Name ctivit Sheet : Deriving the Distance Formula. Use the diagram below to answer the following questions. Date C 6 6 x a. What is C? b. What are the coordinates of and C? c. Use the coordinates of and C to compute C. Show our work. d. What are the coordinates of C and? e. Use the coordinates of C and to compute C. Show our work. f. Draw the segment. What kind of triangle is Δ C C and C called? What is called?? For that kind of triangle, what are g. What theorem can ou use to find the length of the hpotenuse of a right triangle if ou know the lengths of the two legs? h. Use our answer to g (above) to find. Virginia Department of Education 0
. Use the diagram below to answer the following questions. 6 6 x a. On the diagram above, create a right triangle with a horizontal leg, vertical leg, and hpotenuse. Label the vertex of the right triangle C. Is this the onl right triangle ou could have drawn? b. Find C and C. c. Use C and C to find.. The endpoints of a vertical segment are and. Use this diagram for the following questions. (Do not count. Graph is not to scale.) a. Label the lower point and the upper point. Since is a vertical segment, what can ou sa about and? b. Express in terms of and. c. Express in a different wa in terms of and. Virginia Department of Education 0
d. Wh is it necessar to use absolute value for the formulas above? e. Does it matter which of the two formulas above ou use? f. Write a formula for using either formula above. = g. The endpoints of a vertical segment are G( 0, ) and H( 0, ). Use one of our formulas to compute GH.. The endpoints of a horizontal segment CD are and. Use this diagram for the following questions. (Do not count. Graph is not to scale.) a. Label the point on the left and the point on the right. Since CD is a horizontal segment, what can ou sa about and? b. Express CD in two different was in terms of x and x. c. Write a formula for CD using either formula above. CD = d. The endpoints of a horizontal segment are E( 0, ) and F(, ). Use our formula to compute EF. Virginia Department of Education 0
. The endpoints of a segment PQ are and. Use this diagram for the following questions. (Do not count. Graph is not to scale.) a. Use the Pthagorean Theorem on the right triangle, using PQ, PR, and QR. Complete the equation below: (PQ) = Q b. Label the point P as and point Q as. Find the coordinates of the point R. c. Write formulas for PR and QR using either formula above. PR = QR = P R d. Determine whether this equation is true. If it is true, explain wh. If it is false, give a counterexample. (Hint: When ou square a real number, is it ever negative?) e. Use the last three problems to get a formula for (PQ) in terms of,,, and. (PQ) = f. Take the square root of both sides of our last formula to write PQ in terms of,,, and. PQ = g. The endpoints of a segment are I(8, ) and J(, ). Use the formula ou found to compute the distance between I and J. Virginia Department of Education 0 6
ctivit Sheet : Deriving the Midpoint Formula Name Date. The midpoint of a segment is the point on the segment that is the same distance from both endpoints. Use the graph below to answer the following questions. 6 6 x C a. What are the x-coordinates of and C? and b. What is the average of the x-coordinates of and C? c. What number is halfwa between the x-coordinates of and C? d. What are the coordinates of the midpoint of C? Graph and label the midpoint. e. Explain the relationships among the answers to questions b, c, and d.. Use the graph below to answer the following questions. C 6 6 x a. What are the -coordinates of and C? and b. What is the average of the -coordinates of and C? c. What number is halfwa between the -coordinates of and C? d. What are the coordinates of the midpoint of C? Graph and label the midpoint. Virginia Department of Education 0 7
e. Explain the relationships among the answers to questions b, c, and d.. Use the diagram below to answer the following questions. C 6 6 x a. Graph the midpoint of C. Label it P. What is the x-coordinate of this point? b. What is the average of the x-coordinates of and C? How is this related to our answer to a? c. Graph the midpoint of C. Label it Q. What is the -coordinate of this point? d. What is the average of the -coordinates of and C? How is this related to our answer to c? e. What is the average of the x-coordinates of and? f. What is the average of the -coordinates of and? g. What is the midpoint of C? h. How is the midpoint of related to the answers to e and f? Virginia Department of Education 0 8
. 6. Mathematics Enhanced Scope and Sequence Geometr. Use the diagram below to answer the following questions. 6 6 x a. What is the average of the x-coordinates of and? b. What is the average of the -coordinates of and? c. What are the coordinates of the midpoint of? d. Explain the relationships among the answers to questions b, c, and d.. The endpoints of a segment PQ are and. Q P R a. Label the point P as and point Q as. b. Write a formula for the average of the x-coordinates of and. c. Write a formula for the average of the -coordinates of and. d. One wa to think of the midpoint of PQ is as follows: average of the x-coordinates, average of the -coordinates. Use this to derive a formula for the midpoint of PQ. e. The endpoints of a segment are E(8, ) and F(, ). Use our formula to compute the midpoint of EF. Virginia Department of Education 0 9
Name ctivit Sheet : Lake Geometria Date The islands of Lake Geometria are shown below. cabin is marked on each island. The scale, using units called stades, is shown in the lower right. stade measures about 600 feet, so Lake Geometria is not ver big. Use the grid to help ou answer the following questions. ll island measurements should be made from cabin to cabin.. What is the distance in stades from Eudoxus to rchimedes? (Round to the nearest stade.) Describe how ou found our answer.. What is the distance in stades from Thales to Euclid? (Round to the nearest stade.) Describe how ou found our answer. Virginia Department of Education 0 0
. Which is closer to Thales Pthagoras or Heron? Describe how ou found our answer.. Find a point in the water that is the same distance from rchimedes and Eudoxus. Label the point M. Describe how ou found this point.. Now find a point on land that is the same distance from rchimedes and Eudoxus. Label this point N. 6. How man points can ou find that are the same distance from rchimedes and Eudoxus? Explain. 7. Find a point on land that is the same distance from Thales and Pthagoras. Is our point on the mainland or on an island? Does it have to be? 8. Groups staing on an of the islands are provided with solar-charged walkie-talkies, but their ranges are onl about mile. There are about 9 stades in a mile. With which islands could someone staing at the cabin on Thales expect to be able to communicate? Show our work, or explain how ou found our answer. 9. Estimate the shortest distance from Thales to the mainland. (You ma use stades, feet, or miles.) Show our work or explain how ou found our answer. 0. Estimate how man miles wide Lake Geometria is at its widest point. Explain how ou found our answer. Virginia Department of Education 0