10.5 Tangents & Secants

Objectives Use properties of tangents Solve problems using circumscribed polygons

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 1: 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 1: 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 .

Your Turn: is a tangent to at point D. Find a. Answer: 15

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

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

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

Your Turn: a. Determine whether is tangent to Answer: yes

Your Turn: b. Determine whether is tangent to Answer: no

More about Tangents 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 3: 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 3: 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

Your Turn: ALGEBRA Find a. Assume that segments that appear tangent to circles are tangent. Answer: –6

Example 4: Triangle HJK is circumscribed about Find the perimeter of HJK if

Example 4: Use Theorem 10.10 to determine the equal measures. We are given that Definition of perimeter Substitution Answer: The perimeter of HJK is 158 units.

Your Turn: Triangle NOT is circumscribed about Find the perimeter of NOT if Answer: 172 units

Assignment Pre-AP Geometry Pg. 556 #8 – 20, 23 - 26