Lines Here are some equations of straight lines: y+2x=8 2y=x–4 2y+½x+1=0 y+2x+2=0 4y–x=1 y=½x+2 Name______________________________________ y=x–4 y=4–x y=2(x–1) 2y=4–x 1. Which four lines form the four sides of a rectangle? Explain your reasoning carefully. 2. Complete the drawing to show the four lines and the x-­‐ and y-­‐axes. Label the lines clearly. 3. What is the slope of the line joining the points (2,5) and (7,15)? 4. Give the coordinates of two points for which the slope of the line between them is 3. 5. a. Give an equation of a line with a slope of 3. b. Keeping the same equation, write it in a different way. 6. What is the y-­‐intercept of the line y=3x+7? 7. What is the x-­‐intercept of the line 2y=3x-­‐6? 8. What would be the slope of a line perpendicular to y=2x+4? 9. Give the equations of two lines that are perpendicular to each other. Here are some equations of straight lines: 4y=x+3 2y+8=3x y=8x–3 2y+x=4 Here are some properties of equations: y+4x+6=0 2y=8x+3 3y=2x–8 y=6x-­‐4 y+6x=11 y=4x+4 These lines have the same x-­‐
intercept These lines go through the point (1,5) These lines are parallel These lines are perpendicular These lines have the same y-­‐
intercept 10. Find two equations to match each of the properties. 11. Line segment SP has equation y = 2x + 3. Find the equations of the line segments forming the other three sides of the rectangle. 12. Find the equation of a line passing through the origin and parallel to the line 2x – y = 5. 13. If the equation of a line p in the coordinate plane is y = 3x + 2, what is the equation of line q which is a reflection of line p in the x-­‐axis? 14. If the slope of a line is ½ and the y-­‐intercept is 3, what is the x-­‐intercept of the same line? ( ) ( )
15. The vertices of the triangle PQR are the points P 1,2 , Q 4,6 , and R ( −4,12) . Which one of the following statements about triangle PQR must be true? a. PQR is a right triangle with the right angle at P. b. PQR is a right triangle with the right angle at Q. c. PQR is a right triangle with the right angle at R. d. PQR is not a right triangle. Justify your answer. 16. Prove that the slopes of two parallel lines are equal. 17. Prove that the slopes of two perpendicular lines have a product of –1. Quadrilaterals 1. Show that the diagonals of rectangle MNJK are congruent. K(-­‐6,5) J(-­‐5,-­‐1) M(6,7) N(7,1) 3. Show that the diagonals of parallelogram EFGH bisect each other. E(-­‐5,4) F(4,5) G(7,-­‐1) H(-­‐2,-­‐2) Name______________________________________ 2. Show that quadrilateral ABCD is a rhombus. A(-­‐4, 2) B(3,3) C(8, -­‐2) D(1,-­‐3) 4. Is WXYZ an isosceles trapezoid? Why or why not? W(-­‐4,-­‐8) X(-­‐3,-­‐4) Y(2,2) Z(6,3) Use the coordinates of the vertices to classify each figure as a quadrilateral with no special features, or as a trapezoid, parallelogram, or rectangle. Explain your thinking. 5. 6. 7. 8. Three vertices of parallelogram ABCD are given. Find x and y. 9. A(–4,1), B(–1,5), C(6,5), D(x,y) 10. A(–1,0), B(0,–4), C(8,–6), D(x,y) Join the dots to complete these quadrilaterals. Where there are options, find the one on the grid with the largest possible area. 11. Isosceles trapezoid 12. Parallellogram 14. Rhombus 13. Kite 15. Rectangle 16. Trapezoid 17. Quadrilateral PQRS has vertices P(2,2), Q(3,4), R(6,5), and S(5,3). a. A student makes the conclusion given. Describe and correct the error(s) made by the student. b. Show that the opposite sides of PQRS are parallel. 18. Segment AB of square ABCD has endpoints A(0,0) and B(5,2). What are the possible locations of point C? Triangles 1. a. A triangle has vertices (4,7), (7,9), and (10,6). What are the coordinates of the centroid of the triangle? b. Use your results to describe how to calculate the coordinates of the centroid given three vertices of a triangle (a,b), (c,d), and (e,f). Name______________________________________ 2. a. The midpoints of a triangle are (2,6), (5,11), and (10,8). What are the vertices of the triangle? b. Use yours results to describe how to find the vertices of the triangle given three midpoints of a triangle (a,b), (c,d), and (e,f). 3. Find the point of intersection of the angle bisectors of the interior angles of a triangle whose vertices are ( −2, −3 ) , (32, −17) , and ⎛ 1 29 ⎞
⎜ 2 , 2 ⎟ . ⎝
⎠
4. If the lengths of the sides of a triangle ABC are 15, 36, and 39, find the length of the radius of the inscribed circle. Elana and Matt have decided to compete in a "challenge" to see whose powers of observation are the strongest in relation to figures drawn on a coordinate plane. You will be acting as the judge (and supreme keeper of the correct answer) during this challenge. Your task is to prepare a proof showing the correct result for each question and to keep track of Elana's and Matt's scores to determine the winner. 1. Quadrilateral A(1,2), B(2,5), C(5,7) and D(4,4). 2. Triangle A(1,1), B(4,4), C(7,2). Elana’s prediction: ABCD is a rhombus. Elana’s prediction: ABC is a right triangle. Matt’s prediction: ABCD is a parallelogram, but not a rhombus. Matt’s prediction: ABC is not a right triangle. 3. Quadrilateral A(3,1), B(5,6), C(7,6), D(10,2). Elana’s prediction: The diagonals are not perpendicular. Matt’s prediction: The diagonals are perpendicular. 4. Quadrilateral A(0,–2), B(9,1), C(4,6), D(1,5). Elana’s prediction: ABCD is an isosceles trapezoid. Matt’s prediction: ABCD is a trapezoid, but not isosceles. 5. Segment A(1,5) B(2,3) and segment C(4,4) D(–2,1). Elana’s prediction: The segments are perpendicular. Matt’s prediction: The segments are not perpendicular. Area & Perimeter Name______________________________________ For each whole number between 1 and 10, find a square with vertices on the coordinate grid whose area is square units or show that there is no such square. What are the perimeter and area of the polygon? 1. 2. 3. Join the dots to complete these quadrilaterals. Where there are options, find the one on the grid with the largest possible area. What are the perimeter and area of the polygon? 4. Square 5. Rhombus 6. Rectangle KLMNOP is a regular hexagon inscribed in the circle with an apothem
length of !2 3 .
7. What is the area of the hexagon?
8. How can you generalize your results to calculate the area of a regular
pentagon with an apothem length of !2 3 .
9. What is the formula for the area of a regular polygon? 10. A regular nonagon is inscribed in a circle with radius 4 units. What are the perimeter and area of the nonagon?
Circles 1. The endpoints of the diameter of a circle are (6, 0) and (−6, 0). Name______________________________________ a. What are the coordinates of the center of the circle? b. A point on this circle has coordinates (2, m). Write possible values for m. Fully explain your answer. c. What is the equation of this circle? Fully explain your answer. d. Does the point (5,6) lie inside, on, or outside the circle? 2. The center of another circle is (−5,1). Its radius is 14 . What is the equation of this circle? Fully explain your answer. 3. a. What are the coordinates of any point on this circle? b. A point on the circle has coordinates (3, y). Write a value for y. c. Can you think of another point on the circle for which both coordinates are integers? Write all the points as you can think of. d. Now give the coordinates of a point on the circle for which the x value is an integer, but the y value is not an integer. Give the exact value! e. Now give the coordinates of a different point on the circle, for which the y value is an integer, but the x value is not an integer. 4. a. We are now going to move the circle so that its center is at (2, 3). Its radius remains same. Figure out the coordinates of two points on this new circle. Select one point that is easy to figure out and the other difficult. Explain why your points are easy, and difficult to figure out. b. Does the point (7,6) lie inside, on, or outside the circle? 5. Move to page 1.8. Move the center of the circle by dragging point O. Change the radius of the circle by dragging point Q. Drag point P outside the circle, on the circle, and in the interior of the circle. a. What values are being substituted into the equation? b. Describe the location of point P when the inequality statement shows “>”. Describe the location of point P when the inequality statement shows “<”. c. Drag point P until the statement becomes an equality (=). Where is point P? d. Why do the constants within the parentheses in the equation and the coordinates of the center of the circle have opposite signs? 2
2
6. Suppose a circle has the equation (x – 12) + (y + 4) = 25. a. What is the radius of the circle? b. What are the coordinates of the center of the circle? c. How can you determine whether the point (12, –9) lies on the circle? 7. Categorizing Equations: Place an equation in one of the categories in the table. Some of your cards are to go in one of the cells in the final column. You will need to figure out the coordinates for the center of the circle for all equations placed in this column. Make up your own equation for any empty cells. 2
2
(x+2) +(y−1) -­‐100=0 2
2
2
x +(y+1) =100 (x−2) +(y−1) =25 2
(x+2) +(y−1) =10 2
2
2
2
2
(y+1) +x =10 2
2
x +(y+1) =25 2
2
2
2
2
2
2
2
(x−2) +(1+y) =100 2
2
(y+1) +(x−2) =25 (x−2) +(y−1) +15=25 (y+1) +(x−2) =10 2
(y−1) +(x−2) =5 (x−2) +(y+1) +4=9 Center at (2,1) Center at (2,−1) Center at (0,−1) Center (__,__) Radius of ! 5 Radius of ! 10 Radius of 5 Radius of 10 8. Find the equation of a circle with center (2, 1) and radius 5. 9. When an earthquake occurs, energy waves radiate in concentric circles from the epicenter, the point above where the earthquake occurred. Stations with seismographs record the level of that energy and the length of time the energy took to reach the station. a. Suppose that one station determines that the epicenter of an earthquake is about 200 miles from the station. b. A second station, 120 miles east and 160 miles north of the first station, shows the epicenter to be about 235 miles away. Find an equation for the possible location of the epicenter. c. Using the information from parts a and b, find the possible locations of the epicenter. 10. The rectangular coordinates of three points in a plane are Q(–3, –1), R(–2,3), and S(1,–3). A fourth point T is chosen so that segment ST is equal to twice segment QR. What is the y-­‐coordinate of T? A. -­‐11 B. -­‐7 C. -­‐1 D. 1 E. 5 11. How do we represent circles in the x-­‐y coordinate plane? 12. Write the equation of a circle with center (4,0) and radius 3. 13. Write the equation of a circle with center (–2,3) and radius ½. 2
2
14. Where does the point (12, –9) lie relative to the circle with equation (x – 12) + (y + 4) = 25? Translating Circles Name______________________________________ Look for regularity in repeated reasoning. Expand each of the following. 2
2
2
2
2
1. (x+1) 2. (x+2) 3. (x+3) 4. (x+4) 5. (x+5) 2
2
2
2
2
6. (x–6) 7. (x–7) 8. (x–8) 9. (x–9) 10. (x–10) 11. a. Write an equation in standard form for the translation of x 2 + y 2 = 25 one unit left and three units down. b. Expanding the standard form of the equation of the circle and combining like terms will result in the general form of the circle. What is the equation in general form for the given circle? c. If we were given the general form of a circle, how could we rewrite the equation in standard form? 12. Write an equation for the translation of x 2 + y 2 = 9 four units up and three units left. b. What is the equation in general form for the given circle? c. Show how to rewrite the general form of the circle in standard form. Circles in General Form: Rewrite each circle in standard form. State its center and radius. 13. x 2 + 6 x + y 2 = 2 14. x 2 + 12 x + y 2 − 8 y = 5 15. 3x 2 + 12 x + 3 y 2 − 18 y = 2 16. x 2 + y 2 + 14 y = 15 17. x 2 − 10 x + y 2 − y = 1 18. 2 x 2 + 32 x + 2 y 2 + 24 y = 5 19. Write an equation for the translation of x 2 + y 2 = 10 two units right and three units down. (
)
20. Write an equation in standard form given the center 0,−1 with a radius of 7 . (
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21. Write an equation in standard form given center −3,5 and diameter 9. 22. Write the equation in standard form: x 2 + y 2 + 6 x − 8 y + 2 = 0 . 23. What is the distance between T (9, −5) and the center of the circle with equation ( x − 6 ) + ( y + 1) = 10 ? 2
2
24. Circle A has equation ( x + 5 ) + y 2 = 169 . The diameter of circle B is one fourth as long as the diameter of circle A. 2
What is the length of the radius of circle B? 25. Write an equation of a circle with a center at (–3,4) that is tangent to the y-­‐axis. 26. A circle is tangent to the y-­‐axis at y=3 and has one x-­‐intercept at x=1. What is the equation of the circle? 27. Write an equation of a circle with a center at (4,–1) that is tangent to the x-­‐axis. 28. A circle is tangent to the y-­‐axis and has a radius of three units. The center of the circle is in the third quadrant and lies on the graph y–2x=0. What is the product of the coordinates (h,k) of the center of the circle? 29. What are the coordinates of the center of a circle that is tangent to the y-­‐axis and intersects the x-­‐axis at (8,0) and (18,0)? 30. What is the equation for a circle that is tangent to the x-­‐axis at (4,0) and has y-­‐intercepts –2 and –8? 31. What is the equation of the circle tangent to the x-­‐axis at the point A(4,0) and the circle cuts in the y-­‐axis a chord that is 6 units? 32. A moving circle is tangent to the x-­‐axis and to a circle of radius 1 with center at (2,–6). Find the equation of the locus of the center of the moving circle. Polygons Name______________________________________ Draw each figure on a coordinate plane. Assign general coordinates to each vertex of the figure. 1. Square 2. Rectangle 3. Parallelogram 4. Kite Draw each figure on a coordinate plane. Assign general coordinates to each vertex of the figure. Use the midpoint formula, distance formula, or slope formula to prove the following statements using coordinate geometry. 5. Prove that the diagonals of a rectangle are congruent. 6. Prove that the diagonals of a square are congruent and perpendicular bisectors of each other. 7. Prove that the diagonals of a rhombus are perpendicular. 8. Prove that the diagonals of a parallelogram bisect each other. 9. The coordinates of the vertices of a triangle are O(0,0), M(k,!k 3 ), and N(2k,0). Classify ΔOMN. Justify your answer. 11. The coordinate of two vertices of ΔTUV are T(0,4) and U(4,0). Explain why the triangle will always be an isosceles triangle if V is any point on the line y=x except (2,2). 10. The coordinate of ΔOPQ are O(0,0), P(a,a), and Q(2a,0). Classify ΔOPQ by its side lengths. Is ΔOPQ a right triangle? Justify your answer. 12. ΔJKL has vertices J(–3,–2), K(0,–2), and L(–3,–8). ΔRST has vertices R(10,0), S(10,–3), and T(4,0). Graph the triangles in the same coordinate plane and show that they are congruent. Loci 1. Draw a point C. Draw and describe the locus of all points in the plane that are 1 centimeter from C. Name______________________________________ 2. Draw the point (–2,3). Draw and describe the locus of all points in the plane that are 3 cm from the point. Write equations for the locus of points in the coordinate plane that satisfy the given condition. 
3. Equidistant from A and B, where A(2,2) and B(6,2) 4. 4 units from !AB 5. 4 units from !AB 
7. 1 unit from !CD 6. Equidistant from C and D, where C(2,3) and D(4,5) 8. Points A and B lie in a plane. What is the locus of points in the plane that are equidistant from points A and B and are a distance of AB from B? Draw the figure. Then sketch the locus of points on the paper that satisfy the given condition. 9. Point P, the locus of points that are 1 inch from P. 10. Line k, the locus of points that are 1 inch from k. 11. Point C, the locus of points that are at least 1 inch from C. 12. Line j, the locus of points that are no more than 1 inch from j. 13. Point P lies on line l. What is the locus of points on l and 3 cm from P? 14. Point Q lies on line m. What is the locus of points 5 cm from Q and 3 cm from m? 15. Point R is 10 cm from line k. What is the locus of points that are within 10 cm of R, but further than 10 cm from k? 16. Lines l and m are parallel. Point P is 5 cm from both lines. What is the locus of points between l and m and no more than 8 cm from P? 17. A dog’s leash is tied to a stake at the corner of its doghouse, as shown at the right. The leash is 9 ft long. Sketch the locus of points that the dog can reach. 18. A rectangle is rotated three times. • Where is A at the end of the rotations? • What is the locus of points that A travels? • How far does A go? 19. A figure is rotated three times. • Where is A at the end of the rotations? • What is the locus of points that A travels? • How far does A go? 20. A string is wrapped around a regular hexagon, beginning and ending at A. Unwind the string clockwise from the hexagon, beginning at A. • Where is A when you finish unwinding the string? • What is the locus of points that A travels? • How far does A go? Partitioning 1. A 32 foot long piece of rope has a knot tied to divide the rope into a ratio of 3:5. Where is the knot? Name______________________________________ 2. Point P divides !AB in the ratio 3:1. a. What does it mean for point P to divide !AB in the ratio 3:1? b. Do you expect P to be closer to A or B? Why? c. How does the slope of !AP compare to the slope of !AB ? Why? 3. A is at 1, and B is at 10. Find the point, T, so that T is two-­‐
4. A is at –2 and B is at 7. Find the point, T, so that T is one-­‐third thirds of the distance from A to B. of the distance from A to B. 5. A is at 2 and B is at 7. Find the point, T, so that T partitions the 6. A is at –4 and B is at 10. Find the point, T, so that T partitions segment into a 2:3 ratio.
the segment into a 3:4 ratio. 7. Find the point Q along the directed line segment from point R(–3,3) to point S(6,–3) that divides the segment into the ratio 2 to 1. 9. Find the coordinates of point P, which lies along the directed line segment A(3,4) to B(6,10) and partitions the segment in the ratio of 3 to 2. 8. Find the point Q along the directed line segment from point R(–2,4) to point S(18,–6) that divides the segment into the ratio 3 to 7. 10. Find the coordinates of point P, which lies along the directed line segment A(1,1) to B(7,3) and partitions the segment in the ratio of 1 to 4. 11. Find the coordinates of point P, which lies along the directed line segment from M(–3,–3) to N(4,2) and partitions the segment in the ratio of 3 to 2. 13. Find the coordinates of T, which is two-­‐fifths of the distance from A(–2, 1) to B(8, 11).
15. Find the coordinates of T, which divides !AB into a ratio of 3:1 if A is at (-­‐1, -­‐6) and B is at (-­‐5, 2). 12. Find the coordinates of T, which is one-­‐third of the distance from A(2, 3) to B(5, 9). 14. Find the coordinates of T, which divides !AB into a ratio of 5:3 if A is at (-­‐4, -­‐2) and B is at (4, -­‐10). 16. Determine the ratio in which the line segment 3x+y-­‐9=0 divides the segment joining the points (1,3) and (2,7). Mastering Coordinate Geometry Name______________________________________ Join the dots to complete these quadrilaterals. Where there are options, find the one on the grid with the largest possible area. What are the perimeter and area of the polygon? 1. Parallelogram 2. Kite 3. Isosceles Trapezoid Pentagon ELWUD has vertices E(–2,–1), L(–1,2), W(1,2), U(3,1), and D(2,–3). Segment LZ joins L and the midpoint of segment ED. 4. Determine the slopes of the sides of pentagon ELWUD. Which two sides are parallel? 5. What are the coordinates of Z? 6. Which side of ELWUD is congruent to segment LZ? Show the work that leads to your conclusion. Prove that the diagonals of an isosceles trapezoid are congruent – first with the given numerical coordinates and then with the generalized coordinates. 7.
8. 9. Given: !DE bisects !AC and !AB . Prove: DE=½CB. 10. What is the distance from A to D? 11. A dog on a leash is tied to a rectangular-­‐shaped barn 20 m long and 10 m wide. The leash is 8 m long and fastened 6 m from the end of the longer side of the barn. In what region can the dog roam, and what is the area of that region? 12. Find the center and radius and sketch the graph of the 2
2
circle with the equation (x–5) +(y+1) =4. 13. Write an equation of the circle. 14. The center of a circle is at (–2,–1) and the point (2,–4) is on the circle. What is the equation of the circle in standard form? 2
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15. The standard equation of a circle is (x–2) +(y+1) =16. What is the diameter of the circle? 2
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16. Which of the points does not lie on the circle described by the equation (x+2) +(y–4) =25? A. (–2,–1) B. (1,8) C. (3,4) D. (0,5) 2
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17. What are the center and radius of a circle that has the standard equation (x+2) +(y–5) =169? 18. The center of a circle is on (–2,–3) and a point on the circle has coordinates (2,1). What is the equation of the circle in standard form? 19. The coordinates of the endpoints of the diameter of a circle are (3,–4) and (–5,6). What is the equation of the circle in standard form? 20. What is the equation of the secant line that passes through points (5,–3) and (1,5) on a circle? 21. The coordinates of the midpoint of the line joining the point (3p,4) and (–2,2q) are (5,p). What are the values of p and q? 22. Describe and correct the ERROR in writing the equation of a circle: An equation of a circle with center (–3,–5) and 2
2
radius 3 is (x–3) +(y–5) =9. 2
2
23. A horizontal secant passes 6 units below the center of the circle (x–3) +(y+4) =100. At what points does the secant intersect the circle and what is the length of the chord that connects these points? 2
2
24. Complete the square to find the standard form of the equation of a circle: x + 6x + y + 2y = 6. 25. A is at -­‐5 and B is at 5. Find the point, T, so that T is two-­‐fifths of the distance from A to B. 26. Find the coordinate, T, that is two-­‐ 27. Find the coordinate, T, that is thirds of the distance from A(1, 4) to three-­‐eighths of the distance from B(7, 13). A(6, 8) to B(-­‐2, 0). 28. What is the ratio in which the point (11,15) divides the line segment joining the point (15,5) and (9,20)?