Tangent Lines Skill 48

Objective HSG-C.2: Students are responsible for using properties of tangents to circles.

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.

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 𝑶𝑷 ⏊𝒍

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 𝒍 𝒊𝒔 𝒕𝒂𝒏𝒈𝒆𝒏𝒕 𝒕𝒐 ⊙𝑶

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 𝑨𝑩 ≅ 𝑪𝑩

Example 1; Finding Angle Measures a) 𝑀𝐿 and 𝑀𝑁 are tangent to ⊙𝑂. What is the value of x? 𝒎∠𝑳+𝒎∠𝑴+𝒎∠𝑵+𝒎∠𝑶=𝟏𝟖𝟎 𝒏−𝟐 𝟗𝟎+𝒙+𝟗𝟎+𝟏𝟏𝟕=𝟏𝟖𝟎 𝟒−𝟐 O M N L 𝟐𝟗𝟕+𝒙=𝟑𝟔𝟎 𝒙=𝟔𝟑 xᵒ 117ᵒ The measure of 𝒎∠𝑴 is 63ᵒ.

Example 1; Finding Angle Measures b) 𝐸𝐷 is tangent to ⊙𝑂. What is the value of x? 𝒎∠𝑫+𝒎∠𝑬+𝟑𝟖=𝟏𝟖𝟎 O E D 𝟗𝟎+𝒙+𝟑𝟖=𝟏𝟖𝟎 xᵒ 𝟏𝟐𝟖+𝒙=𝟏𝟖𝟎 𝒙=𝟓𝟐 38ᵒ The measure of 𝒎∠𝑬 is 52ᵒ.

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? Earth’s radius is about 6400 km. Convert meters to kilometers 𝟒𝟒𝟕 𝒎=.𝟒𝟒𝟕 𝒌𝒎 Recall the Pythagorean Theorem 𝒂= ? 𝒃=𝟔𝟒𝟎𝟎 𝒄=𝟔𝟒𝟎𝟎.𝟒𝟒𝟕 a c b 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 𝒂 𝟐 + 𝟔𝟒𝟎𝟎 𝟐 = 𝟔𝟒𝟎𝟎.𝟒𝟒𝟕 𝟐 𝒂 𝟐 +𝟒𝟎𝟗𝟔𝟎𝟎𝟎𝟎=𝟒𝟎𝟗𝟔𝟓𝟕𝟐𝟏.𝟖 𝒂 𝟐 =𝟓𝟕𝟐𝟏.𝟖 𝒂=𝟕𝟓.𝟔𝟒 𝒌𝒎

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 Earth’s radius is about 4000 miles. Recall the Pythagorean Theorem 𝒂= ? 𝒃=𝟒𝟎𝟎𝟎 𝒄=𝟒𝟎𝟎𝟐 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 a c b 𝒂 𝟐 + 𝟒𝟎𝟎𝟎 𝟐 = 𝟒𝟎𝟎𝟐 𝟐 𝒂 𝟐 +𝟏𝟔𝟎𝟎𝟎𝟎𝟎𝟎=𝟏𝟔𝟎𝟏𝟔𝟎𝟎𝟒 𝒂 𝟐 =𝟏𝟔𝟎𝟎𝟒 𝒂=𝟏𝟐𝟔.𝟓 𝒎𝒊

Example 3; Finding a Radius a) What is the radius of ⊙𝐴 b 12 A a x 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 8 x c 𝒙 𝟐 + 𝟏𝟐 𝟐 = 𝒙+𝟖 𝟐 𝒙 𝟐 +𝟏𝟒𝟒= 𝒙 𝟐 +𝟏𝟔𝒙+𝟔𝟒 𝟖𝟎=𝟏𝟔𝒙 𝒙=𝟓

Example 3; Finding a Radius b) What is the radius of ⊙𝐵? b 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 10 𝒙 𝟐 + 𝟏𝟎 𝟐 = 𝒙+𝟔 𝟐 B a x 6 𝒙 𝟐 +𝟏𝟎𝟎= 𝒙 𝟐 +𝟏𝟐𝒙+𝟑𝟔 c x 𝟔𝟒=𝟏𝟐𝒙 𝒙= 𝟏𝟔 𝟑

Example 4; Identifying a Tangent a) Is 𝑀𝐿 tangent to ⊙𝑁 at L? Explain. N M L 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 c 25 𝟕 𝟐 + 𝟐𝟒 𝟐 = 𝟐𝟓 𝟐 𝟒𝟗+𝟓𝟕𝟔=𝟔𝟐𝟓 24 b 7 a 𝟔𝟐𝟓=𝟔𝟐𝟓 Yes, 𝑀𝐿 is tangent to ⊙𝑁 at L, because Theorem 70.

Example 4; Identifying a Tangent b) If 𝑁𝐿=4, 𝑀𝐿=7, 𝑎𝑛𝑑 𝑁𝑀=8, is 𝑀𝐿 tangent to ⊙𝑁 at L? Explain. N M L 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 c 8 𝟒 𝟐 + 𝟕 𝟐 = 𝟖 𝟐 𝟏𝟔+𝟒𝟗=𝟔𝟒 7 b 4 a 𝟔𝟓≠𝟔𝟒 No, 𝑀𝐿 does not create a perpendicular with 𝑵𝑳 .

Example 5; Circles Inscribed in Polygons a) ⊙𝑂 is inscribed in ∆𝐴𝐵𝐶. What is the perimeter of ∆𝐴𝐵𝐶. O C B A D E F 𝑩𝑫=𝑩𝑬=𝟏𝟓 𝑨𝑫=𝑨𝑭=𝟏𝟎 15 cm 𝑪𝑭=𝑪𝑬=𝟖 𝑷=𝟏𝟓+𝟏𝟓+𝟖+𝟖+𝟏𝟎+𝟏𝟎 8 cm 𝑷=𝟑𝟎+𝟏𝟔+𝟐𝟎 10 cm 𝑷=𝟔𝟔 𝒄𝒎

Example 5; Circles Inscribed in Polygons b) ⊙𝑂 is inscribed in ∆𝑃𝑄𝑅, which has a perimeter of 88 𝑐𝑚. What is the length of 𝑄𝑌 ? O R Q P Y X Z 𝑷𝑿=𝑷𝒁=𝟏𝟓 𝑹𝒁=𝑹𝒀=𝟏𝟕 𝑷=𝟏𝟓+𝟏𝟓+𝟏𝟕+𝟏𝟕+𝑸𝒀+𝑸𝑿 15 cm 𝟖𝟖=𝟑𝟎+𝟑𝟒+𝟐 𝑸𝒀 𝟖𝟖=𝟔𝟒+𝟐 𝑸𝒀 17 cm 𝟐𝟒=𝟐 𝑸𝒀 𝑸𝒀=𝟏𝟐 𝒄𝒎

#48: Tangent Lines Questions? Summarize Notes Homework Quiz