Determining Lengths of Segments Intersecting Circles Lesson 101 Determining Lengths of Segments Intersecting Circles

New Vocabulary A segment of a secant with at least one endpoint on the circle is a secant segment. 𝐴𝐶 An external secant segment is a secant segment that lies in the exterior of a circle with one endpoint on the circle. 𝐴𝐵 A tangent segment is a segment of a tangent with one endpoint on the circle. 𝐴𝐷

Theorem 101-1 If two secant segments are drawn to a circle from an external point, the product of the length of one secant segment and that of its external segment is equal to the product of the other secant segment length and that of its external segment. In the diagram, 𝐴𝐶 𝐴𝐵 = 𝐴𝐸 𝐴𝐷

Theorem 101-2 If one secant segment and one tangent are drawn to a circle from an external point, the product of the length of one secant segment and that of its external segment is equal to the length of the tangent segment squared. In the diagram, 𝐴𝐶 𝐴𝐵 = 𝐴𝐷 2

Find x 𝑥 2 = 4+12 4 𝑥 2 = 16 4 𝑥 2 =64 𝑥=8 𝑐𝑚

Find x 2+𝑥 2 = 3+7 3 4+2𝑥=30 2𝑥=26 𝑥=13

Find x 𝑥+11 𝑥= 2+4 2 𝑥 2 +11𝑥=12 𝑥 2 +11𝑥−12=0 𝑥+12 𝑥−1 =0 𝑥+12=0 & 𝑥−1=0 𝑥=−12 & 1 𝑥 can’t be negative, lengths are always positive 𝑥=1

If the earth’s diameter is 7920 miles, find the distance to the horizon for an unobstructed view from the top of a 200-foot building. Round answer to the nearest tenth of a mile. What do you notice about the units in this problem? Change 200 𝑓𝑡 to 200 5280 𝑚𝑖𝑙𝑒𝑠 . 𝑥 2 = 7920+ 5 132 5 132 𝑥= 7920+ 5 132 5 132 𝑥≈17.3 miles

Looking Forward Finding lengths of segments intersecting circles will prepare you for: Lesson 104: Relating Arc Lengths and Chords Lesson 106: Circumscribed and Inscribed Figures Lesson 112: Finding Areas of Circle Segments