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Unit 6: Connecting Algebra and Geometry Through CoordinatesUNIT 6: CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATESUse Coordinates to Prove Simple Geometric Theorems AlgebraicallyMGSE9-12.G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle.

MGSE9-12.G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

MGSE9-12.G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

MGSE9-12.G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

KEY IDEAS

1. The distance between points x y,1 1( ) and ,2 2x y( ) is – + – .2 12

2 1

2x x y y( ) ( )

2. The slope of the line through points ,1 1x y( ) and ,2 2x y( ) is–

–.2 1

2 1

y y

x x3. Slopes can be positive, negative, 0, or undefined.

4. Lines and their slopes are related by the following properties:

a. Two nonvertical lines are parallel if and only if they have equal slopes but different y-intercepts.

b. Two nonvertical lines are perpendicular if and only if the product of their slopes is –1.

5. Some useful properties of proportions state that all of the following are equivalent:

=ab

cd

ad = bc =ac

bd

Example:Use the multiplication property of equality to multiply each side of the proportion.

Because =106

36

5, you can also write 5 · 6 = 3 · 10 and

5= .

3 10Multiply each side by 18, a common multiple of 3 and 6.

53

(18) = (18)106

5 · 6 = 3 · 10

Multiply each side by 3

10.

· ·310

53

=3

10106

510

=36

Unit 6: Connecting Algebra and Geometry Through Coordinates6. A directed line segment is a line segment from one point to another point in the coordinate plane.

Example:_

Notice on PQ we can subtract to find the difference of the x- and y-values of Q and P : (9 – 5, 6 – 4) = (4, 2). They tell you that a “route” from P to Q is 4 units right and 2 units up. Note that these are used in the slope:

= =6 – 49 – 5

24

12

Important Tip

When using directed line segments, pay close attention to the beginning and end points of the line. For example, the directed line segments PQ and QP have the same length, but different directions.

7. The perimeter of a polygon is the sum of the lengths of the sides. The area of a polygon is the number of square unitsenclosed by the polygon.

For a triangle with side lengths a, b, c, with side b as the base and height h, the perimeter P and area A are:

P = a + b + c

=A bh12

For a rectangle with length l and width w, the perimeter P and area A are: P = 2l + 2w

A = lwImportant Tip

In a triangle, any side can be used as the base. The corresponding height is the altitude drawn to the line containing that base. In a right triangle, the legs can be used as the base and height.

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Unit 6: Connecting Algebra and Geometry Through Coordinates2. Prove that if two nonvertical lines have equal slopes, then they are parallel.

Solution:

Use a proof by contradiction. Assume that the lines have equal slopes but are not parallel—that is, assume the lines intersect. If you can show this is not true, it is equivalent to proving the original statement. Write the equations for both lines. The slopes are the same, so use m for the slope of each line. The two lines are different, so b1 ≠ b2.

Equation for line n: y = mx + b1Equation for line m: y = mx + b2

Solve a system of equations to find the point of intersection. Both equations are solved for y, so use substitution.mx + b1 = mx + b2

b1 = b2

Because it was assumed that b1 ≠ b2, this is a contradiction. So the original statement is true—if two nonvertical lines haveequal slopes, then they are parallel.

3. The line p is represented by the equation y = 4x + 1. What is the equation of the line that is perpendicular toline p and passes through the point (8, 5)?

4. For what value of n are the lines 7 +x y3 = 8and nx + 3y = 8 perpendicular?

5. Quadrilateral ABCD has vertices A(4, 0), B(3, 3), C(–3, 1),and D(–2, –2). Prove that ABCD is a rectangle.

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Unit 6: Connecting Algebra and Geometry Through Coordinates

7. Find the area of rectangle ABCD with vertices A(–3, 0), B(3, 2), C(4, –1), and D(–2, –3).

SAMPLE ITEMS

1 . An equation of line a is y = –12

– 2x .

Which is an equation of the line that is perpendicular to line a and passes through the point (–4, 0)?

A . = –y x12

+ 2

B . = –y x12

+ 8

C . = 2y x – 2

D . = 2y x + 8

Be careful when using directed line segments. If point P partitions AB in the ratio 1:3, then point P partitions BA in the ratio

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6. Given the points A(–1, 2) and B(7, 8), find the coordinates of the point P on directed line segment AB that partitions AB in the ratio 1:3.

Important Tip

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Unit 6: Connecting Algebra and Geometry Through Coordinates

3 . Given the points P(11, 2) and Q(–13, –5), what are the coordinates of the point

on directed line segment PQ that partitions PQ in the ratio 3:2?

A . (– 17 ___ 5 , – 11 ___ 5 )B . ( 7 ___ 24 , – 29 ___ 24 ) C . (–1, – 3 __ 2 ) D . (–5, – 8 __ 3 )

4 . Triangle ABC has vertices as shown .

What is the area of the triangle?

A . 72 square unitsB . 12 square units

C . 288 square unitsD . 24 square units

2. Triangle KLM has vertices as shown.

Which relationship about the slopes of triangle KLM proves that it’s a right triangle?

A . }LK ⟘ }}

KM because the product of the slopes of these segments is –1.B . }KL ⟘ LM because the product of the slopes of these segments is –1.

} LM because the sum of the slopes of these segments is less C . }KL than 1.

D . } LK } KM because the sum of the slopes of these segments is greaterthan 1.

The Georgia Milestones Assessment SystemGeorgia Milestones End-Of-Course (Eoc) Assessments

How to Use This GuideOverview of the Coordinate Algebra EOC AssessmentItem TypesDepth of Knowledge DescriptorsDepth of Knowledge Example ItemsDescription of Test Format and Organization

Preparing for the Coordinate Algebra EOC AssessmentStudy SkillsOrganization—or Taking Control of Your WorldActive ParticipationTest-Taking StrategiesPREPARING FOR THE COORDINATE ALGEBRA EOC ASSESSMENT

Content of the Coordinate Algebra EOC AssessmentSnapshot of the CourseUnit 1: Relationships Between QuantitiesUnit 2: Reasoning with Equations and InequalitiesUnit 3: Linear and Exponential Functions Unit 4: Describing DataUnit 5: Transformations in the Coordinate PlaneUnit 6: Connecting Algebra and Geometry Through Coordinates

Coordinate Algebra Additional Practice ItemsAdditional Practice Items Answer KeyAdditional Practice Items Scoring Rubrics and Exemplar Responses

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