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

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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 ) t C A D B

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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 ) t C A D B

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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 t C A D B P Q a

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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 t C A D B P Q a e

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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 t C A D B P Q a e g h

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

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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 t A Q S c B C Circle A and Q are internally tangent, one circle is inside the other.

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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 t A Q S c B C 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.

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

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

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

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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.

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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. A C B S AB and AC are tangent to circle S.

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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. A C B S AB and AC are tangent to circle S.

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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. A C B S AB and AC are tangent to circle S. ( 90° - 25° = 65° )

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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. A C B S AB and AC are tangent to circle S. ( 90° - 25° = 65° ) ∆ABC is isosceles from the theorem above about tangents from an exterior point…

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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. A C B S AB and AC are tangent to circle S. ( 90° - 25° = 65° ) ∆ABC is isosceles from the theorem above about tangents from an exterior point… 65°

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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. A C B S AB and AC are tangent to circle S. ( 90° - 25° = 65° ) ∆ABC is isosceles from the theorem above about tangents from an exterior point… 65°

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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 : A C B S AB and AC are tangent to circle S. SC BA AC FIND : 10 26

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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 : A C B S AB and AC are tangent to circle S. SC10 BA AC FIND : 10 26

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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 : A C B S AB and AC are tangent to circle S. SC10 BA24 AC FIND : 10 26 ∆BAS is a right triangle :

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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 : A C B S AB and AC are tangent to circle S. SC10 BA24 AC24 FIND : 10 26 ∆BAS is a right triangle :

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

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

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

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

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

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

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

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

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

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Section 12.1: Lines That intersect Circles

Section 12.1: Lines That intersect Circles

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