Unit 5: Geometric and Algebraic Connections

Vocabulary Builder

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

1

12

2

3 3

y-axis coordinates

x-axis

y-intercept

Parallelogram

Rhombus

Square

Four right angles

Diagonals are Perpendicular

Diagonals are Congruent

Unit 5: Geometric and Algebraic Connections

5.1 Slope and Equations of Lines

Slope

Undefined

Parallel CongruentPerpendicular

-1

−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

𝒚 = 𝟑𝒙 + 𝟔𝒚 =

𝟗

𝟏𝟏𝒙 −

𝟔

𝟏𝟏 𝒚 =𝟒

𝟓𝒙 + 𝟐

𝒚 = −𝟐𝒙𝒚 = −

𝟔

𝟕𝒙 +

𝟒𝟎

𝟕 𝒚 = −𝟏

𝟑𝒙 + 𝟐

𝒚 = −𝟐𝒙 + 𝟏

𝒚 = −𝟐𝒙 + 2

𝒚 = −𝟐𝒙 + 𝟏

𝒚 = −𝟐𝒙 + 𝟐

Unit 5: Geometric and Algebraic Connections

5.2 Coordinate Proofs

Pythagorean Theorem

hypotenuse

Distance Formula

Square root lengths

lengths

(−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

Unit 5: Geometric and Algebraic Connections

5.3 Perimeter and Area of Polygons

Pythagorean Theorem Distance Formula

anglessides

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

Unit 5: Geometric and Algebraic Connections

5.4 Midpoint and Direct Line Segments

Congruent marks

halfway

Midpoint

=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

Completing the Square

Trinomial

To complete the square: Take half, then square it!

16 529

144 49

(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

Center:

Radius:

(𝟒, 𝟎)

𝟑

Center:

Radius:

(𝟒,−𝟏)

𝟕

Center:

Radius:

(−𝟐, 𝟑)

𝟒

Center:

Radius:

(𝟐, 𝟑)

𝟓

𝒙𝟐 + 𝒚𝟐 + 𝟐𝒙 + 𝟔𝒚 − 𝟑𝟗 = 𝟎

Center: Radius:(−𝟏,−𝟑) 𝟕

𝒙𝟐 + 𝒚𝟐 + 𝟏𝟎𝒙 − 𝟏𝟐𝒚 − 𝟓𝟕 = 𝟎

Center: Radius:(−𝟓, 𝟔) 𝟐

yes

yes

no

no