Tangency

Lines of Circles

EXAMPLE 1 Identify special segments and lines Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. AC a. SOLUTION is a radius because C is the center and A is a point on the circle. AC a.

EXAMPLE 1 Identify special segments and lines b. AB is a diameter because it is a chord that contains the center C. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. b. AB SOLUTION

EXAMPLE 1 Identify special segments and lines c. DE is a tangent ray because it is contained in a line that intersects the circle at only one point. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. SOLUTION DE c.

EXAMPLE 1 Identify special segments and lines d. AE is a secant because it is a line that intersects the circle in two points. Tell whether the line, ray, or segment is best described as a radius, chord, diameter, secant, or tangent of C. SOLUTION AE d.

SOLUTION GUIDED PRACTICE for Example 1 Is a chord because it is a segment whose endpoints are on the circle. AG CB is a radius because C is the center and B is a point on the circle. 1. In Example 1, what word best describes AG ? CB ?

SOLUTION GUIDED PRACTICE for Example 1 2. In Example 1, name a tangent and a tangent segment. A tangent is DE A tangent segment is DB

EXAMPLE 2 Find lengths in circles in a coordinate plane b. Diameter of A Radius of B c. Diameter of B d. Use the diagram to find the given lengths. a.Radius of A SOLUTION a.The radius of A is 3 units. b.The diameter of A is 6 units. c. The radius of B is 2 units. d. The diameter of B is 4 units.

SOLUTION GUIDED PRACTICE for Example 2 a.The radius of C is 3 units. b.The diameter of C is 6 units. c. The radius of D is 2 units. d. The diameter of D is 4 units. 3. Use the diagram in Example 2 to find the radius and diameter of C and D.

Diagram Common tangents

EXAMPLE 3 Draw common tangents Tell how many common tangents the circles have and draw them. a.b. c. SOLUTION a. 4 common tangents 3 common tangents b.

EXAMPLE 3 Draw common tangents c. 2 common tangents Tell how many common tangents the circles have and draw them. c. SOLUTION

GUIDED PRACTICE for Example 3 Tell how many common tangents the circles have and draw them common tangents

SOLUTION GUIDED PRACTICE for Example 3 Tell how many common tangents the circles have and draw them. 1 common tangent 5.

SOLUTION GUIDED PRACTICE for Example 3 Tell how many common tangents the circles have and draw them. No common tangents 6.

Theorem Tangent Perpendicular

Theorem Tangent Perpendicular Converse

EXAMPLE 4 Verify a tangent to a circle SOLUTION Use the Converse of the Pythagorean Theorem. Because = 37 2, PST is a right triangle and ST PT. So, ST is perpendicular to a radius of P at its endpoint on P. By Theorem 10.1, ST is tangent to P. In the diagram, PT is a radius of P. Is ST tangent to P ?

EXAMPLE 5 Find the radius of a circle In the diagram, B is a point of tangency. Find the radius r of C. SOLUTION You know from Theorem 10.1 that AB BC, so ABC is a right triangle. You can use the Pythagorean Theorem. AC 2 = BC 2 + AB 2 (r + 50) 2 = r r r = r r = 3900 r = 39 ft. Pythagorean Theorem Substitute. Multiply. Subtract from each side. Divide each side by 100.

Theorem Common Point Tangency

EXAMPLE 6 Find the radius of a circle RS is tangent to C at S and RT is tangent to C at T. Find the value of x. SOLUTION RS = RT 28 = 3x = x Substitute. Solve for x. Tangent segments from the same point are

GUIDED PRACTICE for Examples 4, 5 and 6 7. Is DE tangent to C ? ANSWER Yes

GUIDED PRACTICE for Examples 4, 5 and 6 8. ST is tangent to Q.Find the value of r. ANSWER r = 7

GUIDED PRACTICE for Examples 4, 5 and 6 9. Find the value(s) of x. + 3 = x ANSWER