Distance on the Coordinate Plane
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1 6 7 Distance on the Coordinate Plane What You ll Learn You ll learn to find the distance between two points on the coordinate plane. Wh It s Important Transportation Knowing how to find the distance between two points can help ou determine distance traveled. See Eample 3. California Standards Standard 15.0 Students use the Pthagorean theorem to determine distance and find missing lengths of sides of right triangles. In Lesson 1, ou learned how to find the distance between two points on a number line. In this lesson, ou will learn how to find the distance between two points on the coordinate plane. When two points lie on a horizontal line or a vertical line, the distance between the two points can be found b subtracting one of the coordinates. In the coordinate plane below, points A and B lie on a horizontal line, and points C and D lie on a vertical line. A( 4, 3) B(1, 3) 5 C (1, 1) 4 D(1, 3) The distance between A and B is 4 1 or 5. The distance between C and D is 3 1 or 4. In the following activit, ou will learn how to find the distance between two points that do not lie on a horizontal or vertical line. Graphing rdered Pairs: Lesson 4 Materials: grid paper straightedge Step 1 Graph A( 3, 1) and C(, 3). Step Draw a horizontal segment from A and a vertical segment from C. Label the intersection B and find the coordinates of B. A( 3, 1) C(, 3) B Tr These 1. What is the measure of the distance between A and B?. What is the measure of the distance between B and C? 3. What kind of triangle is ABC? 4. If AB and BC are known, what theorem can be used to find AC? 5. What is the measure of A C? 6 Chapter 6 More About Triangles
2 In the activit, ou found that (AC) (AB) (BC). B taking the square root of each side of the equation, ou find that AC (AB) ) (BC. AC measure of the distance between points A and C AB difference of the -coordinates of A and C BC difference of the -coordinates of A and C This formula can be generalized to find the distance between an two points. Theorem 6 11 Distance Formula Words: Model: If d is the measure of the distance between two points with coordinates ( 1 ) and ), then d ( 1 ) ( 1. ) A( 1 ) d B ) Eample 1 Use the Distance Formula to find the distance between J( 8, 6) and K(1, 3). Round to the nearest tenth, if necessar. Use the Distance Formula. Replace ( 1 ) with ( 8, 6) and ) with (1, 3). d ( 1 ) ( 1 ) Distance Formula JK [1 ( 8)] ( 3 6) Substitution JK (9) ( 9) JK JK 16 JK 1.7 Simplif. Your Turn tenth, if necessar. a. M(0, 3), N(0, 6) b. G( 3, 4), H(5, 1) Lesson 6 7 Distance on the Coordinate Plane 63
3 You can use the Distance Formula to determine whether a triangle is isosceles given the coordinates of its vertices. Eamples Hint: Draw a picture on a coordinate plane. Determine whether ABC with vertices A( 3, ), B(6, 5), and C(3, 1) is isosceles. An isosceles triangle has at least two congruent sides. Use the Distance Formula to find the measures of the sides of ABC. Then determine if an two are equal. AB [6 ( 3)] ) (5 BC (3 ) 6 ( 1 5) 9 3 ( 3) ) ( AC [3 ( 3)] ( 1 ) 6 3) ( B C and A C have equal measures. Therefore, ABC is isosceles. Transportation Link Real World 3 Lena takes the bus from Mill s Market to the Candle Shop. Mill s Market is 3 miles west and miles north of Blendon Park. The Candle Shop is miles east and 4 miles south of Blendon Park. How far is the Candle Shop from Mill s Market? Let Mill s Market be represented b ( 1 ) and the Candle Shop b ). Then 1 3,, and 4. d ( 1 ) ( 1 ) d [ ( 3)] ( 4 ) d (5) ( 6) d d 61 d 7.8 The Candle Shop is about 7.8 miles from Mill s Market. West Mill s Market ( 3, ) North Blendon Park East South Candle Shop (, 4) 64 Chapter 6 More About Triangles
4 Check for Understanding Communicating Mathematics 1. State the Distance Formula for points represented b ( 1 ) and ).. Name the theorem that is used to determine the Distance Formula in the coordinate plane. 3. Ana sas that to find the distance from A( 3, ) to B( 7, 5), ou must evaluate the epression [ 7 ( 3)] (5. ) Emil disagrees. She sas that ou must evaluate the epression [ 3 ( 7)] (. 5) Who is correct? Eplain our answer. Guided Practice Getting Read Find the value of each epression. Sample: ( 7 4) [3 ( 6)] Solution: ( 7 4) [3 ( 6)] ( 3) [3 6] 4. (6 ) ( 5 3) 5. [ ( 3)] ( 3) 6. [ 5 ( 6)] (4 ) ( 3) or 90 tenth, if necessar. (Eample 1) 7. E(1, ), F(3, 4) 8. R( 6, 0), S(, 0) 9. P(5, 6), Q( 3, 1) 10. Determine whether FGH with vertices F(, 1), G(1, 6), and H(4, 1) is isosceles. (Eample ) 11. Hiking Tamika and Matthew are going to hike from Cedar Creek Cave to the Ford Nature Center. Cedar Creek Cave is located 3 kilometers west of the ranger s station. The Ford Nature Center is located kilometers east and 4 kilometers north of the ranger s station. (Eample 3) a. Draw a diagram on a coordinate grid to represent this situation. b. What is the distance between Cedar Creek Cave and Ford Nature Center? Lesson 6 7 Distance on the Coordinate Plane 65
5 Eercises Practice Homework Help For Eercises See Eamples Etra Practice See page 738. tenth, if necessar. 1. A(5, 0), B(1, 0) 13. M(, 3), N(5, 7) 14. D( 1, ), E( 3, 4) 15. X( 4, 0), Y(3, 3) 16. P( 6, 4), Q(6, 8) 17. T(6, 4), U(, ) 18. B(0, 0), C( 5, 6) 19. G( 6, 8), H( 6, 4) 0. J( 3, ), K(3, 1) 1. S( 6, 4), T( 3, 7). Find the distance between A( 1, 5) and C(3, 5). 3. What is the distance between E( 3, 1) and F(4, )? 4. Is MNP with vertices M(1, 4), N( 3, ), and P(4, 3) an isosceles triangle? Eplain. 5. Determine whether RST with vertices R(1, 5), S( 1, 1), and T(5, 4) is scalene. Eplain. 6. Triangle FGH has vertices F(, 4), G(0, ), and H(3, 1). Determine whether FGH is a right triangle. Eplain. Applications and Problem Solving Real World 7. Gardening At Memorial Flower Garden, the rose garden is located 5 ards west and 30 ards north of the gazebo. The herb garden is located 35 ards west and 15 ards south of the gazebo. a. Draw a diagram on a coordinate grid to represent this situation. b. How far is the herb garden from the rose garden? c. What is the distance from the rose garden to the gazebo? 8. Communication To set long-distance rates, telephone companies superimpose an imaginar coordinate plane over the United States. Each ordered pair on this coordinate plane represents the location of a telephone echange. The phone compan calculates the distances between the echanges in miles to establish long-distance rates. Suppose two echanges are located at (53, 187) and (19, 71). What is the distance between these echanges to the nearest mile? The location units are in miles. 9. Critical Thinking In ABC, the coordinates of the vertices are A(, 4), B( 3, 6), and C( 5, ). To the nearest tenth, what is the measure of the median drawn from A to B C? Include a drawing on a coordinate plane of the triangle and the median. Mied Review 30. Music The frame of the music stand shown contains several triangles. Find the length of the hpotenuse of right triangle MSC if the length of one leg is 10 inches and the length of the other leg is 8.5 inches. Round to the nearest tenth, if necessar. (Lesson 6 6) C 8.5 in. M 10 in. S 66 Chapter 6 More About Triangles
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