t Circles – Tangent Lines A tangent line touches a circle at exactly one point. In this case, line t is tangent to circle A. A t

t Circles – Tangent Lines A tangent line touches a circle at exactly one point. In this case, line t is tangent to circle A. The point at which the line is tangent is called the point of tangency ( point C ) A D C t B

t Circles – Tangent Lines A tangent line touches a circle at exactly one point. In this case, line t is tangent to circle A. The point at which the line is tangent is called the point of tangency ( point C ) Rays can also be tangent to circles. Ray CD ( segment CD ) Ray CB ( segment CB ) A D C t B

t a Circles – Tangent Lines A tangent line touches a circle at exactly one point. In this case, line t is tangent to circle A. The point at which the line is tangent is called the point of tangency ( point C ) Rays can also be tangent to circles. Ray CD ( segment CD ) Ray CB ( segment CB ) Common Tangent Line - tangent to two coplanar circles A D C t B a P Q

t a e Circles – Tangent Lines A tangent line touches a circle at exactly one point. In this case, line t is tangent to circle A. The point at which the line is tangent is called the point of tangency ( point C ) Rays can also be tangent to circles. Ray CD ( segment CD ) Ray CB ( segment CB ) Common Tangent Line - tangent to two coplanar circles - common external tangent lines do not intersect ( lines “a” and “e” ) the segment joining the circles A D C t B a P Q e

t g h a e Circles – Tangent Lines A tangent line touches a circle at exactly one point. In this case, line t is tangent to circle A. The point at which the line is tangent is called the point of tangency ( point C ) Rays can also be tangent to circles. Ray CD ( segment CD ) Ray CB ( segment CB ) Common Tangent Line - tangent to two coplanar circles - common external tangent lines do not intersect ( lines “a” and “e” ) the segment joining the circles common internal tangent lines intersect the segment joining the circles A D C t B g h a P Q e

c t Circles – Tangent Lines Tangent circles - two coplanar circles that are tangent to the same line at the same point c A C Q S B t

c t Circles – Tangent Lines Tangent circles - two coplanar circles that are tangent to the same line at the same point circle A tangent to circle Q circle A tangent to circle S c A C Q S B t Circle A and Q are internally tangent, one circle is inside the other.

c t Circles – Tangent Lines Tangent circles - two coplanar circles that are tangent to the same line at the same point circle A tangent to circle Q circle A tangent to circle S c A C Q S B t Circle A and Q are internally tangent, one circle is inside the other. Circle A and S are externally tangent, not one point of one circle is in the interior of the other.

t Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. A B t

t Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. A B t Theorem - tangents to a circle from an exterior point are congruent D A P E

t Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. A B t Theorem - tangents to a circle from an exterior point are congruent D A P E

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent Let’s use these two theorems to solve some problems.

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent Let’s use these two theorems to solve some problems. AB and AC are tangent to circle S. B A S C

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent Let’s use these two theorems to solve some problems. AB and AC are tangent to circle S. B A S C

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent Let’s use these two theorems to solve some problems. AB and AC are tangent to circle S. B A S C ( 90° - 25° = 65° )

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent Let’s use these two theorems to solve some problems. AB and AC are tangent to circle S. ∆ABC is isosceles from the theorem above about tangents from an exterior point… B A S C ( 90° - 25° = 65° )

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent Let’s use these two theorems to solve some problems. AB and AC are tangent to circle S. ∆ABC is isosceles from the theorem above about tangents from an exterior point… B 65° A S 65° C ( 90° - 25° = 65° )

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent Let’s use these two theorems to solve some problems. AB and AC are tangent to circle S. ∆ABC is isosceles from the theorem above about tangents from an exterior point… B 65° A S 65° C ( 90° - 25° = 65° )

SC BA AC Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent EXAMPLE # 2 : AB and AC are tangent to circle S. FIND : B SC BA AC 10 A S 26 C

SC 10 BA AC Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent EXAMPLE # 2 : AB and AC are tangent to circle S. FIND : B SC 10 BA AC 10 A S 26 C

SC 10 BA 24 AC Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent EXAMPLE # 2 : AB and AC are tangent to circle S. FIND : B SC 10 BA 24 AC 10 A S 26 ∆BAS is a right triangle : C

SC 10 BA 24 AC Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. Theorem - tangents to a circle from an exterior point are congruent EXAMPLE # 2 : AB and AC are tangent to circle S. FIND : B SC 10 BA 24 AC 10 A S 26 ∆BAS is a right triangle : C

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C EXAMPLE # 3 : 10.5 AD is tangent to circle S. Find CD if CS = 10.5 and AD = 25 S A 25 D

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C EXAMPLE # 3 : 10.5 AD is tangent to circle S. Find CD if CS = 10.5 and AD = 25 S 10.5 ( both are radii ) A 25 D

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C EXAMPLE # 3 : 10.5 AD is tangent to circle S. Find CD if CS = 10.5 and AD = 25 S 10.5 ( both are radii ) A 25 D

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C EXAMPLE # 3 : 10.5 AD is tangent to circle S. Find CD if CS = 10.5 and AD = 25 S 10.5 ( both are radii ) A 25 D

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C D EXAMPLE # 4 : 14 24 CD is a common tangent to circle S and circle R. Find CD if CS = 24, DR = 14 and SR = 26 R 26 S

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C D EXAMPLE # 4 : 14 24 E CD is a common tangent to circle S and circle R. Find CD if CS = 24, DR = 14 and SR = 26 R 26 S We can sketch in a parallel line to CD that creates a right triangle SRE. Since CD was perpendicular to CS and DR, the new line will be perpendicular as well.

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C D EXAMPLE # 4 : 14 24 E CD is a common tangent to circle S and circle R. Find CD if CS = 24, DR = 14 and SR = 26 10 R 26 S We can sketch in a parallel line to CD that creates a right triangle SRE. Since CD was perpendicular to CS and DR, the new line will be perpendicular as well.

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C D EXAMPLE # 4 : 14 24 E CD is a common tangent to circle S and circle R. Find CD if CS = 24, DR = 14 and SR = 26 10 R 26 S

Circles – Tangent Lines Theorem – the radius of a circle is perpendicular to a tangent line of that circle drawn to the point of tangency. C D EXAMPLE # 4 : 14 24 E CD is a common tangent to circle S and circle R. Find CD if CS = 24, DR = 14 and SR = 26 10 R 26 S