5-Minute Check on Lesson 10-45-Minute Check on Lesson 10-45-Minute Check on Lesson 10-45-Minute Check on Lesson 10-4 Transparency 10-5

Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.

Refer to the figure and find each measure.

1. m1

2. m2

3. m3

4. m4

5. In ⊙B, find x if mA = 3x + 9and mB = 8x – 4.

6. If an inscribed angle has a measure of 110, what is the measure of its intercepted arc?Standardized Test Practice:

A CB D55 70 110 220

20°

60°

20°

100°

x = 11

D

Lesson 10-5

Tangents

Objectives

• Use properties of tangents

• Solve problems involving circumscribed polygons

Vocabulary

• Tangent – a line that intersects a circle in exactly one point

• Point of tangency – point where a tangent intersects a circle

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.

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 .

Answer: 15

is a tangent to at point D. Find a.

First determine whether ABC is a right triangle by using the converse of the Pythagorean Theorem.

Determine whether is tangent to

Pythagorean Theorem

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 .

First determine whether EWD is a right triangle by using the converse of the Pythagorean Theorem.

Determine whether is tangent to

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.

Answer: no

Determine whether is tangent to

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

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

Triangle HJK is circumscribed about Find the perimeter of HJK if

Use Theorem 10.10 to determine the equal measures.

We are given that

Answer: The perimeter of HJK is 158 units.

Definition of perimeter

Substitution

Triangle NOT is circumscribed about Find the perimeter of NOT if

Answer: 172 units

Summary & Homework

• Summary:– A line that is tangent to a circle intersects the

circle in exactly one point.– A tangent is perpendicular to a radius of a circle– Two segments tangent to a circle form the same

exterior point are congruent

• Homework: – pg 556-558; 8-11, 12-17