1. What measure is needed to find the circumference 1. What measure is needed to find the circumference or area of a circle? ANSWER radius or diameter 2. Find the radius of a circle with diameter 8 centimeters. ANSWER 4 cm 3. A right triangle has legs with lengths 5 inches and 12 inches. Find the length of the hypotenuse. ANSWER 13 in. 4. Solve 6x + 15 = 33. ANSWER 3 5. Solve (x + 18)2 = x2 + 242. ANSWER 7

Use properties of segments that intersect circles. Target Use properties of segments that intersect circles. You will… Use properties of a tangent to a circle.

Vocabulary circle = set of all points equidistant from a given point chord radius center tangent diameter point of tangency secant

Vocabulary circle – the set of all points in a plane equidistant from a given point called the center of the circle radius – a segment whose endpoints are the center and any point on the circle chord – a segment whose endpoints are on the circle diameter – a chord that contains the center secant – a line that intersects a circle in two points tangent – a line, segment, or ray in the plane of a circle that intersects the circle in exactly one point called the point of tangency

Vocabulary circle = set of all points equidistant from a given point chord radius center tangent diameter point of tangency secant

Vocabulary Properties of Tangents Theorem 10.1 - 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 (point of tangency) Q T P Theorem 10.2 – Tangent segments from a common external point are congruent R S

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

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. DE c. SOLUTION c. DE is a tangent ray because it is contained in a line that intersects the circle at only one point.

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. AE d. SOLUTION d. AE is a secant because it is a line that intersects the circle in two points.

GUIDED PRACTICE for Example 1 1. In Example 1, what word best describes AG ? CB ? SOLUTION 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.

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

EXAMPLE 2 Find lengths in circles in a coordinate plane Use the diagram to find the given lengths. a. Radius of A b. Diameter of A Radius of B c. Diameter of B d. 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.

GUIDED PRACTICE for Example 2 3. Use the diagram to find the radius and diameter of C and D. SOLUTION 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.

EXAMPLE 3 Draw common tangents Tell how many common tangents the circles have and draw them. a. b. c. SOLUTION a. 4 3 b. c. 2

GUIDED PRACTICE for Example 3 Tell how many common tangents the circles have and draw them. 4. SOLUTION 2 common tangents

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

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

EXAMPLE 4 Verify a tangent to a circle In the diagram, PT is a radius of P. Is ST tangent to P ? SOLUTION Use the Converse of the Pythagorean Theorem. Because 122 + 352 = 372, 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.

Find the radius of a circle 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. AC2 = BC2 + AB2 Pythagorean Theorem (r + 50)2 = r2 + 802 Substitute. r2 + 100r + 2500 = r2 + 6400 Multiply. 100r = 3900 Subtract from each side. r = 39 ft . Divide each side by 100.

Find the radius of a circle 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 Tangent segments from the same point are 28 = 3x + 4 Substitute. 8 = x Solve for x.

GUIDED PRACTICE for Examples 4, 5 and 6 Is DE tangent to C? Explain your reasoning. ANSWER Since the radius of the circle is 3 units, CE = 3 + 2 = 5. Using the converse of Pythagorean’s Theorem, since CD2 + DE2 = CE2 , a Pythagorean Triple, angle D is right. Therefore DE CD and tangent to C.

GUIDED PRACTICE for Examples 4, 5 and 6 8. ST is tangent to Q. Find the value of r. 9. Find the value(s) of x. +3 = x ANSWER ANSWER r = 7