Unit 5: Geometric and Algebraic Connections · Unit 5: Geometric and Algebraic Connections 5.3...
Transcript of Unit 5: Geometric and Algebraic Connections · Unit 5: Geometric and Algebraic Connections 5.3...
Identify opposite sides
Find the slope of 𝑨𝑫.
Find the slope of 𝑨𝑫.
Find the slope of 𝑨𝑩.
Find the slope of 𝑫𝑪.
Compare slopes, opposite sides have the
Same slope, then they are parallel.
ABCD is a parallelogram
−𝟏𝟎 − (−𝟐𝟎)
𝟏𝟏 − (−𝟏𝟑)
=𝟏𝟎
𝟐𝟒=
𝟓
𝟏𝟐
(𝟑, 𝟏)
(𝟎, 𝟐)𝟏 − (−𝟐)
𝟑 − 𝟎=𝟑
𝟑= 𝟏
Slope: 3
Parallel Slope: 3
Perpendicular Slope: −𝟏
𝟑
Slope: −𝟒
𝟓
Parallel Slope: −𝟒
𝟓
Perpendicular Slope: 𝟓
𝟒
−𝟏𝟓 − 𝟖
−𝟏𝟏 − (−𝟐𝟎)
=𝟕
𝟗
(−𝟑, 𝟑)
(𝟏, −𝟑)
−𝟑 − 𝟑
𝟏 − (−𝟑)
= −𝟔
𝟒= −
𝟑
𝟐
Slope: 𝟐
𝟓
Parallel Slope: 𝟐
𝟓
Perpendicular Slope: −𝟓
𝟐
Slope: −𝟏
Parallel Slope: −𝟏
Perpendicular Slope: 𝟏
(−𝟐 − 𝟏)𝟐 + (𝟑 − 𝟒)𝟐 = (−𝟑)𝟐 + (−𝟏)𝟐
= 𝟗 + 𝟏= 𝟏𝟎
(𝟒𝟎 − 𝟏𝟎)𝟐 + (𝟒𝟓 − 𝟓)𝟐 = (𝟑𝟎)𝟐 + (𝟒𝟎)𝟐 = 𝟗𝟎𝟎 + 𝟏𝟔𝟎𝟎
= 𝟓𝟎
𝑩𝑪 = (𝟓 − 𝟏)𝟐 + (𝟔 − 𝟑)𝟐 = (𝟑)𝟐 + (𝟒)𝟐 = 𝟓
Use Distance Formula to prove congruent length
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=𝟑
𝟒𝑨𝑪 =
−𝟏 − 𝟑
𝟓 − 𝟏= −
𝟒
𝟑
Use Slope Formula to prove a right triangle
𝑩𝑪 =−𝟏 − 𝟑
𝟐 − 𝟐=−𝟒
𝟎𝑨𝑪 =
−𝟏 − (−𝟏)
𝟐 − (−𝟑)=𝟎
𝟓
Undefined 0
𝑨𝑩 = (𝟐 − 𝟏)𝟐 + (𝟓 − 𝟐)𝟐
Use Distance Formula to prove congruent length
𝑪𝑫 = (𝟒 − 𝟓)𝟐 + (𝟒 − 𝟕)𝟐
= 𝟏𝟎
= 𝟏𝟎
𝑩𝑪 = (𝟓 − 𝟐)𝟐 + (𝟕 − 𝟓)𝟐 = 𝟏𝟑
𝑨𝑫 = (𝟒 − 𝟏)𝟐 + (𝟒 − 𝟐)𝟐 = 𝟏𝟑
𝑨𝑩 = (−𝟐 − (−𝟑))𝟐 + (𝟔 − 𝟐)𝟐
𝑪𝑫 = (𝟏 − 𝟐)𝟐 + (𝟑 − 𝟕)𝟐
= 𝟏𝟕
= 𝟏𝟕
𝑩𝑪 = (𝟐 − (−𝟐))𝟐 + (𝟕 − 𝟔)𝟐 = 𝟏𝟕
𝑨𝑫 = (𝟏 − (−𝟑))𝟐 + (𝟑 − 𝟐)𝟐 = 𝟏𝟕
Use Slope Formula to prove a right triangle
𝑨𝑩 =𝟐 − 𝟎
𝟑 − (−𝟑)=𝟏
𝟑𝑨𝑫 =
−𝟑 − 𝟎
−𝟐 − (−𝟑)= −𝟑
𝑨𝑩 = (𝟑 − (−𝟑))𝟐 + (𝟐 − 𝟎)𝟐= (𝟔)𝟐 + (𝟐)𝟐
Use Distance Formula to prove congruent length
= 𝟒𝟎
𝑪𝑫 = (−𝟐 − 𝟒)𝟐 + (−𝟑 − (−𝟏))𝟐 = (−𝟔)𝟐 + (−𝟐)𝟐 = 𝟒𝟎
Use Slope Formula to prove a right triangle
𝑨𝑩 =𝟒 − 𝟎
𝟎 − (−𝟑)=𝟒
𝟑𝑨𝑫 =
−𝟑 − 𝟎
−𝟏 − (−𝟑)
𝑨𝑩 = (𝟎 − (−𝟑))𝟐 + (𝟒 − 𝟎)𝟐= (𝟑)𝟐 + (𝟒)𝟐
Use Distance Formula to prove congruent length
= 𝟓
𝑩𝑪 = (𝟒 − 𝟎)𝟐 + (𝟏 − 𝟒)𝟐 = (𝟒)𝟐 + (−𝟑)𝟐 = 𝟓
= −𝟑
𝟒
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
(−𝟏, 𝟒)
(−𝟒, 𝟏)
(𝟐, −𝟐)
P = 𝟑𝟒 + 𝟑𝟕 + 𝟒𝟓 = 𝟏𝟖. 𝟔𝟐
(−𝟑, 𝟐) (𝟑, 𝟐)
(−𝟒,−𝟐) (𝟒, −𝟐)
𝑬𝑯 = (𝟑 − (−𝟑))𝟐 + (𝟐 − 𝟐)𝟐= (𝟔)𝟐 + (𝟎)𝟐 = 𝟔
𝑯𝑮 = (𝟒 − 𝟑)𝟐 + (−𝟐 − 𝟐)𝟐 = (𝟏)𝟐 + (−𝟒)𝟐= 𝟏𝟕
𝑭𝑮 = (𝟒 − (−𝟒))𝟐 + (−𝟐 − (−𝟐))𝟐= (𝟖)𝟐 + (𝟎)𝟐 = 𝟖
𝑬𝑭 = (𝟒 − (−𝟑))𝟐 + (−𝟐 − 𝟐)𝟐 = (−𝟏)𝟐 + (−𝟒)𝟐 = 𝟏𝟕
P = 8 + 𝟔 + 𝟏𝟕 + 𝟒𝟕 =
= 𝟐𝟐. 𝟐𝟒
= 14 +𝟐 𝟏𝟕
𝑩𝑫 = (𝟐 − 𝟎)𝟐 + (−𝟒 − 𝟑)𝟐 = (𝟐)𝟐 + (−𝟕)𝟐 = 𝟓𝟑
𝑨𝑪 = (𝟔 − (−𝟐))𝟐 + (−𝟑 − (−𝟐))𝟐= (𝟖)𝟐 + (−𝟏)𝟐 = 𝟔𝟓
𝑨 =𝟏
𝟐𝒃𝒉
=𝟏
𝟐𝟓𝟑 𝟔𝟓
= 𝟐𝟗. 𝟑𝟓
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
𝟐𝒃𝒉 = 𝟒𝟎 𝟏𝟎 = 𝟐𝟎
(𝒙 − 𝟑)𝟐+(𝒚 − (−𝟐))𝟐 = 𝟒𝟐
(𝒙 − 𝟑)𝟐+(𝒚 + 𝟐)𝟐 = 𝟏𝟔
Radius: = (𝟏 − 𝟒)𝟐 + (−𝟏 − −𝟏 )𝟐= (−𝟓)𝟐 + (𝟎)𝟐 = 𝟓
(𝒙 − 𝟒)𝟐+(𝒚 − (−𝟏))𝟐 = 𝟓𝟐
(𝒙 − 𝟒)𝟐+(𝒚 + 𝟏)𝟐 = 𝟐𝟓
(𝒙 − 𝟐)𝟐+(𝒚 − (−𝟗))𝟐 = 𝟏𝟏𝟐
(𝒙 − 𝟐)𝟐+(𝒚 + 𝟗)𝟐 = 𝟏𝟏
Radius: = (−𝟑 − 𝟐)𝟐 + (𝟏𝟔 − 𝟒)𝟐 = (−𝟓)𝟐 + (𝟏𝟐)𝟐= 𝟏𝟑
(𝒙 − 𝟐)𝟐+(𝒚 − 𝟒)𝟐 = 𝟏𝟑𝟐
(𝒙 − 𝟐)𝟐+(𝒚 − 𝟒)𝟐 = 𝟏𝟔𝟗
Center:
Radius:
(𝟔,−𝟑)
𝟓
Center:
Radius:Center:
Radius:
(−𝟑, 𝟑)
𝟔
=−𝟔 + 𝟐
𝟐,𝟑𝟐 + 𝟐𝟔
𝟐= −𝟐, 𝟐𝟗
= (−𝟐 − 𝟐)𝟐 + (𝟐𝟗 − 𝟐𝟔)𝟐
= (−𝟒)𝟐 + (𝟑)𝟐 = 𝟓
Center:
Radius:
(𝟒,−𝟑)
𝟔
Center:
Radius:Center:
Radius:
(𝟓, 𝟓)
𝟒
=𝟐 + 𝟏𝟐
𝟐,𝟖 + 𝟐𝟒
𝟐= 𝟕, 𝟏𝟔
= (𝟕 − 𝟐)𝟐 + (𝟏𝟔 − 𝟖)𝟐
= (𝟓)𝟐 + (𝟖)𝟐 = 𝟖𝟗
𝒙𝟐 + 𝒚𝟐 + 𝟐𝒙 + 𝟔𝒚 − 𝟑𝟗 = 𝟎
Center: Radius:(−𝟏,−𝟑) 𝟕
𝒙𝟐 + 𝒚𝟐 + 𝟏𝟎𝒙 − 𝟏𝟐𝒚 − 𝟓𝟕 = 𝟎
Center: Radius:(−𝟓, 𝟔) 𝟐