lines, and Circles in Geometry

Learn about tangent and secant lines with examples of finding angles, segment lengths, and rules involving circles and line intersections.

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Understanding Tangent, Lines, and Circles in Geometry

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  1. Tangent Lines and Secant Lines Feb. 8, 2019

  2. If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent.

  3. Example Finding Angle Measures • Segment MN and Segment ML are tangent to Circle O. Find the value of X • Since the segments are tangent, Angle L and Angle N are right angles. • LMNO is a quadrilateral whose interior angle measures is 3600 • Therefore, 360 – 90 – 90 – 117 = X0 • X = 630 O N L 1170 X0 M

  4. Circumscribed and Inscribed Circles • Previously, we learned that a circle is circumscribed about a triangle if all the vertices (corners) of the triangle lie on points of the circle. In this case, the triangle is inscribed in the circle. • Similarly, when a circle is inscribed inside a triangle, then we can say the triangle is circumscribed about the circle. Each side of the triangle would then be considered tangent to the circle.

  5. The two segments tangent to a circle from one point outside the circle are congruent. Or: • AD = AF • BD = BE • CE = CF B D E A C F

  6. Find the Perimeter B 8 cm D E • Therefore, the perimeter of triangle ABC = 2(10) + 2(15) + 2(8) = 66 cm O C A 15 cm F 10 cm

  7. Find a Segment Length B • Perimeter (88) = 2(17) + 2(15) + 2(BE) • 2(BE) = 88 – 34 – 30 • 2(BE) = 24 • BE = 12 cm D E 17 cm A C F 15 cm

  8. Example: Given: Each side of quadrilateral ABCD is a tangent to the circle. AB = 10, BC = 15, AD = 18. Find CD. 15 - x C x 15 - x B x E 18 – (10 – x) 10 - x A D 10 - x 18 – (10 – x) Let BE = x and “walk around” the figure using the given information and the Two-Tangent Theorem. CD = 15 – x + 18 – (10 – x) = 15 – x + 18 – 10 + x = 23

  9. Finding Segment Lengths

  10. Definition • Tangent – a line in the plane of a circle that intersects the circle in exactly one point.

  11. Definition • Secant – a line that intersects a circle in two points.

  12. Definition • Chord – a segment whose endpoints are points on the circle.

  13. Chord-Chord Rule: Two chords intersect INSIDE the circle a ab = cd d c b

  14. Example 1: 9 12 6 3 x x 2 2 X = 3 X = 8 x 3 6 2 X = 1

  15. 12 2x 8 3x Example 2: Find x 2x  3x = 12  8 6x2 = 96 x2 = 16 x = 4

  16. Secant-Secant Rule: OW-OW

  17. Two secants intersect OUTSIDE the circle Secant-Secant Rule: E A B C D EA•EB = EC•ED

  18. Example 3: B 13 A 7 E 4 C x D 7 (7 + 13) = 4 (4 + x) x = 31 140 = 16 + 4x 124 = 4x

  19. Example 4: B x A 5 D 8 6 C E 6 (6 + 8) = 5 (5 + x) x = 11.8 84 = 25 + 5x 59 = 5x

  20. Notice that on the tangent segment, the outside is the whole! Secant Segment External Segment Tangent Segment

  21. Secant-Tangent Rule: A secant and tangent originate from the same point OUTSIDE the circle.

  22. A secant and tangent originate from the same point OUTSIDE the circle Secant-Tangent Rule: C B E A EA2= EB • EC

  23. Example 5: C B x 12 E 24 A (12 + x) 242 = 12 x = 36 576 = 144 + 12x

  24. Example 6: 5 B E 15 C x A (5 + 15) x2 = 5 x = 10 x2 = 100

  25. What you should know by now… Given two chords Given two secants ORa tangent and a secant

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