Distance Formula Date 3/6/19 We are taking notes, not doing a task. Copy down the EQ Work on Warm Up Essential question How is the Distance formula related to Pythagorean Theorem? Warm up: Copy down the purpose Students will formulate a way to calculate the distance between two points. In the process, they will make the connection between the distance formula and the Pythagorean Theorem.

Purpose Students will formulate a way to calculate the distance between two points. In the process, they will make the connection between the distance formula and the Pythagorean Theorem.

We Label it as an order pair ( x, y) Introduction: Label Coordinate Plane Where is the Sports Store? Answer: It is at a location What is a location in terms of Geometry? The answer is a point. How do we label a point? We Label it as an order pair ( x, y) Example : The Sports Store is at ( 3, 9) 3 on the x-axis and 9 on the y-axis

Introduction: Label Coordinate Plane Jewel Accessories = ( 8, 4) Baby Clothes = ( 13, 4) Movie Store = (7, 13) Bathrooms = ( 7 , 9 ) Teacher Store = ( 7 , 7) Clothes = ( 11, 10 ) Jewel accessories = ( , ) Baby clothes = ( , ) Movie Store = ( , ) Bathrooms = ( , ) Teacher Store = ( , ) Clothes = ( , )

Part 1) Lesson Question on the X-Axis 1. What is the distance between the Jewel accessories and the Baby clothes? Answer: 5 Units Using the Coordinate Points. What is the distance between two points on the x-axis?

Part 1) Lesson Question on the X-Axis 1. What is the distance between the Jewel accessories and the Baby clothes? Solution: Jewel Accessories is ( 8,4) Baby Clothes is (13, 4) I started at Jewel Accessories (8 ,4), then Baby Clothes (13,4) ( 𝑋 1 , 𝑌 1 ) ( 𝑋 2 , 𝑌 2 ) Solution: 13 - 8 = 5 2. What is the distance between two points on the x-axis? 𝑋 2 - 𝑋 1 = Distance in the x-Axis

Part 1) Lets Practices 3. What is the distance between the Print shop and the Baby Clothes? Print Shop ( 10, 4) 𝑋 1 = 10 Baby Clothes (13, 4) 𝑋 2 =13 𝑋 2 - 𝑋 1 = Distance in the x-Axis 13 - 10 = Distance in the x-Axis 3= Distance in the x-Axis 4. What is the distance between the Cell Phones and the Animal Store? Cell Phones( 17, 7) 𝑋 1 = 17 Animal Store (23, 7) 𝑋 2 =23 23 - 17 =6 6= Distance in the x-Axis 5. !!!Not On the Map!!! What is the distance between the Game Stop and The Gap? Game Stop ( 1,4) GAP(9, 4) 6 = Distance in the x-Axis

Part 3) Lesson Question on the Y-Axis 6. What is the distance between the Movie Store and the Bathrooms? I walked from the bathrooms to the Movie Store Answer: 4 Units Using the Coordinate Points. What is the distance between two points on the x-axis?

Part 3) Lesson Question on the Y-Axis 6. What is the distance between the Movie store and the Bathrooms? Movie Store is ( 7,13) Bathroom is (7, 9) I started at Bathroom( 7, 9 ) , then Movie Store (7, 13) ( 𝑋 1 , 𝑌 1 ) ( 𝑋 2 , 𝑌 2 ) Solution: 13 - 9 = 4 7. What is the distance between two points on the Y-axis? 𝑌 2 - 𝑌 1 = Distance in the Y-Axis

Part 4) Lets Practice 8. What is the distance between the Party Store and the hair Salon? Party Store ( 3, 7) 𝑌 1 = 7 Hair Salon (3, 13) 𝑌 2 =13 𝑌 2 - 𝑌 1 = Distance in the Y-Axis 13 - 7 = Distance in the Y-Axis 6= Distance in the Y-Axis 9 . !!!Not On the Map!!! What is the distance between Target and Subway? Target( 11,4) Target (21, 4) 10= Distance in the Y-Axis

Part 5) Can we have negative distance? Answer: NO 10. What is the distance between the Music Store and the Teacher Store? I started at the Music Store then we to the Teacher store Music Store ( 7, 11) 𝑌 1 = 11 Teacher Store (7,7) 𝑌 2 =7 𝑌 2 - 𝑌 1 = Distance in the Y-Axis 7 - 11 = Distance in the Y-Axis - 4= Distance in the Y-Axis Wait, we can’t have negative number? So what do we do. We take the absolute Value −4 =4

Part 6) Distance Formula Let Create the Distance Formula Combine the formula for the X-axis and the Y-Axis 𝑋 2 - 𝑋 1 = Distance in the x-Axis 𝑌 2 - 𝑌 1 = Distance in the Y-Axis (𝑋 2 - 𝑋 1 )+ 𝑌 2 − 𝑌 1 =𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 ???? !!! NO!!! Because, The issue is; how are going to deal with the negative values like in part 5? Math Trick: if we Square both side, it fix it!!!!! 𝑑 2 =( 𝑋 2 − 𝑋 1 ) 2 +( 𝑌 2 − 𝑌 1 ) 2 or 𝑑= ( 𝑋 2 − 𝑋 1 ) 2 + (𝑌 2 − 𝑌 1 ) 2

Core Concept

Part 7) Practice the Distance Formula 11. What is the distance between the Teacher Store and Clothes? Teacher Store( 7,7) 𝑋 1 = 7 𝑌 1 = 7 Clothes (11, 10) 𝑋 2 = 11 𝑌 2 = 10 𝑑= ( 𝑋 2 − 𝑋 1 ) 2 + (𝑌 2 − 𝑌 1 ) 2 𝑑= (11−7) 2 + (10−7) 2 𝑑= (4) 2 + (3) 2 𝑑= 16+9 𝑑= 25 𝑑=5

Part 7) Practice the Distance Formula 12. What is the distance between the Jewel Accessories and Clothes? Jewel Accessories ( 8, 4) 𝑋 1 = 8 𝑌 1 = 4 Clothes (11, 10) 𝑋 2 = 11 𝑌 2 = 10 𝑑= ( 𝑋 2 − 𝑋 1 ) 2 + (𝑌 2 − 𝑌 1 ) 2 𝑑= (11−8) 2 + (10−4) 2 𝑑= (3) 2 + (6) 2 𝑑= 9+36 𝑑= 45 𝑑= 9∙5 𝑑= 9 ∙ 5 𝑑=3 5

Practice the Distance Formula 13. What is the distance between points P(1,4) and Q(13,9)? How do you know? Solution: I started at point P(1,4) and I ended at point Q(13,9) Let 𝑋 1 =1 𝑌 1 =4 𝑋 2 =13 𝑌 2 =9 𝑑= ( 𝑋 2 − 𝑋 1 ) 2 + (𝑌 2 − 𝑌 1 ) 2 𝑑= (13−1) 2 + (9−4) 2 𝑑= 144+25 𝑑= 169 𝑑=13

Practice the Distance Formula 14. What is the distance between M(9, -5) and N(-11,10)? Solution: I started at point M(9,-5) and I ended at point N(-11,10) Let 𝑋 1 =9 𝑌 1 =−5 𝑋 2 =−11 𝑌 2 =10 𝑑= ( 𝑋 2 − 𝑋 1 ) 2 + (𝑌 2 − 𝑌 1 ) 2 𝑑= (−11−9) 2 + (10−(−5)) 2 𝑑= (−20) 2 + (15) 2 𝑑= 400+225 𝑑= 625 𝑑=25

Perimeter and Area in the Coordinated Plane Date 3/7/19 We are taking notes, not doing a task. Copy down the EQ Work on Warm Up Essential question How can you find the perimeter of a polygon in a coordinate plane? Warm up: Copy down the purpose Students will calculate the perimeter of polygon by using the distance formula that they created in a prior lesson. Through this process, students will solidify definition of different quadrilaterals.

Exploration Activity Work with a partner. 1. On a piece of graph paper, draw quadrilateral 𝐴𝐵𝐶𝐷 in a coordinate plane. Label the point 𝐴 1,4 , 𝐵 −3,1 , 𝐶 0,−3 , 𝑎𝑛𝑑 𝐷(4,0). B. Find the perimeter of quadrilateral 𝐴𝐵𝐶𝐷. C. Are adjacent sides of quadrilateral 𝐴𝐵𝐶𝐷 perpendicular to each other? How can you tell? D. What is the definition of a square? Is a quadrilateral 𝐴𝐵𝐶𝐷 a square? Justify your answer.

Exploration Activity

Exploration Activity Work with a partner. 1. On a piece of graph paper, draw quadrilateral 𝐴𝐵𝐶𝐷 in a coordinate plane. Label the point 𝐴 1,4 , 𝐵 −3,1 , 𝐶 0,−3 , 𝑎𝑛𝑑 𝐷(4,0). B. Find the perimeter of quadrilateral 𝐴𝐵𝐶𝐷. Perimeter: 25 C. Are adjacent sides of quadrilateral 𝐴𝐵𝐶𝐷 perpendicular to each other? How can you tell? Yes, they all have the same distance of 5 D. What is the definition of a square? Is a quadrilateral 𝐴𝐵𝐶𝐷 a square? Justify your answer. Square: All equal side, and 90°

Core Concept You can use the formulas given below and the Distance Formula to find the perimeters and areas of polygons in the coordinate plane.

Finding Perimeter in the Coordinate Plane Find the perimeter of ∆𝐴𝐵𝐶 with vertices 𝐴 −2,3 , 𝐵 3,−3 , 𝑎𝑛𝑑 𝐶 −2,−3 Step 1 Draw the triangle in a coordinate plane. Then find the length of each side.

Finding Perimeter in the Coordinate Plane

Identifying a Triangle in the Coordinate Plane Prove that 𝐴 −2,3 , 𝐵 3,−3 , 𝑎𝑛𝑑 𝐶 −2,−3 forms a right triangle? Step 1 Label the three points and then find the length of each side.

Identifying a Triangle in the Coordinate Plane Remember If it is a right triangle then 𝑎 2 + 𝑏 2 = 𝑐 2 . Likewise If 𝑎 2 + 𝑏 2 = 𝑐 2 then it forms a right triangle 𝑎=5, 𝑏=6, 𝑐= 61 𝑎 2 + 𝑏 2 = 𝑐 2 Formula (5) 2 + (6) 2 = ( 61 ) 2 Substitute 25+36=61 Simplify 61=61 The values are true hence, the 3 points form a right triangle.

Identifying a Parallelogram in the Coordinate Plane Show that equadrilateral ABCD is a parllelogram 𝐴(−3,3), 𝐵(2,5), 𝐶(5,2), 𝐷( 0,0) Step 1 Label the points then connect the point using a line segment. Then find the length of each side.

Core Concept Ways to Prove a Quadrilateral is a Parallelogram

Identifying a Parallelogram in the Coordinate Plane

Identifying a Parallelogram in the Coordinate Plane

8.3 Prove It! Task Date 3/8/19 Grab your blinder Copy down the EQ Work on Warm Up Essential question How is the Distance formula related to Pythagorean Theorem? Warm up: Explain if the two lines are perpendicular to each other.

Introduction of Task In this task you need to use all the things you know about quadrilaterals, distance, and slope to prove that the shapes are parallelograms, rectangles, rhombi, or squares. Be systematic and be sure that you give all the evidence necessary to verify your claim. Is 𝐴𝐵𝐶𝐷 a parallelogram? Explain how you know. Is 𝐸𝐹𝐺𝐻 a parallelogram? Explain how you know.

Core Concept Ways to Prove a Quadrilateral is a Parallelogram

1a. SOLUTION Method: Show that pair of sides are congruent and parallel. Then apply the Opposite Sides Parallel and Congruent Theorem. First, use the Distance Formula to show that 𝐴𝐵 𝑎𝑛𝑑 𝐷𝐶 are congruent. 𝐴𝐵 = (−4− −10 ) 2 + (12−12) 2 = 36 𝐴𝐵 =6 𝐷𝐶 = (−12− −6 ) 2 + (8−8) 2 = 36 𝐷𝐶 =6 Because 𝐴𝐵 = 𝐷𝐶 =6 , 𝐴𝐵 = 𝐷𝐶 Second, use the slope formula to show that 𝐴𝐵 ∥ 𝐷𝐶 Slope of 𝐴𝐵 = 12−12 −4−(−10) =0 Slope of 𝐷𝐶 = 8−8 −6−(−12) =0 Because 𝐴𝐵 and 𝐷𝐶 have the same slope, they are parallel. Since. 𝐴𝐵 and 𝐷𝐶 are congruent and parallel. So 𝐴𝐵𝐶𝐷 is a parallelogram by the Opposite Sides Parallel and Congruent Theorem.

1b. SOLUTION Method: Show that pair of sides are congruent and parallel. Then apply the Opposite Sides Parallel and Congruent Theorem. First, use the Distance Formula to show that 𝐴𝐵 𝑎𝑛𝑑 𝐷𝐶 are congruent. 𝐸𝐹 = (15−5) 2 + (0−2) 2 = 104 𝐻𝐺 = (13−2) 2 + (−9−(−7)) 2 = 125 𝐴𝐵 ≠ 𝐷𝐶 Then, use the slope formula to show that 𝐴𝐵 ∥ 𝐷𝐶 Slope of 𝐸𝐹 = 10−2 15−5 = 8 10 Slope of 𝐻𝐺 = −9−(−7) 13−2 = −2 11 𝐴𝐵 and 𝐷𝐶 don’t have slope So 𝐸𝐹𝐺𝐻 is not a parallelogram

Rectangle on Coordinate Grid Is 𝐴𝐵𝐶𝐷 a rectangle? Explain how you know. Is 𝐸𝐹𝐺𝐻 a rectangle? Explain how you know.

Core Concept Ways to Prove a Quadrilateral is a Rectangle

2a. SOLUTION Method: Show that the diagonal are congruent. Then apply the Rectangle Diagonals Theorem. 𝐴𝐶 = (2−(−8)) 2 + (9−13) 2 = 116 𝐷𝐵 = (2−(−8)) 2 + (13−9) 2 = 116 𝐴𝐶 = 𝐷𝐵 The diagonals are congruent So 𝐸𝐹𝐺𝐻 a rectangle

2b. SOLUTION Method: Show that the diagonal are congruent. Then apply the Rectangle Diagonals Theorem. 𝐸𝐺 = (−9−6) 2 + (7−6) 2 = 226 𝐻𝐹 = (14−(−1)) 2 + (0−(−3)) 2 = 241 𝐴𝐶 ≠ 𝐷𝐵 The diagonals are not congruent So 𝐸𝐹𝐺𝐻 is not a rectangle

Rhombus on Coordinate Grid Is 𝐴𝐵𝐶𝐷 a rhombus? Explain how you know. Is 𝐸𝐹𝐺𝐻 a rhombus? Explain how you know.

Core Concept Ways to Prove a Quadrilateral is a Rhombus

3a. SOLUTION Method: Show that the diagonal are perpendicular. Then apply the Rhombus Diagonals Theorem. Slope of 𝐸𝐺 = −6−3 −6−(−6) = −9 0 it is a straight vertical line Slope of 𝐻𝐹 = −2−(−2) −4−(−8) = 0 4 its is a straight horizontal line A vertical line always intersection perpendicular to a horizontal line So 𝐸𝐹𝐺𝐻 is rhombus

3b. SOLUTION Method: Show that the diagonal are perpendicular. Then apply the Rhombus Diagonals Theorem. Slope of 𝐷𝐵 = 2−9 9−3 = −7 6 Slope of 𝐶𝐴 = 8−3 9−3 = 5 6 A vertical line always intersection perpendicular to a horizontal line 𝐷𝐴𝐵𝐶 is not a rhombus

Sqaure on Coordinate Grid Is 𝐴𝐵𝐶𝐷 a square? Explain how you know.

Core Concept Ways to Prove a Quadrilateral is a Square

4a.