Geometry Lines That Intersect Circles CONFIDENTIAL

Warm Up A Point is chosen randomly on . Use the diagram to find the probability of each event. 1) The point is not on 2) The point is on 3) The point is or 4) The point is on 1) ¾ 2) ½ 3) 13/20 4) 1/40 CONFIDENTIAL

Lines That Intersect Circles A Circle is the set of all points in a plane that are equidistant from a given point , called the center of the circle . A circle with center C is called circle C, or C. The Interior of a circle is the set of all points inside the circle . The exterior of a circle is the set of all points outside the circle. CONFIDENTIAL

Lines and Segments That Intersects circles TERM DIAGRAM A Chord is a segment whose end points lie on a circle. A Secant is a line that intersects a circle at two points. A Tangent is a line in the same plane as a circle that intersects it at exactly one point. The point where the tangent and a circle intersect is called the point of tangency. CONFIDENTIAL

Identifying Lines and Segments That Intersect Circles Identify each line or segment that intersect A. chords: and tangent: L radii: and secant: diameter: CONFIDENTIAL

Identify each line or segment that intersects P. Now you try Identify each line or segment that intersects P. chord: QR tangent: UV radii: SP and PT secant: ST diameter: ST CONFIDENTIAL

Remember that the terms radius and diameter may refer to line segments, or to the lengths of segments CONFIDENTIAL

TERM CONDITION Two circles are congruent circles if and only if they have congruent radii. Concentric circles are coplanar circles with the same center. CONFIDENTIAL

Two coplanar circles that intersect at exactly one point are called tangent circles CONFIDENTIAL

Identifying Tangents of a circles Find the length of each radius . Identify the point of tangency and write the equation of the tangent line at this point. A B X Y -4 4 -2 2 CONFIDENTIAL

A B X Y -4 4 -2 2 CONFIDENTIAL

Find the length of each radius. Now you try Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point -4 4 Point of tangency: (2, -1) equation of the tangent: y = -1 CONFIDENTIAL

A common tangent is a line that is tangent to two circles CONFIDENTIAL

B A p q CONFIDENTIAL

Tangent to a circle at a point Construction Tangent to a circle at a point 2) 1) CONFIDENTIAL

3) l CONFIDENTIAL

The tangent line is perpendicular to the radius at the point of tangency. This fact is the basis for the following theorems . CONFIDENTIAL

THEOREM HYPOTHESIS CONCLUSION 1) 2) B l A CONFIDENTIAL

THEOREM HYPOTHESIS CONCLUSION 3) CONFIDENTIAL

Using Properties of Tangents CONFIDENTIAL

Now you try 1) R S T X- 6.3 RS = 2.1 CONFIDENTIAL

Now you try 2) S T R n+3 2n-1 RS = 7 CONFIDENTIAL

Now some problems for you to practice. CONFIDENTIAL

Identify each line or segment that intersects each circle. ASSESSMENT Identify each line or segment that intersects each circle. 1) F D E m l 2) T Q S CONFIDENTIAL

3) Find the length of each radius . Identify the point of tangency and write the equation of the tangent line at this point. X Y A B -4 4 Point of tangency: (-1, 4) equation of the tangent: y = 4 CONFIDENTIAL

4) Find the length of each radius . Identify the point of tangency and write the equation of the tangent line at this point. X Y R S -4 4 length of radius OS = 2 Point of tangency: (2, 0) equation of the tangent: y = 0 length of radius OR = 2 Point of tangency: (-1, 0) equation of the tangent: y = 0 CONFIDENTIAL

The segments in each figure are tangent to the circle. Find length JK. 5) The segments in each figure are tangent to the circle. Find length JK. K C L J 4x-1 2x+9 JK = 7 CONFIDENTIAL

The segments in each figure are tangent to the circle. Find length ST. 6) The segments in each figure are tangent to the circle. Find length ST. T P U S Y- 4 3 4 y ST = 12 CONFIDENTIAL

Lines and Segments That Intersects circles REVIEW Lines and Segments That Intersects circles TERM DIAGRAM A Chord is a segment whose end points lie on a circle. A Secant is a line that intersects a circle at two points. A Tangent is a line in the same plane as a circle that intersects it at exactly one point. The point where the tangent and a circle intersect is called the point of tangency. CONFIDENTIAL

Identifying Lines and Segments That Intersect Circles Identify each line or segment that intersect A. chords: and tangent: L radii: and secant: diameter: CONFIDENTIAL

TERM CONDITION Two circles are congruent circles if and only if they have congruent radii. Concentric circles are coplanar circles with the same center. CONFIDENTIAL

Two coplanar circles that intersect at exactly one point are called tangent circles CONFIDENTIAL

Identifying Tangents of a circles Find the length of each radius . Identify the point of tangency and write the equation of the tangent line at this point. A B X Y -4 4 -2 2 CONFIDENTIAL

A B X Y -4 4 -2 2 CONFIDENTIAL

Find the length of each radius. Now you try Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point -4 4 CONFIDENTIAL

A common tangent is a line that is tangent to two circles CONFIDENTIAL

B A p q CONFIDENTIAL

THEOREM HYPOTHESIS CONCLUSION 1) 2) B l A CONFIDENTIAL

THEOREM HYPOTHESIS CONCLUSION 3) CONFIDENTIAL

Using Properties of Tangents CONFIDENTIAL

You did a great job today! CONFIDENTIAL