Lesson 6.5 Proving Properties of Triangles and Quadrilaterals Concept: 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 4.2.4: Fitting Linear Functions to Data 1 Types of Triangles Types of Quadrilaterals Complete the Frayer Diagram with your knowledge of

shapes and their properties. Properties of Triangles Properties of Quadrilaterals 2 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Types of Triangles -Scalene -Isosceles -Equilateral -Right Properties of Triangles -Three sides and three angles -Scalene No sides the same

-Isosceles Two sides equal length -Equilateral All sides equal length -Right Right angle Types of Quadrilaterals -Square -Rectangle -Rhombus -Trapezoid Properties of Quadrilaterals -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 3 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Introduction In 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. 4 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: (x2 - x1)2 +(y2 - y1)2 5 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice Example 1 A 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. 6 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. 7 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, = . Slope of AB = 6 (2) (8 ) 3

( 1 ) ( 4 ) 2 (6) ( 2) 4 1 (7 ) ( 1 ) 8 2 (6 ) ( 8) 2 2 Slope of AC = (7 ) ( 4 ) 11 11 Slope of BC =

8 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 1, continued 3. Observe the slopes of each side. The slope of AB is 2 and the slope of BC is 1 . 2 The slopes of AB and BC are opposite reciprocals of each other which means they are perpendicular.

9 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. 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance 10 Guided Practice Example 2 A 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. 11 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 2, continued 1. Plot the quadrilateral on a coordinate plane. 12 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 2, continued 2. Show the figure has two pairs of parallel opposite sides. Calculate the slope of each side using the general slope formula, m= y2 - y1 x2 - x1 13 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 2, continued 3 ( 5 ) ( 2)

Slope of AB = (1 ) ( 1 ) 2 (3) (5) 2 2 Slope of BC = 3 (4 ) (1) 3 Slope of CD = 3 ( 0) (3 ) 3 2 ( 2) (4 ) 2

2 2 ( 0 ) ( 2 ) Slope of AD = 3 3 ( 2 ) ( 1 ) 14 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance

Guided Practice: Example 2, continued 3. Observe the slopes of each side. The side opposite AB is CD. The slopes of these sides are the same. The side opposite BC is AD. The slopes of these sides are the same. This shows that the quadrilateral has two pairs of parallel opposite sides. 15 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 2, continued Sides AB and BC 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.

16 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 2, continued 4. Show that the quadrilateral has four congruent sides. Find the length of each side using the distance 2 2 formula, d = (x2 - x1) +(y2 - y1) . length of AB = (1- (- 1))2 +(5 - 2)2 = (2)2 +(3)2 = 4 + 9 = 13 length of BC = (4 - 1)2 +(3 - 5)2 = (3)2 +(- 2)2 = 9 + 4 = 13 17 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 2, continued

length of CD = (2 - 4)2 +(0 - 3)2 = (- 2)2 +(- 3)2 = 4 + 9 = 13 length of AD = (2 - (- 1))2 +(0 - 2)2 = (3)2 +(- 2)2 = 9 + 4 = 13 Thus, the lengths of all four sides are congruent. 18 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 2, continued 5. 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.

6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance 19 Guided Practice Example 3 Use the distance formula and slope to determine the shape of the figure. A (1, 4 ) B (3, 2) ( 4, 2) D C

(0, 4) 20 Guided Practice: Example 3, continued 1. First find the slope of each side. Calculate the slope of each side using the general slope formula, m= y2 - y1 x2 - x1 21 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 3, continued

1 (2) ( 4) 2 Slope of AB: 4 2 (3) ( 1) 6 Slope of BC: 3 2 2 1 Slope of CD: 4

2 A (1, 4 ) B (3, 2) (2)( 4) 6 Slope of AD: 2 3 (3) (1)

( 4, 2) D C 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance (0, 4) 22 Guided Practice: Example 3, continued 1 Slope of AB: 2 2. Observe the slopes of each side. The slopes for AB and CD are Slope of BC: 2 the same and BC and DA are 1 the same; this makes these

Slope of CD: 2 parallel lines! The slopes for AB and CD are Slope of AD: 2 the opposite reciprocal of BC and AD. This makes these lines perpendicular, which means they meet at a angle! 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance 23 Guided Practice: Example 3, continued 3. Find and observe the side lengths of the quadrilateral. Find the length of each side using the distance 2 2

formula, d = (x2 - x1) +(y2 - y1) . 24 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: A (1, 4 ) B (3, 2) 20 4.47 ( 4, 2) D

C (0, 4) 25 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 3, continued The length from B to C is: A (1, 4 ) B (3, 2) 45 6.71

( 4, 2) D C (0, 4) 26 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 3, continued The length from C to D is: A (1, 4 ) B (3, 2)

20 4.47 ( 4, 2) D C (0, 4) 27 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 3, continued The length from B to C is: A (1, 4 ) B

(3, 2) 45 6.71 ( 4, 2) D C (0, 4) 28 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Guided Practice: Example 3, continued Distance of AB 4. Observe the distances Distance of BC Distance of CD

Distance of AD 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. 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance 29 Guided Practice: Example 3, continued 5. 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. 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance 30 Guided Practice

You 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. 31 Since it is a three sided figure, we can eliminate the possibility of a quadrilateral. Label each point and find the slope of each side: KL LM K MK (3, 2) Notice that none of the

slopes are the same or M L opposite reciprocals. This means there are no (1, 0) (5, 0) parallel or perpendicular lines. 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance 32 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. K M (1, 0) (3, 2) L (5, 0)

33 6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance Use the words listed below summarize the lesson for today in a few sentences. Distance Perpendicular Same Opposite Length Parallel Slope Side Consecutive

6.1.2: Using Coordinates to Prove Geometric Theorems with Slope and Distance 34