Lesson 6.5 Proving Properties of Triangles and Quadrilaterals

Lesson 6.5Proving Properties of Triangles and

QuadrilateralsConcept: Proofs in the Coordinate Plane

EQs: -How do we prove the properties of triangles and quadrilaterals in the coordinate plane? (G.GPE.4)

Vocabulary: Square, Rhombus, Rectangle, Parallelogram, Trapezoid, Isosceles Triangle, Right Triangle, Equilateral Triangle

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4.2.4: Fitting Linear Functions to Data

Complete the Frayer Diagram with your knowledge of shapes and their properties.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Types of Triangles Types of Quadrilaterals

Properties of Triangles Properties of Quadrilaterals

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Types of Triangles Types of Quadrilaterals-Scalene-Isosceles-Equilateral-Right

-Square-Rectangle-Rhombus-Trapezoid

Properties of Triangles Properties of Quadrilaterals-Three sides and three angles-Scalene – No sides the same-Isosceles – Two sides equal length-Equilateral – All sides equal length-Right – Right angle

-Four sides and four angles -Square – Equal & Parallel sides and four right angles-Rectangle – Two sets of equal and parallel lines; four right angles-Rhombus - Two sets of equal and parallel lines-Trapezoid – One set of parallel lines

IntroductionIn this lesson, we will use the distance formula and what we know about parallel and perpendicular lines to prove the properties of triangles and quadrilaterals in the coordinate plane.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Key Concepts to Recall:• Formula to find slope:

• Parallel lines have the SAME slope and NEVER meet

• Perpendicular lines have the OPPOSITE

RECIPROCAL slopes and meet at a angle

• The Distance Formula:

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided PracticeExample 1A right triangle is defined as a triangle with 2 sides that are perpendicular. Triangle ABC has vertices A (–4, 8), B (–1, 2), and C (7, 6). Determine if this triangle is a right triangle. When disproving a figure, you only need to show one condition is not met.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 1, continued

1. Plot the triangle on a coordinate plane.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 1, continued

2. Calculate the slope of each side using the

general slope formula, .

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Slope of AB = ¿−63

(2 )− ( 8 )(−1 )− (−4 ) ¿−2

Slope of BC = ¿48

(6 )− (2 )(7 )− (−1 )

¿12

Slope of AC = ¿−211

(6 )− (8 )(7 )− (−4 ) ¿− 2

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𝒎=𝒚𝟐−𝒚𝟏

𝒙𝟐−𝒙𝟏

Guided Practice: Example 1, continued

3. Observe the slopes of each side.The slope of is –2 and the slope of BC is .

• The slopes of AB and BC are opposite reciprocals of each other which means they are perpendicular.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 1, continued

4. Make connections.• Right triangles have two sides that are

perpendicular.

• Triangle ABC has two sides that are perpendicular; therefore, it is a right triangle.

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✔6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided PracticeExample 2A square is a quadrilateral with two pairs of parallel opposite sides, consecutive sides that are perpendicular, and all sides congruent, meaning they have the same length. Quadrilateral ABCD has vertices A (–1, 2), B (1, 5), C (4, 3), and D (2, 0). Determine if this quadrilateral is a square.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 2, continued1. Plot the

quadrilateral on a coordinate plane.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 2, continued2. Show the figure has two pairs of parallel

opposite sides.Calculate the slope of each side using the general

slope formula,

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 2, continued

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Slope of AB = ¿32

(5 )− (2 )(1 )− (−1 )

Slope of BC = ¿−23

(3 )− (5 )(4 )− (1 )

¿− 23

Slope of CD = ¿−3−2

(0 )− (3 )(2 )− (4 )

¿32

Slope of AD = ¿−23

(0 )− (2 )(2 )− (−1 )

¿− 23

Guided Practice: Example 2, continued3. Observe the slopes of each side.

• The side opposite is . The slopes of these sides are the same.

• The side opposite is . The slopes of these sides are the same.

This shows that the quadrilateral has two pairs of parallel opposite sides.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 2, continued• Sides and are consecutive sides and their

slopes are opposite reciprocals.

• This is the case for sides BC and CD ; CD and AD; as well as AB and AD.

Thus, the consecutive sides are perpendicular.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 2, continued4. Show that the quadrilateral has four

congruent sides.Find the length of each side using the distance

formula, .

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 2, continued

Thus, the lengths of all four sides are congruent.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 2, continued5. Make connections.Recall: A square is a quadrilateral with two pairs of parallel opposite sides, consecutive sides that are perpendicular, and all sides congruent.

Quadrilateral ABCD has two pairs of parallel opposite sides, the consecutive sides are perpendicular, and all the sides are congruent. It is a square.

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✔6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided PracticeExample 3Use the distance formula and slope to determine the shape of the figure.

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A

B

C

D

(−1 , 4)(3 ,2)

(0 ,−4 )(−4 ,−2)

Guided Practice: Example 3, continued1. First find the slope of each side.

Calculate the slope of each side using the general

slope formula,

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 3, continued

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

A

B

C

D

(−1 , 4)(3 ,2)

(0 ,−4 )(−4 ,−2)

Slope of AB:

Slope of BC:

Slope of CD:

Slope of AD:

(2)−(4)(3)−(−1)

¿−24¿−

12

¿−6−3¿2

¿− 12

¿2−4

(2)−(4)(3)−(−1)

¿−6−3¿2

Guided Practice: Example 3, continued

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Slope of AB:

Slope of BC:

Slope of CD:

Slope of AD:

¿− 12

¿2

¿− 12

¿2

2. Observe the slopes of each side.• The slopes for AB and CD are

the same and BC and DA are the same; this makes these parallel lines!

• The slopes for AB and CD are the opposite reciprocal of BC and AD. This makes these lines perpendicular, which means they meet at a angle!

Guided Practice: Example 3, continued3. Find and observe the side lengths of the quadrilateral.

Find the length of each side using the distance

formula, .

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 3, continued

The length from A to B is:

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

A

B

C

D

(−1 , 4)(3 ,2)

(0 ,−4 )(−4 ,−2)

√20≈ 4.47

Guided Practice: Example 3, continued

The length from B to C is:

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

A

B

C

D

(−1 , 4)(3 ,2)

(0 ,−4 )(−4 ,−2)

√ 45≈6.71

Guided Practice: Example 3, continued

The length from C to D is:

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

A

B

C

D

(−1 , 4)(3 ,2)

(0 ,−4 )(−4 ,−2)

√20≈ 4.47

Guided Practice: Example 3, continued

The length from B to C is:

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

A

B

C

D

(−1 , 4)(3 ,2)

(0 ,−4 )(−4 ,−2)

√ 45≈6.71

Guided Practice: Example 3, continued

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Distance of AB

Distance of BC

Distance of CD

Distance of AD

4. Observe the distances of each side.• The distances of AB and CD

are both 4.47 and the distances of BC and DA are both 6.71.

This shows us that the opposite sides are of equal length.

Guided Practice: Example 3, continued5. Make connections.

• A rectangle and a square are both quadrilaterals with two pairs of parallel opposite sides and consecutive sides that are perpendicular

• A rectangle is a quadrilateral with two pairs of opposite sides of equal length.

• Quadrilateral ABCD has two pairs of parallel opposite sides, the consecutive sides are perpendicular, and the opposite sides are of equal length.

• This shape is a rectangle.30✔

6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided PracticeYou Try!

What shape is formed by connecting the points and ?

Prove your answer using your knowledge of parallel and perpendicular lines and the

distance and midpoint formulas.

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Since it is a three sided figure, we can eliminate the possibility of a quadrilateral.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

K

LM

(1 ,0) (5 ,0)

(3 ,2)

Label each point and find the slope of each side:KL LM MK Notice that none of the slopes are the same or opposite reciprocals. This means there are no parallel or perpendicular lines.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

K

LM

(1 ,0) (5 ,0)

(3 ,2)

Find the length of each side:KL LM MK

Since there were no parallel or perpendicular lines and since two sides are equal, we know this must be an isosceles triangle.

Use the words listed below summarize the lesson for today in a few sentences.

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6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Distance

Same

Perpendicular Slope

Parallel

SideOpposite

Length

Consecutive