6 - Discovering and Proving Circle Properties 6.1- Tangent Properties Basic terms from Chapter 1 F E C A D B J G I H JRLeon Geometry Chapter 6.1 – 6.2 HGSH

JRLeon Geometry Chapter 6.1 – 6.2 HGSH 6.1- Tangent Properties In this lesson you will investigate the relationship : between a tangent line to a circle the radius of the circle between two tangent segments to a common point outside the circle. JRLeon Geometry Chapter 6.1 – 6.2 HGSH

JRLeon Geometry Chapter 6.1 – 6.2 HGSH 6.1- Tangent Properties Relationship between a tangent line to a circle Points of Tangency Rails act as tangent lines to the wheels of a train. Each wheel of a train theoretically touches only one point on the rail. The point where the rail and the wheel meet is a point of tangency. Why can’t a train wheel touch more than one point at a time on the rail? How is the radius of the wheel to the point of tangency related to the rail? Let’s investigate.... JRLeon Geometry Chapter 6.1 – 6.2 HGSH

JRLeon Geometry Chapter 6.1 – 6.2 HGSH 6.1- Tangent Properties The relationship between a tangent line and the radius drawn to the point of tangency Tanget Line and Radius Construct a large circle. Label the center O and using a straightedge, draw a line that appears to touch the circle at only one point. Label the point T. Now connect O to T. Using your protractor measure the angles at T. What can you conclude about the radius OT and the tangent line at T? JRLeon Geometry Chapter 6.1 – 6.2 HGSH

The converse of the Tangent Conjecture is true. 6.1- Tangent Properties The converse of the Tangent Conjecture is true. Tanget Line and Radius If you construct a line perpendicular to a radius at the point where it touches the circle, the line will be tangent to the circle. The Tangent Conjecture has important applications related to circular motion. For example, a satellite maintains its velocity in a direction tangent to its circular orbit. This velocity vector is perpendicular to the force of gravity, which keeps the satellite in orbit. What happens if gravity is lost? The objects would travel off into space on a straight line tangent to their orbits, and not continue in a curved path. JRLeon Geometry Chapter 6.1 – 6.2 HGSH

JRLeon Geometry Chapter 6.1 – 6.2 HGSH 6.1- Tangent Properties Lengths of segments tangent to a circle from a point outside the circle Construct a circle. Label the center E. Choose a point outside the circle and label it N. Draw two lines through point N tangent to the circle. Mark the points where these lines appear to touch the circle and label them A and G. Use your compass to compare the lengths of segments NA and NG. Segments such as these are called tangent segments. Tangents through an external point JRLeon Geometry Chapter 6.1 – 6.2 HGSH

JRLeon Geometry Chapter 6.1 – 6.2 HGSH 6.1- Tangent Properties Tangents to two circles Given two circles, there are lines that are tangents to both of them at the same time. If the circles are separate (do not intersect), there are four possible common tangents: If the two circles touch at just one point, there are three possible tangent lines that are common to both: JRLeon Geometry Chapter 6.1 – 6.2 HGSH

JRLeon Geometry Chapter 6.1 – 6.2 HGSH 6.1- Tangent Properties Tangents to two circles Given two circles, there are lines that are tangents to both of them at the same time. If the two circles touch at just one point, with one inside the other, there is just one line that is a tangent to both: If the circles overlap - i.e. intersect at two points, there are two tangents that are common to both: JRLeon Geometry Chapter 6.1 – 6.2 HGSH

JRLeon Geometry Chapter 6.1 – 6.2 HGSH 6.1- Tangent Properties In the figure the central angle, BOA, determines the minor arc, AB. BOA is said to intercept AB because the arc is within the angle*. The measure of a minor arc is defined as the measure of its central angle, so mAB 40°. The measure of a major arc is the reflex measure of BOA, or 360° minus the measure of the minor arc, so mBCA320°. Intercepted Arc * When two straight lines from the center of a circle, cross the circle, the part of the circle between the intersection points is called the intercepted arc. The lines intercept, or 'cut off', the arc. Let’s look at an example involving arc measures and tangent segments. JRLeon Geometry Chapter 6.1 – 6.2 HGSH

JRLeon Geometry Chapter 6.1 – 6.2 HGSH 6.1- Tangent Properties Intercepted Arc In the figure at right, TA and TG are both tangent to circle N. If the major arc formed by the two tangents measures 220°, find the measure of T. JRLeon Geometry Chapter 6.1 – 6.2 HGSH

JRLeon Geometry Chapter 6.1 – 6.2 HGSH 6.1- Tangent Properties Classwork: Pages: 313-315: 1-7, 16 Homework: #8 JRLeon Geometry Chapter 6.1 – 6.2 HGSH