10-5 Tangents A tangent is a line in the same place as a circle that intersects that circle in exactly one point. That point is called the point of tangency. In the circle to the right, what would be the point of tangency? A common tangent is a line, ray or segment that is tangent to two circles in the same place. In each figure below, line l is a common tangent of circle F and G. A B C F G l l F G

The shortest distance from a tangent to the center of the circle is the radius drawn to the point of tangency. Theorem In a plane, a line is tangent to a circle IFF it is perpendicular to a radius drawn to the point of tangency. Example: Line l is tangent to circle S IFF l ST. Τ l S Example 2: Identify a Tangent JL is a radius of circle J. Determine whether KL is tangent to circle J. Justify your answer J L K

Example 3: Use a tangent to find missing values JH is tangent to circle G at J. Find the value of x. x 8 12 x G H J

More than one line can be tangent to a circle. Theorem If two segments from the same exterior point are tangent to a circle, then they are congruent. Example: If AB and CB are tangent to circle D, then AB ≅ CB. B A C D Example 4: Use congruent tangents to find measures AB and CB are tangent to circle D. Find the value of x. B A C D x+15 2x-5

Circumscribed Polygons A polygon is circumscribed about a circle if every side of the polygon is tangent to the circle. Polygons CircumscribedPolygons NOT Circumscribed.

Real-World Example 5: Find measures in circumscribed polygons. A B F 10ft C D E 7ft 8ft G A graphic designer is giving directions to create a larger version of the triangular logo shown. If ΔABC is circumscribed about circle G, find the perimeter of ΔABC.

10-6 Secants, Tangents, and Angle Measures Li nes j and k are secants of circle c. A secant is a line that intersects a circle in exactly 2 points. Theorem If 2 secants or chords intersect in the interior of a circle, then the measure of an angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle. Example: m<1 = 1/2 (m AB + m CD) and m<2 = 1/2 (m DA + m BC) 1 2 B C D A j k C

Example 1: Use intersecting chords or secants Find x. a. b. c. R S T U 84o 130o V xo A E 75o D 143o B G xo H 110o K J L 97o

Theorem If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one half the measure of its intercepted arc. Example: m<1 = 1/2 m AB and m<2 = 1/2 m ACB Example 2: Use intersecting secants and tangents Find each measure. a. m<QPR b. m DEF B A C 2 1 P P D R S C Q F 64o 148o E

Intersections outside a circle Secants and tangents can also meet outside a circle. The measure of the angle formed also involves half of the measures of arcs they intercept. Theorem If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle then, the measure of the angle formed is one half the difference of the measures of the intercepted arcs. Examples Two SecantsSecant-TangentTwo Tangents m<A = 1/2 (mDE - mBC)m<A = 1/2 (mDC-mBC)m<A = 1/2(mBDC - mBC) D E B C A A D C B B C A D

Example 3: Use tangents and secants that intersect outside a circle Find each measure. a. m< L b. m CD J 102o L K H 95o 56o D C B A

Key Concept: Circle and Angle Relationships Vertex of Angle Model(s)Angle Measure on the circle one half the measure of the intercepts arc m<1=1/2 x inside the circle one half the measure of the sum of the intercepted arc m<1 = 1/2(x+y) outside the circle one half the measure of the difference of the intercepted arcs m<1 = 1/2 (x-y) xo yo xo 1 yo xo 1 yo xo 1 yo xo