Distance Formula
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
Distance Formula
𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
Midpoint Formula
𝑥1 + 𝑥2
𝑦1 + 𝑦2
𝑥=
,𝑦 =
2
2
Midpoint Formula
𝑥1 + 𝑥2
𝑦1 + 𝑦2
𝑥=
,𝑦 =
2
2
Division of a Line Segment
𝑟1 𝑥2 + 𝑟2 𝑥1
𝑟1 𝑦2 + 𝑟2 𝑦1
𝑥=
,𝑦 =
𝑟1 + 𝑟2
𝑟1 + 𝑟2
Division of a Line Segment
𝑟1 𝑥2 + 𝑟2 𝑥1
𝑟1 𝑦2 + 𝑟2 𝑦1
𝑥=
,𝑦 =
𝑟1 + 𝑟2
𝑟1 + 𝑟2
Linear Equation (General Form)
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
Linear Equation (General Form)
𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
Slope
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
Slope
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
Two-Point Form
𝑦2 − 𝑦1 𝑦 − 𝑦1
=
𝑥2 − 𝑥1 𝑥 − 𝑥1
Two-Point Form
𝑦2 − 𝑦1 𝑦 − 𝑦1
=
𝑥2 − 𝑥1 𝑥 − 𝑥1
Point Slope Form
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Point Slope Form
𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
y-Intercept Form
𝑦 = 𝑚𝑥 + 𝑏
y-Intercept Form
𝑦 = 𝑚𝑥 + 𝑏
Intercept Form
𝑥 𝑦
+ =1
𝑎 𝑏
Intercept Form
𝑥 𝑦
+ =1
𝑎 𝑏
Parallel and Perpendicular Lines
Parallel and Perpendicular Lines
𝐼𝑓 𝑚1 = 𝑚2 , 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙
𝐼𝑓 𝑚1 = 𝑚2 , 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙
1
,
𝑚1
𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟
1
,
𝑚1
𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠 𝑎𝑟𝑒 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟
𝐼𝑓 𝑚2 = −
𝐼𝑓 𝑚2 = −
Equation of a Circle (Standard Form)
Equation of a Circle (Standard Form)
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒 = (ℎ, 𝑘)
𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒 = 𝑟
𝑐𝑒𝑛𝑡𝑒𝑟 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒 = (ℎ, 𝑘)
𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒 = 𝑟
𝑖𝑓 𝑟 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑧𝑒𝑟𝑜, 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑎 𝑝𝑜𝑖𝑛𝑡
𝑖𝑓 𝑟 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑧𝑒𝑟𝑜, 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑎 𝑝𝑜𝑖𝑛𝑡
Equation of a Circle (General Form)
Equation of a Circle (General Form)
𝐴𝑥 2 + 𝐵𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
𝐴𝑥 2 + 𝐵𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝐵 𝑎𝑛𝑑 𝑖𝑠 𝑁𝑂𝑇 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑧𝑒𝑟𝑜
𝑤ℎ𝑒𝑟𝑒 𝐴 = 𝐵 𝑎𝑛𝑑 𝑖𝑠 𝑁𝑂𝑇 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑧𝑒𝑟𝑜
𝑡ℎ𝑖𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑐𝑎𝑛 𝑏𝑒 𝑟𝑒𝑑𝑢𝑐𝑒𝑑 𝑡𝑜:
𝑥 2 + 𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
𝑡ℎ𝑖𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑐𝑎𝑛 𝑏𝑒 𝑟𝑒𝑑𝑢𝑐𝑒𝑑 𝑡𝑜:
𝑥 2 + 𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
Point-Line Distance Formula
Point-Line Distance Formula
𝑑=
|𝐴𝑥1 + 𝐵𝑦1 + 𝐶|
√𝐴2 + 𝐵2
𝑑=
|𝐴𝑥1 + 𝐵𝑦1 + 𝐶|
√𝐴2 + 𝐵2