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Tangent Lines Skill 48
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Objective HSG-C.2: Students are responsible for using properties of tangents to circles.
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Definitions A tangent to a circle is a line in the plane of the circle that intersects the circle in exactly one point. The point of where a circle and a tangent intersect is the point of tangency.
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Theorem 79: Tangent Perpendicular to Radius
If a line is tangent to a circle, then the line is perpendicular to the radius at the point of tangency. P l O πΆπ· βπ
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Theorem 80: Radius Perpendicular to Tangent
If a line in the plane of a circle is perpendicular to a radius at its endpoint on the circle, then the line is tangent to the circle. P l O π ππ πππππππ ππ βπΆ
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Theorem 81: Congruent Tangents
If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent. A B C O π¨π© β
πͺπ©
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Example 1; Finding Angle Measures
a) ππΏ and ππ are tangent to βπ. What is the value of x? πβ π³+πβ π΄+πβ π΅+πβ πΆ=πππ πβπ ππ+π+ππ+πππ=πππ πβπ O M N L πππ+π=πππ π=ππ xα΅ 117α΅ The measure of πβ π΄ is 63α΅.
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Example 1; Finding Angle Measures
b) πΈπ· is tangent to βπ. What is the value of x? πβ π«+πβ π¬+ππ=πππ O E D ππ+π+ππ=πππ xα΅ πππ+π=πππ π=ππ 38α΅ The measure of πβ π¬ is 52α΅.
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Example 2; Finding Distance
a) The CN Tower in Toronto, Canada, has an observation deck 447 m above ground level. About how far is it from the observation deck to the horizon? Earthβs radius is about 6400 km. Convert meters to kilometers πππ π=.πππ ππ Recall the Pythagorean Theorem π= ? π=ππππ π=ππππ.πππ a c b π π + π π = π π π π + ππππ π = ππππ.πππ π π π +ππππππππ=ππππππππ.π π π =ππππ.π π=ππ.ππ ππ
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Example 2; Finding Distance
b) What is the distance to the horizon that a person can see on a clear day from an airplane 2 mi above Earth? The Earthβs radius is about 4000 miles. Recall the Pythagorean Theorem π= ? π=ππππ π=ππππ π π + π π = π π a c b π π + ππππ π = ππππ π π π +ππππππππ=ππππππππ π π =πππππ π=πππ.π ππ
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Example 3; Finding a Radius
a) What is the radius of βπ΄ b 12 A a x π π + π π = π π 8 x c π π + ππ π = π+π π π π +πππ= π π +πππ+ππ ππ=πππ π=π
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Example 3; Finding a Radius
b) What is the radius of βπ΅? b π π + π π = π π 10 π π + ππ π = π+π π B a x 6 π π +πππ= π π +πππ+ππ c x ππ=πππ π= ππ π
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Example 4; Identifying a Tangent
a) Is ππΏ tangent to βπ at L? Explain. N M L π π + π π = π π c 25 π π + ππ π = ππ π ππ+πππ=πππ 24 b 7 a πππ=πππ Yes, ππΏ is tangent to βπ at L, because Theorem 70.
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Example 4; Identifying a Tangent
b) If ππΏ=4, ππΏ=7, πππ ππ=8, is ππΏ tangent to βπ at L? Explain. N M L π π + π π = π π c 8 π π + π π = π π ππ+ππ=ππ 7 b 4 a ππβ ππ No, ππΏ does not create a perpendicular with π΅π³ .
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Example 5; Circles Inscribed in Polygons
a) βπ is inscribed in βπ΄π΅πΆ. What is the perimeter of βπ΄π΅πΆ. O C B A D E F π©π«=π©π¬=ππ π¨π«=π¨π=ππ 15 cm πͺπ=πͺπ¬=π π·=ππ+ππ+π+π+ππ+ππ 8 cm π·=ππ+ππ+ππ 10 cm π·=ππ ππ
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Example 5; Circles Inscribed in Polygons
b) βπ is inscribed in βπππ
, which has a perimeter of 88 ππ. What is the length of ππ ? O R Q P Y X Z π·πΏ=π·π=ππ πΉπ=πΉπ=ππ π·=ππ+ππ+ππ+ππ+πΈπ+πΈπΏ 15 cm ππ=ππ+ππ+π πΈπ ππ=ππ+π πΈπ 17 cm ππ=π πΈπ πΈπ=ππ ππ
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#48: Tangent Lines Questions? Summarize Notes Homework Quiz
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