Unit 5: Geometric and Algebraic Connections Vocabulary Builder

Identify opposite sides Find the slope of 𝑨𝑫 . Find the slope of 𝑩𝑪 . Find the slope of 𝑨𝑩 . Find the slope of 𝑫𝑪 . Compare slopes, opposite sides have the Same slope, then they are parallel. LMNO is a parallelogram

3 1 1 2 2 3

y-axis coordinates x-axis y-intercept

Diagonals are Congruent Parallelogram Rhombus Four right angles Square Diagonals are Perpendicular Diagonals are Congruent

Unit 5: Geometric and Algebraic Connections 5.1 Slope and Equations of Lines

Slope Undefined Parallel Congruent Perpendicular -1

−𝟏𝟎 −(−𝟐𝟎) 𝟏𝟏−(−𝟏𝟑) 𝟏 −(−𝟐) 𝟑−𝟎 (𝟑, 𝟏) (𝟎,−𝟐) = 𝟑 𝟑 =𝟏 = 𝟏𝟎 𝟐𝟒 = 𝟓 𝟏𝟐 −𝟏𝟎 −(−𝟐𝟎) 𝟏𝟏−(−𝟏𝟑) 𝟏 −(−𝟐) 𝟑−𝟎 (𝟎,−𝟐) = 𝟑 𝟑 =𝟏 = 𝟏𝟎 𝟐𝟒 = 𝟓 𝟏𝟐 Slope: − 𝟒 𝟓 Slope: 3 Parallel Slope: − 𝟒 𝟓 Parallel Slope: 3 Perpendicular Slope: 𝟓 𝟒 Perpendicular Slope: − 𝟏 𝟑

−𝟑 −𝟑 𝟏−(−𝟑) 𝟏𝟓 −𝟖 −𝟏𝟏−(−𝟐𝟎) (−𝟑, 𝟑) (𝟏,−𝟑) =− 𝟔 𝟒 =− 𝟑 𝟐 = 𝟕 𝟗 𝟏𝟓 −𝟖 −𝟏𝟏−(−𝟐𝟎) =− 𝟔 𝟒 =− 𝟑 𝟐 (𝟏,−𝟑) = 𝟕 𝟗 Slope: −𝟏 Slope: 𝟐 𝟓 Parallel Slope: −𝟏 Parallel Slope: 𝟐 𝟓 Perpendicular Slope: 𝟏 Perpendicular Slope: − 𝟓 𝟐

𝒚= 𝟗 𝟏𝟏 𝒙− 𝟔 𝟏𝟏 𝒚= 𝟒 𝟓 𝒙+𝟐 𝒚=𝟑𝒙+𝟔

𝒚=− 𝟔 𝟕 𝒙+ 𝟒𝟎 𝟕 𝒚=−𝟐𝒙 𝒚=−𝟑𝒙+𝟐

𝒚=−𝟐𝒙+𝟏 𝒚=−𝟐𝒙+𝟐

𝒚=𝟑𝒙+𝟏𝟑 𝒚= 𝟏 𝟑 𝒙+𝟑

Unit 5: Geometric and Algebraic Connections 5.2 Coordinate Proofs

Pythagorean Theorem hypotenuse Distance Formula difference difference Square root difference difference

= 𝟗+𝟏 = 𝟏𝟎 = 𝟗𝟎𝟎+𝟏𝟔𝟎𝟎 =𝟓𝟎 ( −𝟐−𝟏) 𝟐 +( 𝟑−𝟒) 𝟐 = ( −𝟑) 𝟐 +(−𝟏 ) 𝟐 = ( −𝟑) 𝟐 +(−𝟏 ) 𝟐 = 𝟗+𝟏 = 𝟏𝟎 ( 𝟒𝟎−𝟏𝟎) 𝟐 +(𝟒𝟓−𝟓 ) 𝟐 = ( 𝟑𝟎) 𝟐 +(𝟒𝟎 ) 𝟐 = 𝟗𝟎𝟎+𝟏𝟔𝟎𝟎 =𝟓𝟎

Use Distance Formula to prove congruent length 𝑩𝑪 = (𝟓−𝟏 ) 𝟐 +(𝟔− 𝟑) 𝟐 = ( 𝟑) 𝟐 +(𝟒 ) 𝟐 =𝟓 𝑨𝑪 = ( 𝟏−𝟒) 𝟐 +( 𝟑−(−𝟏)) 𝟐 = ( −𝟑) 𝟐 +(𝟒 ) 𝟐 =𝟓 Use Slope Formula to prove a right triangle 𝑩𝑪 = 𝟔−𝟑 𝟓−𝟏 𝑨𝑪 = −𝟏−𝟑 𝟓−𝟏 = 𝟑 𝟒 =− 𝟒 𝟑

Use Slope Formula to prove a right triangle 𝑩𝑪 = −𝟏−𝟑 𝟐−𝟐 𝑨𝑪 = −𝟏−(−𝟏) 𝟐−(−𝟑) = −𝟒 𝟎 = 𝟎 𝟓 Undefined

𝑨𝑩 = (𝟐−𝟏 ) 𝟐 +(𝟓− 𝟐) 𝟐 = 𝟏𝟎 𝑪𝑫 = ( 𝟒−𝟓) 𝟐 +(𝟒−𝟕 ) 𝟐 = 𝟏𝟎 Use Distance Formula to prove congruent length 𝑨𝑩 = (𝟐−𝟏 ) 𝟐 +(𝟓− 𝟐) 𝟐 = 𝟏𝟎 𝑪𝑫 = ( 𝟒−𝟓) 𝟐 +(𝟒−𝟕 ) 𝟐 = 𝟏𝟎 𝑩𝑪 = (𝟓−𝟐 ) 𝟐 +(𝟕−𝟓 ) 𝟐 = 𝟏𝟑 𝑨𝑫 = ( 𝟒−𝟏) 𝟐 +( 𝟒−𝟐) 𝟐 = 𝟏𝟑

𝑨𝑩 = (−𝟐−(−𝟑) ) 𝟐 +(𝟔− 𝟐) 𝟐 = 𝟏𝟕 𝑪𝑫 = ( 𝟏−𝟐) 𝟐 +(𝟑−𝟕 ) 𝟐 = 𝟏𝟕 𝑩𝑪 = (𝟐−(−𝟐) ) 𝟐 +(𝟕−𝟔 ) 𝟐 = 𝟏𝟕 𝑨𝑫 = ( 𝟏−(−𝟑)) 𝟐 +( 𝟑−𝟐) 𝟐 = 𝟏𝟕

𝑨𝑩 = 𝟐−𝟎 𝟑−(−𝟑) 𝑨𝑫 = −𝟑−𝟎 −𝟐−(−𝟑) 𝑨𝑩 = (𝟑−(−𝟑) ) 𝟐 +(𝟐−𝟎 ) 𝟐 Use Slope Formula to prove a right triangle 𝑨𝑩 = 𝟐−𝟎 𝟑−(−𝟑) = 𝟏 𝟑 𝑨𝑫 = −𝟑−𝟎 −𝟐−(−𝟑) =−𝟑 Use Distance Formula to prove congruent length 𝑨𝑩 = (𝟑−(−𝟑) ) 𝟐 +(𝟐−𝟎 ) 𝟐 = ( 𝟔) 𝟐 +(𝟐 ) 𝟐 = 𝟒𝟎 𝑪𝑫 = (−𝟐−𝟒 ) 𝟐 +(−𝟑−(−𝟏) ) 𝟐 = ( −𝟔) 𝟐 +(−𝟐 ) 𝟐 = 𝟒𝟎

𝑨𝑩 = 𝟒−𝟎 𝟎−(−𝟑) 𝑨𝑫 = −𝟑−𝟎 −𝟏−(−𝟑) 𝑨𝑩 = (𝟎−(−𝟑) ) 𝟐 +(𝟒−𝟎 ) 𝟐 Use Slope Formula to prove a right triangle 𝑨𝑩 = 𝟒−𝟎 𝟎−(−𝟑) = 𝟒 𝟑 𝑨𝑫 = −𝟑−𝟎 −𝟏−(−𝟑) =− 𝟑 𝟒 Use Distance Formula to prove congruent length 𝑨𝑩 = (𝟎−(−𝟑) ) 𝟐 +(𝟒−𝟎 ) 𝟐 = ( 𝟑) 𝟐 +(𝟒 ) 𝟐 =𝟓 𝑩𝑪 = (𝟒−𝟎 ) 𝟐 +(𝟏−𝟒 ) 𝟐 = ( 𝟒) 𝟐 +(−𝟑 ) 𝟐 =𝟓

Unit 5: Geometric and Algebraic Connections 5.3 Perimeter and Area of Polygons

sides angles Pythagorean Theorem Distance Formula

P = 𝟑𝟒 + 𝟑𝟕 + 𝟒𝟓 = 𝟏𝟖.𝟕 𝑫𝑬 = (𝟏−(−𝟒) ) 𝟐 +(𝟒−𝟏 ) 𝟐 = ( 𝟓) 𝟐 +(𝟑 ) 𝟐 (𝟏, 𝟒) (−𝟒, 𝟏) (𝟐, −𝟐) 𝑫𝑬 = (𝟏−(−𝟒) ) 𝟐 +(𝟒−𝟏 ) 𝟐 = ( 𝟓) 𝟐 +(𝟑 ) 𝟐 = 𝟑𝟒 𝑬𝑭 = (𝟐−𝟏 ) 𝟐 +(−𝟐−𝟒 ) 𝟐 = ( 𝟏) 𝟐 +(−𝟔 ) 𝟐 = 𝟑𝟕 𝑫𝑭 = (𝟐−(−𝟒) ) 𝟐 +(−𝟐−𝟏 ) 𝟐 = ( 𝟔) 𝟐 +(−𝟑 ) 𝟐 = 𝟒𝟓 P = 𝟑𝟒 + 𝟑𝟕 + 𝟒𝟓 = 𝟏𝟖.𝟕

= 14 +𝟐 𝟏𝟕 =𝟐𝟐.𝟐𝟒 P = 8+𝟔+ 𝟏𝟕 + 𝟏𝟕 = 𝑬𝑯 = (𝟑−(−𝟑) ) 𝟐 +(𝟐−𝟐 ) 𝟐 (−𝟑, 𝟐) (𝟑, 𝟐) P = 8+𝟔+ 𝟏𝟕 + 𝟏𝟕 = = 14 +𝟐 𝟏𝟕 (−𝟒, −𝟐) (𝟒, −𝟐) =𝟐𝟐.𝟐𝟒 𝑬𝑯 = (𝟑−(−𝟑) ) 𝟐 +(𝟐−𝟐 ) 𝟐 = ( 𝟔) 𝟐 +(𝟎 ) 𝟐 =𝟔 𝑯𝑮 = (𝟒−𝟑 ) 𝟐 +(−𝟐−𝟐 ) 𝟐 = ( 𝟏) 𝟐 +(−𝟒 ) 𝟐 = 𝟏𝟕 = ( 𝟖) 𝟐 +(𝟎 ) 𝟐 𝑭𝑮 = (𝟒 −(−𝟒) ) 𝟐 +(−𝟐−(−𝟐) ) 𝟐 =𝟖 𝑬𝑭 = (𝟒−(−𝟑) ) 𝟐 +(−𝟐−𝟐 ) 𝟐 = (−𝟏 ) 𝟐 +(−𝟒 ) 𝟐 = 𝟏𝟕

𝑨= 𝟏 𝟐 𝒃𝒉 = 𝟏 𝟐 𝟓𝟑 𝟔𝟓 𝑫(𝟐, −𝟒) =𝟐𝟗.𝟑𝟓 𝑩𝑫 = (𝟐−𝟎 ) 𝟐 +(−𝟒−𝟑 ) 𝟐 = ( 𝟐) 𝟐 +(−𝟕 ) 𝟐 = 𝟓𝟑 𝑨𝑪 = (𝟔−(−𝟐) ) 𝟐 +(−𝟑−(−𝟐) ) 𝟐 = ( 𝟖) 𝟐 +(−𝟏 ) 𝟐 = 𝟔𝟓

=𝟐𝟎 = 𝟒𝟎 = 𝟏𝟎 𝑨= 𝟏 𝟐 𝒃𝒉 = 𝟒𝟎 𝟏𝟎 𝑨𝑩 = (−𝟑−(−𝟏) ) 𝟐 +(−𝟒−𝟑 ) 𝟐 𝑨𝑩 = (−𝟑−(−𝟏) ) 𝟐 +(−𝟒−𝟑 ) 𝟐 = ( −𝟐) 𝟐 +(−𝟔 ) 𝟐 = 𝟒𝟎 𝑨𝑪 = (𝟎−(−𝟑) ) 𝟐 +(−𝟓−(−𝟒) ) 𝟐 = ( 𝟑) 𝟐 +(−𝟏 ) 𝟐 = 𝟏𝟎 𝑨= 𝟏 𝟐 𝒃𝒉 = 𝟒𝟎 𝟏𝟎 =𝟐𝟎

Unit 5: Geometric and Algebraic Connections 5.4 Midpoint and Direct Line Segments

Congruent marks Midpoint halfway

= 𝟑+(−𝟐) 𝟐 , 𝟕+𝟒 𝟐 = 𝟏 𝟐 , 𝟏𝟏 𝟐 = 𝟓+𝟔 𝟐 , −𝟐+𝟏𝟒 𝟐 = 𝟏𝟏 𝟐 , 𝟔

OR 𝒁(−𝟕, 𝟏) 𝑩(𝟒, 𝟏𝟎)

OR 𝒁(−𝟕, − 𝟖 𝟓 ) 𝑩(𝟑, 𝟏𝟑)

OR

OR

Unit 5: Geometric and Algebraic Connections 5.5 Equations of Circles

Trinomial 16 529 144 49 Completing the Square To complete the square: Take half, then square it! 16 529 144 49

(𝒙−𝟑) 𝟐 +( 𝒚−(−𝟐)) 𝟐 = 𝟒 𝟐 (𝒙−𝟑) 𝟐 +( 𝒚+𝟐) 𝟐 =𝟏𝟔 = (𝟏−𝟒 ) 𝟐 +(−𝟏− −𝟏 ) 𝟐 = ( −𝟓) 𝟐 +(𝟎 ) 𝟐 Radius: =𝟓 (𝒙−𝟒) 𝟐 +( 𝒚−(−𝟏)) 𝟐 = 𝟓 𝟐 (𝒙−𝟒) 𝟐 +( 𝒚+𝟏) 𝟐 =𝟐𝟓

(𝒙−𝟐) 𝟐 +( 𝒚−(−𝟗)) 𝟐 = 𝟏𝟏 𝟐 (𝒙−𝟐) 𝟐 +( 𝒚+𝟗) 𝟐 =𝟏𝟏 = (−𝟑−𝟐 ) 𝟐 +(𝟏𝟔−𝟒 ) 𝟐 = ( −𝟓) 𝟐 +(𝟏𝟐 ) 𝟐 Radius: =𝟏𝟑 (𝒙−𝟐) 𝟐 +( 𝒚−𝟒) 𝟐 = 𝟏𝟑 𝟐 (𝒙−𝟐) 𝟐 +( 𝒚−𝟒) 𝟐 =𝟏𝟔𝟗

(𝟔, −𝟑) 𝟓 = −𝟔+𝟐 𝟐 , 𝟑𝟐+𝟐𝟔 𝟐 = −𝟐, 𝟐𝟗 (−𝟑, 𝟑) 𝟔 Center: Radius: = −𝟔+𝟐 𝟐 , 𝟑𝟐+𝟐𝟔 𝟐 = −𝟐, 𝟐𝟗 Center: (−𝟑, 𝟑) Radius: = (−𝟐 −𝟐 ) 𝟐 +(𝟐𝟗−𝟐𝟔 ) 𝟐 Radius: 𝟔 = ( −𝟒) 𝟐 +(𝟑 ) 𝟐 =𝟓

(𝟒, −𝟑) 𝟔 = 𝟐+𝟏𝟐 𝟐 , 𝟖+𝟐𝟒 𝟐 = 𝟕, 𝟏𝟔 (𝟓, 𝟓) 𝟒 Center: Radius: Center: = 𝟐+𝟏𝟐 𝟐 , 𝟖+𝟐𝟒 𝟐 = 𝟕, 𝟏𝟔 Radius: Center: (𝟓, 𝟓) = (𝟕 −𝟐 ) 𝟐 +(𝟏𝟔−𝟖 ) 𝟐 = ( 𝟓) 𝟐 +(𝟖 ) 𝟐 Radius: 𝟒 = 𝟖𝟗

Center: (𝟒, 𝟎) Center: (𝟒, −𝟏) Radius: 𝟑 Radius: 𝟕

Center: (−𝟐, 𝟑) Center: (𝟐, 𝟑) Radius: 𝟒 Radius: 𝟓

(−𝟏, −𝟑) 𝟕 𝒙 𝟐 + 𝒚 𝟐 +𝟐𝒙+𝟔𝒚−𝟑𝟗=𝟎 (−𝟓, 𝟔) 𝟐 𝒙 𝟐 + 𝒚 𝟐 +𝟏𝟎𝒙−𝟏𝟐𝒚−𝟓𝟕=𝟎 Center: (−𝟏, −𝟑) Radius: 𝟕 𝒙 𝟐 + 𝒚 𝟐 +𝟐𝒙+𝟔𝒚−𝟑𝟗=𝟎 Center: (−𝟓, 𝟔) Radius: 𝟐 𝒙 𝟐 + 𝒚 𝟐 +𝟏𝟎𝒙−𝟏𝟐𝒚−𝟓𝟕=𝟎

yes yes

no no