Finding Distance between Two Points on the Coordinate Plane
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1 Lesson.G. Finding Distance between Two Points on the Coordinate Plane Getting the idea You can use the Pthagorean theorem to find the distance between two points in the coordinate plane. The diagram below shows how the distance, d, between two points can be represented b the length of the hpotenuse of a right triangle. d (, ) (, ) Find the horizontal and vertical distances that are the lengths of the legs of the right triangle. Then substitute into the formula a b c to find the distance, d. (, ) d (, ) horizontal distance from (, ) to (, ) vertical distance from (, ) to (, ) a b c c 9 c c c The distance between the points is units. Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC Domain : Geometr
2 Eample Find the distance between the points (, ) and (, ). Round our answer to the nearest tenth. Strateg Use the Pthagorean theorem. Step Plot the points, and connect them with a segment. Then draw a right triangle that has the segment as its hpotenuse. (, ) d (, ) Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC Step Identif a, b, and c. The length of the hpotenuse is d, so let c d. Use absolute value to find the horizontal and vertical distances that are the lengths of the legs. horizontal distance from (, ) to (, ) vertical distance from (, ) to (, ) The lengths of the legs are units and units. So, a and b. Step Substitute into a b c. Then solve the equation for d. Solution a b c d d d d. < d The distance between the points (, ) and (, ) is approimatel. units. Lesson : Finding Distance between Two Points on the Coordinate Plane
3 Eample Find the distance between the points (, ) and (, 0). Round our answer to the nearest tenth. Strateg Use the Pthagorean theorem. Step Plot the points, and connect them with a segment. Then draw a right triangle that has the segment as its hpotenuse. 0 (, ) d (, 0) Step Identif a, b, and c. The length of the hpotenuse is d, so let c d. Use absolute value to find the horizontal and vertical distances that are the lengths of the legs. horizontal distance from (, ) to (, 0) vertical distance from (, ) to (, 0) 0 The lengths of the legs are units and units. So, a and b. Step Substitute into a b c. Then solve the equation for d. Solution a b c d 9 d d d. d The distance between the points (, ) and (, 0) is approimatel. units. Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC Domain : Geometr
4 Eample Cali drew the map below showing her house and the houses of two of her friends. On the map, each unit square represents square mile. What is the distance between Jenna s house and Mira s house to the nearest tenth of a mile? Jenna s house Cali s house 0 Mira s house Strateg Use the Pthagorean theorem. Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC Step Connect the points representing Jenna s house and Mira s house with a segment. Then draw a right triangle that has the segment as its hpotenuse. Jenna s house Step Identif a, b, and c. The length of the hpotenuse is d, so let c d. 0 Use absolute value to find the horizontal and vertical distances that are the lengths of the legs. horizontal distance from (, ) to (, ) vertical distance from (, ) to (, ) () The legs are both units long. So, a and b. Step Substitute into a b c. Then solve the equation for d. a b c Solution d d d d. d The distance between Jenna s house and Mira s house is approimatel. miles. Cali s house d Mira s house Lesson : Finding Distance between Two Points on the Coordinate Plane
5 Eample Find the perimeter of the trapezoid A B C D 9 0 Strateg Find the length of each side, using the Pthagorean theorem where necessar. Then find the sum of the lengths of the sides. Step Find the lengths of the horizontal and vertical sides. Use absolute value to find the horizontal and vertical distances that are the lengths of sides AB, BC, and AD. AB units BC units AD units Step Use the Pthagorean theorem to find the length of the slanted side. To find the length of side CD, draw a right triangle that has CD as its hpotenuse. Identif a, b, and c. The length of the hpotenuse is, so let c. The vertical distance is equal to AB, so is units. Use absolute value to find the horizontal distance. horizontal distance from (, ) to (, ) The lengths of the legs are units and units. So, a and b. a b c 9. So, the length of side CD is approimatel. units A B C D 9 0 Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC Domain : Geometr
6 Step Find the sum of the lengths of the sides... Solution The perimeter is approimatel. units. COACHED EXAMPLE Find the approimate length of the longer diagonal of the kite ABCD to the nearest tenth of a unit. Draw the diagonals of the kite, AC and BD. Which is the longer diagonal? Draw a right triangle using the longer diagonal as its hpotenuse. Label the diagonal. Look at the right triangle ou drew. Identif the lengths of a, b, and c. a units b units c units Write and solve an equation to find the value of. Round our answer to the nearest tenth. a b c A B 9 0 C D Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC 000 The length of the longer diagonal is approimatel units. Lesson : Finding Distance between Two Points on the Coordinate Plane 9
7 LESSON PRACTICE Find the perimeter of rhombus ABCD below. A 0 B D C The perimeter is units. A trapezoid has vertices (, ), (, ), (, ), and (, ). Part A Graph the trapezoid on the grid below. 0 Part B What is the perimeter of the trapezoid? Show our work. Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC 0 Domain : Geometr
8 The map below shows Tler s house and the houses of two of his friends, Pedro and Jake. On the map, each unit square represents square mile. Select True or False for each statement. Pedro Jake Tler A. Tler lives closer to Jake than to Pedro. True False B. Pedro lives closer to Jake than to Tler. True False C. Jake lives about 0. mile closer to Tler than to Pedro. True False D. The sum of the distances between the bos houses is less than miles. E. The sum of the distances between the bos houses is greater than 0 miles. True True False False Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC A kite has vertices (, ), (, ), (, ), and (, ). What is the difference between the lengths of the two diagonals of the kite? Round our answer to the nearest hundredth of a unit. Show our work. Lesson : Finding Distance between Two Points on the Coordinate Plane
9 Use epressions from the bo to complete the table. Points (, ) and (, ) (, ) and (, ) (, ) and (, ) (, ) and (, ) Distance between the Points 9 9 A coordinate grid is superimposed on a map of the count park. The grid shows a rectangular plaground with vertices (, ), (., ), (, ), and (., ). Each unit on the grid represents 0 feet. Part A Graph the rectangle that represents the plaground on the grid below. 0 9 Part B What is the area of the plaground? Round our answer to the nearest square foot. Show our work. Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC Domain : Geometr
10 Circle the numbers that make the sentence true The distance between the points (, ) and (, ) is units. A plan for a mosaic made from tiles calls for triangles of different sizes and shapes. Adrianna is drawing patterns for the different shapes on grid paper. The figure below shows one side of what will be a tile in the shape of an equilateral triangle. 0 Duplicating an part of this book is prohibited b law. 0 Triumph Learning, LLC If the third verte of the triangle is located in the first quadrant, what are the coordinates of the verte? Round the coordinates to the nearest hundredth. Use words, numbers, and diagrams to justif our answer. Lesson : Finding Distance between Two Points on the Coordinate Plane
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