Welcome to Interactive Chalkboard Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc., Cincinnati, Ohio Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio and 10.6 Tangents and Secants

Objectives Use properties of tangents Solve problems using circumscribed polygons Find measures of angles formed by lines intersecting on, inside, or outside a circle.

Tangents and Secants A tangent is a line in the plane of a circle that intersects the circle in exactly one point. Line j is a tangent. A secant is a line that intersects a circle in two points. Line k is a secant. A secant contains a chord. k j

Tangents Theorem 10.9: If a line is tangent to a, then it is ┴ to the radius drawn to the point of tangency. The converse is also true. j r r ┴ j

Example 5-1a ALGEBRA is tangent to at point R. Find y. Because the radius is perpendicular to the tangent at the point of tangency,. This makes a right angle and  a right triangle. Use the Pythagorean Theorem to find QR, which is one-half the length y.

Example 5-1b Pythagorean Theorem Simplify. Subtract 256 from each side. Take the square root of each side. Because y is the length of the diameter, ignore the negative result. Answer: Thus, y is twice.

Example 5-1c Answer: 15 is a tangent to at point D. Find a.

Example 5-2a First determine whether  ABC is a right triangle by using the converse of the Pythagorean Theorem. Determine whether is tangent to

Example 5-2b Pythagorean Theorem Simplify. Because the converse of the Pythagorean Theorem did not prove true in this case,  ABC is not a right triangle. Answer: So, is not tangent to.

Example 5-2c First determine whether  EWD is a right triangle by using the converse of the Pythagorean Theorem. Determine whether is tangent to

Example 5-2d Pythagorean Theorem Simplify. Answer: Thus, making a tangent to Because the converse of the Pythagorean Theorem is true,  EWD is a right triangle and  EWD is a right angle.

Example 5-2e Answer: yes a. Determine whether is tangent to

Example 5-2f Answer: no b. Determine whether is tangent to

Tangents (continued) Theorem 10.11: If two segments from the same exterior point are tangent to a circle, then they are congruent. W X Y Z XW  XY

Example 5-3a ALGEBRA Find x. Assume that segments that appear tangent to circles are tangent. are drawn from the same exterior point and are tangent to so are drawn from the same exterior point and are tangent to

Example 5-3b Definition of congruent segments Substitution. Use the value of y to find x. Definition of congruent segments Substitution Simplify. Subtract 14 from each side. Answer: 1

Example 5-3d ALGEBRA Find a. Assume that segments that appear tangent to circles are tangent. Answer: –6

Example 5-4a Triangle HJK is circumscribed about Find the perimeter of  HJK if

Example 5-4b Use Theorem to determine the equal measures. We are given that Answer: The perimeter of  HJK is 158 units. Definition of perimeter Substitution

Example 5-4c Triangle NOT is circumscribed about Find the perimeter of  NOT if Answer: 172 units

Assignment Pre-AP GeometryPre-AP Geometry Pg. 556 #8 – 20, Geometry:Geometry: Pg. 556 #8 – 18,

Secants and Interior Angles Theorem 10.12: Theorem 10.12: If two secants intersect in the interior of a, then the measure of an  formed is ½ the measure of the sum of the arcs intercepted by the secants that created the . A B C D O m  AOC = ½ (m arc AC + m arc DB) m  AOD = ½ (m arc AD + m arc CB)

Example 6-1a Find if and Method 1

Example 6-1b Method 2 Answer: 98

Example 6-1d Answer: 138 Find if and

Secants, Tangents and Angles Theorem 10.13: Theorem 10.13: If a secant and a tangent intersect at the point of tangency, then the measure of each  formed is ½ the measure of its intercepted arc. m  DOB = ½ (m arc AB) m  COB = ½ (m arc CFB) A B C D O F

Example 6-2a Find if and Answer: 55

Example 6-2c Answer: 58 Find if and

Secants – Tangents and Exterior Angles Theorem 10.14: Theorem 10.14: If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the  formed is ½ the measures of the difference of the intercepted arcs. m  BOC = ½ (m arc BC – m arc HI) m  AOC= ½ (m arc AC – m arc AI) m  AED= ½ (m arc ABD – m arc AD) A C G B F D E O H I

Example 6-3a Find x. Theorem Multiply each side by 2. Add x to each side. Subtract 124 from each side. Answer: 17

Example 6-3c Find x. Answer: 111

Example 6-4a JEWELRY A jeweler wants to craft a pendant with the shape shown. Use the figure to determine the measure of the arc at the bottom of the pendant. Let x represent the measure of the arc at the bottom of the pendant. Then the arc at the top of the circle will be 360 – x. The measure of the angle marked 40° is equal to one-half the difference of the measure of the two intercepted arcs.

Example 6-4b Multiply each side by 2 and simplify. Add 360 to each side. Divide each side by 2. Answer: 220

Example 6-4c Answer: 75 PARKS Two sides of a fence to be built around a circular garden in a park are shown. Use the figure to determine the measure of

Example 6-5a Find x. Multiply each side by 2. Add 40 to each side. Divide each side by 6. Answer: 25

Example 6-5c Find x. Answer: 9

Assignment Pre-AP GeometryPre-AP Geometry Pg. 564 # Geometry:Geometry: Pg. 564 #3 – 7,