1. Unit 5: Geometric and Algebraic Connections
Vocabulary Builder

4. 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

5. 3 1 1 2 2 3

7. y-axis
coordinates
x-axis
y-intercept

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

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

10. Slope
Undefined
Parallel
Congruent
Perpendicular -1

11. −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

12. −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

13. 𝒚= 𝟗 𝟏𝟏 𝒙− 𝟔 𝟏𝟏
𝒚= 𝟒 𝟓 𝒙+𝟐
𝒚=𝟑𝒙+𝟔

14. 𝒚=− 𝟔 𝟕 𝒙+ 𝟒𝟎 𝟕
𝒚=−𝟐𝒙
𝒚=−𝟑𝒙+𝟐

15. 𝒚=−𝟐𝒙+𝟏
𝒚=−𝟐𝒙+𝟐

16. 𝒚=𝟑𝒙+𝟏𝟑
𝒚= 𝟏 𝟑 𝒙+𝟑

19. Unit 5: Geometric and Algebraic Connections
5.2 Coordinate Proofs

20. Pythagorean Theorem hypotenuse Distance Formula difference difference
Square root
difference
difference

21. = 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

23. 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

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

25. 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

26. 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

27. 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

28. 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

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

32. sides
angles
Pythagorean Theorem
Distance Formula

33. 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

34. = 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

35. 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

36. =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

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

40. Congruent marks
Midpoint
halfway

41. = 𝟑+(−𝟐) 𝟐 , 𝟕+𝟒 𝟐
= 𝟏 𝟐 , 𝟏𝟏 𝟐
= 𝟓+𝟔 𝟐 , −𝟐+𝟏𝟒 𝟐
= 𝟏𝟏 𝟐 , 𝟔

43. OR
𝒁(−𝟕, 𝟏)
𝑩(𝟒, 𝟏𝟎)

44. OR
𝒁(−𝟕, − 𝟖 𝟓 )
𝑩(𝟑, 𝟏𝟑)

45. OR

46. OR

47. Unit 5: Geometric and Algebraic Connections
5.5 Equations of Circles

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

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

50. (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

51. (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

52. (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

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

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

55. (−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

56. yes
yes

57. no no