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

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

3. = 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

5. 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 𝟒−𝟏
= 𝟑 𝟒
=− 𝟒 𝟑

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

7. 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

8. 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

9. 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

10. 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