Warm Up Section 3.1 Draw and label each of the following: A point, N, in the exterior of  KLP 6. A point, W, in the interior of  KLP

Warm Up Section 3.1 Draw and label each of the following: A point, N, in the exterior of  KLP 6. A point, W, in the interior of  KLP A B C D G H K L P N W

Intro to Circles and Properties of Tangents Section 3.1 Standard: MCC9-12.G.C.1,2,5 Essential Question: How are tangents used to solve problems?

Introduction: A circle is the set of all points in a plane at a given distance from a given point. A. A circle is named by its center. The circle shown below has center C so it is called circle C. This is symbolized by writing C. C D A F E B

B. Draw a line segment by connecting points C and D in the circle above. The segment you have drawn is called a radius. The plural of radius is radii. Name three other radii of the circle. Be sure to use the correct notation for a line segment: ______, ______, ______ Note: one endpoint of the radius is the center of the circle and the other endpoint is a point on the circle. Also, all radii of a circle are congruent. So, if AC = 2 cm, then CE = _____ cm. C D A F E B

C. Now, make another segment by connecting points A and E. The segment you have drawn is called a chord. Name five other chords of the circle. ______, ______, ______, ______, ______ Note: both endpoints of a chord are points on the circle. Chords of a circle do not necessarily have the same length. C D A F E B

D. If a chord passes through the center of a circle it is given a special name. It is called a diameter. Name the diameter pictured in C. _________ Now, draw that diameter. Note: a diameter is the longest chord of a circle. Its length is twice that of the radius. So, if AC = 2 cm, then AB = _____ cm. C D A F E B

E. Next, draw a line passing through points B and F. The line you have drawn is called a tangent. A tangent lies in the plane of the circle and intersects the circle in only one point. The point of intersection is called the point of tangency. What is the point of tangency for BF? ________ C D A F E B

F. Now, draw the line passing through points A and F. This line is called a secant. Any line that contains a chord of a circle is called a secant. C D A F E B

Use the circle below to identify the following: Name the circle ________ Name all radii pictured. _____________ Name all of chords pictured. __________ Name a diameter. ________ Name a tangent. ________ Name a secant ________ Y K W Z X K

The points located inside a circle are called interior points. The points located on the circle are points of the circle itself. The points located outside the circle are called exterior points. Draw four lines that are tangent to both of the circles below at the same time. These lines are called common tangents.

The two common tangents that pass between the two circles above are called internal common tangents. The other two tangents are called external common tangents.

Draw all the common tangents possible for the problems below

Summary: Circle: The set of all points in a plane at a given distance from a given point is a circle. P is the set of all points in the plane that are 2 units from P. The given point P is the center of the circle. Radius: The given distance is the radius of the circle. A radius is also a segment joining the center of the circle to a point of the circle. (The plural of radius is radii.) Chord: A segment whose endpoints lie on a circle is a chord. Diameter: A chord that contains the center of a circle is a diameter. A diameter is also the length equal to twice a radius. P

Secant: A line that contains a chord of a circle is a secant. Tangent: A line in the plane of a circle that intersects the circle in exactly one point is a tangent. The point of tangency is the point of intersection. radius diameter secant tangent chord

Identify each of the following from the diagram below. 1.Center 2.3 radii 3.3 chords 4.Secant 5.Tangent 6.Point of Tangency C A B D E G H F J Sticky Note Problem

Identify each of the following from the diagram below. 1.Center 2.3 radii 3.3 chords 4.Secant 5.Tangent 6.Point of Tangency C A B D E G H F J Sticky Note Problem

REMEMBER: a tangent is a line in the plane of a circle that intersects the circle in exactly one point, the point of tangency. A tangent ray and a tangent segment are also called tangents.

Theorem 1: In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle (the point of tangency). For the figure at right, identify the center of the circle as O and the point of tangency as P. Mark a square corner to indicate that the tangent line is perpendicular to the radius. O P

Theorem 2 : Tangent segments from a common external point are congruent. Measure and with a straightedge to the nearest tenth of a cm. RS = _______ cm RT = ______ cm T S R 2.6 cm 2.6

Example 1: In the diagram below, is a radius of circle R. If TR = 26, is tangent to circle R? S R T Right Triangle? = = = 676 Therefore, ∆RST is a right triangle. So, is tangent to R.

Example 2: is tangent to C at R and is tangent to C at S. Find the value of x. S R Q 32 3x = 3x = 3x 9 = x

Example 3: Find the value(s) of x: S R Q x2x2 16 x 2 = 16 x = ±4

Example 4: In the diagram, B is a point of tangency. Find the length of the radius, r, of C. B C 50 r 70 r r = (r + 50) 2 r = (r + 50)(r + 50) r = r r + 50r r = r r = 100r = 100r 24 = r

Recall: Two polygons are similar polygons if corresponding angles are congruent and corresponding sides are proportional. In the statement  ABC   DEF, the symbol  means “is similar to.” A B C D E F

Triangle Similarity Postulates and Theorems: Angle-Angle (AA) Similarity Postulate: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar. Side-Side-Side (SSS) Similarity Theorem: If the corresponding side lengths of two triangles are proportional, then the triangles are similar. Side-Angle-Side (SAS) Similarity Theorem: If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

Example 5: In the diagram, the circles are concentric with center A. is tangent to the inner circle at B and is tangent to the outer circle at C. Use similar triangles to show that. A B C D E

1. 1. ________________ 2. _____________________2. Definition of  3. _____________________3. All right angles are  4.  CAD   BAE4. _________________ 5. _____________________5. AA Similarity Postulate 6. _____________________6. Corresponding lengths of similar triangles are in proportion Tangent if  to radius A B C D E  ABE   ACD

Example 6: In the diagram, is a common internal tangent to M and P. Use similar triangles to show that M N T P S

1.1. ________________ 2. _____________________2. Definition of  3. _____________________3. All right angles are  4.  MNS   PNT4. ________________ 5. _____________________5. AA Similarity Postulate 6. _____________________6. Corresponding lengths of similar triangles are in proportion M N T P S Tangent if  to radius  MNS   PNT

Radius and tangent are perpendicular, hence their slopes are opposite reciprocals Example 7: Use the diagram at right to find each of the following: 1. Find the length of the radius of A. 2. Find the slope of the tangent line, t. A (3, 1) (5, -1) t