6.4 Some Constructions and Inequalities for the Circle. Constructing a tangent to a circle Theorem 6.4.1: The line that is perpendicular to the radius of a circle at its endpoint on the circle is a tangent to the circle. Strategy: Draw a radius and extend it beyond the circle. At the point on the circle where the radius touches the circle, construct a line perpendicular to the radius. 2/23/2019 Section 6.4 Nack

Constructing a Tangent to a Circle Draw radius PX and extend it to form PX. 2. Using X as the center, construct a line perpendicular to PX.  3. That line, XW is the tangent to Circle P. X P W ( X ) P 2/23/2019 Section 6.4 Nack

Construct a tangent to a circle from an external point Given Q and external point E Construct: Tangent ET with T as the point of tangency. Draw EQ. Construct the bisector EQ to intersect EQ at its midpoint M. With ME as the length of the radius, construct a circle. Draw ET, the desired tangent. Note: EV would also be a tangent. E M Q  T E M Q  V 2/23/2019 Section 6.4 Nack

Inequalities in the Circle Theorem 6.4.2: In a circle ( or in congruent circles) containing two unequal central angles, the larger angle corresponds to the larger intercepted arc. Proof p. 307 Theorem 6.4.3: In a circle ( or in congruent circles) containing two unequal arcs, the larger arc corresponds to the larger central angle. Theorem 6.4.4: In a circle ( or in congruent circles) containing two unequal chords, the shorter chord is at a greater distance from the center of the circle. Fig. 6.59 Theorem 6.4.5 : In a circle ( or in congruent circles) containing two unequal chords, the chord nearer the center of the circle has the greater length. Fig. 6.60 2/23/2019 Section 6.4 Nack

Theorems on Minor Arcs Theorem 6.4.6: In a circle ( or in congruent circles) containing two unequal chords, the longer chord corresponds to the greater minor arc. Theorem 6.4.7: In a circle ( or in congruent circles) containing two unequal chords, the greater minor arc corresponds to the longer of the chords related to these arcs. Fig. 6.61 (four cases) 2/23/2019 Section 6.4 Nack