THE DISTANCE FORMULA


 Harriet Davis
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1 THE DISTANCE FORMULA In this activity, you will develop a formula for calculating the distance between any two points in a coordinate plane. Part 1: Distance Along a Horizontal or Vertical Line To find the distance between points on the same horizontal or vertical line, count the number of boxes between the points. 1. Find the distances: AB = CD = Distances along diagonal lines work differently. 2. At right, a square has a quarter circle inside it. EE is one side of the square and the radius of the circle. How does this diagram show why the vertical distance EG is not the same as the diagonal distance EH? Discuss this with someone near you. Part 2: Review of Pythagorean Theorem and Simplifying Radicals 3. Recall the Pythagorean Theorem from Prealgebra. Write the Pythagorean Theorem here: 4. Use the Pythagorean Theorem to find the length of the missing side of each right triangle. Leave your answers to the last two in radical form
2 Your last two answers in #4 did not come out as perfect squares. These radicals are irrational numbers; as decimals, they would continue infinitely without a pattern. We sometimes simplify numbers like these to make it easier for us to compare. When a square root is in simplest radical form when it no longer contains perfect square factors. Example: Express 40 in simplest radical form. 40 = 4 10 Rewrite as a product of 2 smaller square roots. = 2 10 Simplify the radical that is a perfect square. 5. In the example, 40 could have been written as 2 20 or 5 8 instead. Why wouldn t these products help us simplify the radical? 6. How do we know that 2 10 is in simplest radical form? 7. How can you use a calculator to check that 40 and 2 10 have the same value? Try this now. 8. Simplify each of the following square roots: Rewrite your answers to the last two triangles from #4 in simplest radical form. 2
3 Part 3: Distance Along a Diagonal Line M2 GEOMETRY PACKET 2 FOR UNIT 1 SECTIONS 09, 13, AND 16 The Pythagorean Theorem is used for finding the length of one side of a right triangle. We can treat a diagonal distance as the hypotenuse of a right triangle. 10. Graph the points A (3, 1) and B (7, 1). Then make a right triangle by drawing a vertical line through A and a horizontal line through B. 11. Count boxes to find the length of the horizontal and vertical sides of the triangle. Label them in the diagram. 12. Use the Pythagorean Theorem to find AB. Write your answer in simplest radical form. 13. How could you use the coordinates of A and B to find the length of the horizontal side without counting boxes? What about the vertical side? 14. Use vertical and horizontal lines to draw a right triangle with hypotenuse AA. Then use your answers in #910 to write formulas about x1, x2, y1, and y2 for the lengths below. Horizontal side = Vertical side = 15. Write these formulas in the parentheses below: By the Pythagorean Theorem, ( ) 2 + ( ) 2 = d 2, where d is the distance from A(x1, y1) to B(x2, y2). Then d =. This is the Distance Formula. Draw a box around it. An animation similar to #1415 can be found at Use the distance formula to find the distance between each pair of points. Write irrational answers in simplest radical form. a. ( 1,7) and (8,4) b. (2, 3) and (6, 8) c. ( 8, 5) and ( 2, 1) 3
4 MIDPOINTS Part 1: Midpoint 1. In the figure at right, use the distance formula to show that AC = BC. Point C is called the midpoint of AB. 2. Use the distance formula to confirm that BD = AD. Even though this is true, why isn t D considered to be the midpoint of AB? 3. Add midpoint to your online glossary now. Include a diagram with congruence markings and a congruence statement. 4. Write the coordinates of each point below: A (, ) B (, ) C (, ) 5. How is the xcoordinate of C related to the x s of A and B? What about the ycoordinate? 6. You should have answered that the coordinates of C are halfway between the coordinates of A and B. Find the number that is halfway between each pair of numbers: a. 8 and 12 b. 10 and 15 c. 2 and 8 What is a formula for finding a number that is halfway between two others? 4
5 The Midpoint Formula states that the midpoint between any two points ( x, y ) and ( x, y ) can be found by,. 7. Find the midpoint between each pair of points: 1. ( 1,4 ), ( 7,10) b. ( 5, 2 ), ( 6,8) Suppose point C is the midpoint between A and B. The coordinates of point A are ( 7,5), and the coordinates of C are ( 2,3 ). a. Sketch and label a diagram to show this. Label the coordinates of B as (x, y). b. Since we weren t given the coordinates of B, we used the variables (x, y) for now. To find x, write and solve an equation showing that the average of x and 7 equals 2. (Where did the 7 and 2 come from?) c. Repeat part (b) for the ycoordinates. d. The location of point B is (, ). 9. In the figure at right, C is the midpoint of A and B. Write an equation, and solve to find the value of x. 5
6 PERIMETER AND AREA 1. Find each distance in the graph. Give irrational answers in simplest radical form. AB = GH = G to HI = AD = FE = HI = IG = 2. Complete these formulas. Use the internet or another student to help you. Area of a rectangle = Area of a circle = Area of a square = Circumference of a circle = Area of a triangle = 3. Use your formulas to calculate each. Show your work. Area of rectangle ABCD: Perimeter of ABCD: Area of circle with center F and radius FE: Circumference of the same circle: Area of triangle GHI: Perimeter of triangle GHI: 4. In your own words, define area: 5. In your own words, define perimeter (and circumference): 6. Add these terms to your online glossary: area, perimeter, circumference 6
7 7. Find perimeter given area: Suppose a rectangle s length is 3 times its width, and its area is 75 cm 2. Follow the steps to find the perimeter of this rectangle. a. Write an equation, using L for length and W for width to describe a rectangle s length is 3 times its width. b. Write the formula for the area of a rectangle below, and then replace the letters in the formula with L, W, and 75 as appropriate. c. Use the substitution method to combine your equation from part (a) with your equation from part (b). Ask for help if you don t remember what this means. d. Solve your equation from part (c), and use your answer to find the length and width of the rectangle. e. Find the perimeter of the rectangle. Include units with your answer. 8. Find circumference given area: Suppose a circle has area 81π cm 2. Follow the steps below to find its area. a. Write the formula for the area of a circle below, and replace A its value given above. b. Solve your equation for r. c. Use your answer in part (b) to find the circumference of the circle. Include units with your answer. 7
8 9. Find area given perimeter: Suppose a rectangle s length is twice its width, and its perimeter is 27 in. Follow the steps to find its perimeter. a. Write an equation describing a rectangle s length is twice its width. b. Write the formula for the perimeter of a rectangle below, and replace the letters in the formula with L, W, and 27 as appropriate. c. Use the substitution method to combine your equations from parts (a) and (b). d. Solve your equation from part (c), and use the answer to find the length and width of the rectangle. e. Find the area of the rectangle. Include units with your answer. 10. Find area given circumference: Suppose a circle has circumference 36π in. Follow the steps to find its area. a. Write the formula for the circumference of a circle below, and replace C with its given value. b. Solve your equation for r. c. Use your answer to find the area of the circle. Include units with your answer. 8
9 PRACTICE FOR 09, 13, & 16 Find the distance between each pair of points. Express irrational answers in simplest radical form A ( 7, 3), B (5, 2) 3. W ( 2, 2), R (5, 2) 4. A cargo ship must travel from New York City (N) to Southampton, England (S). The two points are N ( 3250, 2816) and S ( 72, 3505), where x represents miles west of the prime meridian and y represents miles north of the equator. Calculate the distance from NYC to Southampton. Note: These calculations will be slightly off because the distance formula is used on flat planes, while the Earth is round. 5. Use the distance formula to write an equation, and solve: The points A ( 2, 1) and B (x, 5) are 5 units apart. What are two possible locations of point B? Find the coordinates of the midpoint of a segment with the given endpoints. 6. T (3, 1), U (5, 3) 7. J ( 4, 2), F (5, 2) 9
10 Find the coordinates of the missing endpoint L if K is the midpoint of JJ. 8. J (2, 0), K (5, 2) 9. J (5, 4), K (6, 3) 10. The coordinates of the vertices of quadrilateral RSTU are R ( 1, 3), S (3, 3), T (5, 1), and U ( 2, 1). Find the perimeter of RSTU. Round lengths to the nearest tenth. 11. Find the area and perimeter of the rectangle at right. 12. Find the area and circumference of the circle at right. Express answers in terms of π. 13. Find the area and perimeter of the triangle at right. 10
11 Use formulas and equations to solve. Include units with your answers. 14. Find the perimeter of a square that has area 25 cm The length of a rectangle is 5 times its width, and its perimeter is 24 ft. Find the area of the rectangle. 16. The length of a rectangle is 3 times its width, and its area is 300 cm 2. Find the perimeter of the rectangle. 17. The area of a circle is 64π in 2. Find the circumference of the circle. 18. The circumference of a circle is 100π in. Find the area of the circle. 11
12 REVIEW FOR 09, 13, & 16 Find the distance between each pair of points. Express irrational answers in simplest radical form C ( 3, 1), D (2, 6) 3. A linear portion of the roller coast Boulder Dash at Lake Compounce starts at the highest point (64, 82) and then drops to (110, 32). To the nearest foot, determine the distance of this linear section of the track. 4. Find the midpoint between each pair of points in #13 above. Find the coordinates of endpoint B if C is the midpoint of AB. 5. A (3, 9), C ( 1, 5) 6. A (6, 2), C ( 8, 7) 12
13 In #79, find the perimeter and area of the figure. Express irrational answers in simplest radical form. 7. Rectangle QRST, where Q ( 3, 2), R (1, 2), S (1, 4), T ( 3, 4) 8. Triangle ABC at right 9. Triangle DEF at right 10. Jessica plans to sew fringe around the circular pillow shown in the diagram. Use π a. How many inches of fringe does she need to purchase? Round to the nearest tenth of an inch. b. If Jessica doubles the radius of the pillow, what is the new area of the top of the pillow? Round to the nearest tenth. 13
14 Use formulas and equations to solve. Include units with your answers. 11. Find the perimeter of a square that has area 36 cm The length of a rectangle is 4 times its width, and its perimeter is 20 ft. Find the area of the rectangle. 13. The length of a rectangle is 2 times its width, and its area is 162 cm 2. Find the perimeter of the rectangle. 14. The area of a circle is 144π in 2. Find the circumference of the circle. 15. The circumference of a circle is 36π in. Find the area of the circle. 14
15 Answers to oddnumbered problems in Review for 09, 13, and 16: ft 5. ( 5,1) 7. perimeter = 18, area = perimeter = cm cm π in 2 15
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