10-6 The Meaning of Locus (Day 1) Definition – Locus (plural Loci): 1. Sketch and describe each locus. a. The locus of points 2 units from line n. b. The locus of points equidistant from Is n part of the locus? two parallel lines m and p. 2. Sketch and describe the locus of points equidistant from points A and B. 1 4. Use a ruler for these! 5. You may want to use a string or shoelace and tape or thumbtacks. 6. You may want to use a coin. End 2 10.7 Intersection of Loci (Day 2) 1. Suppose points P and Q are 4 units apart. Sketch and describe the locus of points equidistant from both points and 3 units from Q. 2. In this problem, you will consider the locus of points equidistant from points P, Q, and R. Suppose P, Q, and R are not on the same line (i.e., non-collinear). P R Q (a) Sketch the locus of points equidistant from P and Q. (b) Sketch the locus of points equidistant from Q and R. (c) Let point A be the intersection of the two perpendicular bisectors you drew in (a) and (b). Explain why this point must be equidistant from P, Q, and R. (d) Draw the perpendicular bisector of PR. Explain why this line must also pass through point A. 3. Now, suppose points P, Q, and R lie on the same line (i.e., collinear). Can you find the locus of points equidistant from the three points? Why or why not? 3 4. Consider ΔABC shown below. (a) Sketch the locus of points equidistant from AB and AC. What is this line called? (b) Sketch the locus of points equidistant from AC and BC. (c) Let point P be the intersection of the two angle bisectors you drew in (a) and (b). Explain why this point must be equidistant from AB, AC, and BC. (Hint: Draw in the segments that represent the distance from P to each side and think about congruent triangles.) (d) Draw the bisector of ∠B. Explain why this line must also pass through point P. (e) Draw a circle inscribed in ΔABC with point P as its center. Why must this circle fit perfectly inside the triangle? 5. Consider ΔABC shown below. (a) Sketch and describe the locus of points equidistant from vertices A, B, and C. See Problem 2. Call this point P. Note: Point P is not equidistant from the sides of the triangle. (b) Draw a circle circumscribed about ΔABC with point P as its center. Why does the triangle fit perfectly inside this circle? 4 6. (a) The point where the angle bisectors of a triangle meet is called the incenter. See Problem 4(e) Explain the significance of this point. (b) The point where the perpendicular bisectors of a triangle meet is called the circumcenter. See Problem 5(b). Explain the significance of this point. (c) Are the incenter and the circumcenter of a triangle necessarily the same point? Can they ever be the same point? Explain. 7. Sketch and describe the locus of points that are equidistant from parallel lines l and m and also 3 units from point A. You will need to consider three cases. (Hint: The locus could contain 0, 1, or 2 points depending on the location of point A.) 8. Consider a square whose sides have length 4. Segment AB of length 3 is placed so that its endpoints are always on the side of the square. Describe the locus formed by all midpoints of AB as it moves throughout the square. [If you have trouble, cut a 3-unit segment out of graph paper and mark its midpoint. Then move the paper segment around a 4 x 4 graph paper square marking where its midpoint falls.] 5 9. Sketch and describe the following loci. (a) Equidistant from intersecting lines l and m. (b) Equidistant from the sides of ∠A and 2 units from vertex A. (c) Given ΔRST, equidistant from RS and RT and also equidistant from R and S. 10. (a) Draw a triangle whose circumcenter lies outside the triangle. (b) Can you draw a triangle whose incenter lies outside the triangle? Explain. 11. Given point E and line k, sketch and describe the locus of points 3 cm from E and 2 cm from k. You will need to consider five cases. 12. Given points A, B, C, and D, can you always find point P equidistant from the four points? If not, how must the four points be arranged so that you can find the equidistant point? Also, describe how you would go about finding this point. 13. Given quadrilateral ABCD, can you always find point P equidistant from AB, BC, CD, and AD? Explain. End 6 11.1 Points in a Plane & 11.2 Distance and Midpoint (Day 3) The Distance Formula The Midpoint Formula 7 End 8 11.3 Slope of a Line (Day 4) - Do work on a separate piece of paper. 9 This should be a coordinate proof, not a typical 2-column proof. 1. Give coordinates to the points A, B, and C. i.e. A is (-a, 0). 2. Calculate slopes, midpoints, distances, etc. in order to reach your “prove” statement. End 10 11.4 Proofs Using a Coordinate System (Day 5) - Do work on a separate piece of paper. 11 End 12 Quiz Review (Day 6) Read the following problem and student answer. Problem: Find the locus of points that are equidistant from points A and B and 4 units from the midpoint of AB. Student Answer: A B Question: If you were a Geometry Advanced teacher, how would you grade this student’s answer for this problem? What would they need to add or change to make it better? Part 1: Practice sketching and describing the following loci. Use both words and pictures. 1. Locus of points equidistant from 2 points. 2. Locus of points equidistant from 2 intersecting lines. 3. Locus of points equidistant from 2 parallel lines. 4. Locus of points equidistant from 2 perpendicular lines. 5. Locus of points equidistant from 1 point. 6. Locus of points equidistant from 1 line. 7. Locus of points equidistant from 2 perpendicular lines and 6 units from their point of intersection. 8. Locus of points equidistant from the vertices of an isosceles triangle and also k units from the base of the triangle. 13 Part 2: Describe how you would prove that 4 connected points in the coordinate plane form: 1. An isosceles trapezoid? 2. A square? 3. A rectangle? 4. A rhombus? Part 3: Describe how you would prove that 3 connected points in the coordinate plane form: 1. An isosceles triangle? 2. An equilateral triangle? 3. An obtuse triangle? 4. A right triangle? 5. An acute triangle? 6. A straight line? Part 4: True or False? 1. There are 2 lines through point E(-2, 5) with slope m = 3. 2. There is more than one line perpendicular to a different line that goes through the origin and has a slope of 2. 14 Quiz Review (continued) – Write answers on a separate sheet of paper. 1. Sketch and describe the following loci. (a) Equidistant from intersecting lines l and m. (b) Equidistant from the sides of ∠A and 2 units from vertex A. (c) Given ∆RST, equidistant from RS and RT and also equidistant from R and S. 2. (a) Draw a triangle whose circumcenter lies outside the triangle. (b) Can you draw a triangle whose incenter lies outside the triangle? Explain. 3. Given point E and line k, sketch and describe the locus of points 3 cm from E and 2 cm from k. You will need to consider five cases! 4. Given points A, B, C, D, can you always find point P equidistant from the four points?______ If not, how must the four points be arranged so that you can find the equidistant point?______ Also, describe how you would go about finding this point. 5. Given quadrilateral ABCD, can you always find point P equidistant from AB, BC,CD,and AD ? Explain. 6. ∆RST has vertices R(0, 0), S(2a, 0), and T(2b, 2c). (a) Find the coordinates of the midpoint of each side of ∆RST. (b) Use coordinate geometry to show that the perimeter of the triangle formed by joining the midpoints is one half the perimeter of ∆RST. 7. P is a point on a circle. Sketch and describe the locus of points traced by P as the circle rolls along line l. l P 8. Find the perimeter and area of a triangle with the following vertices: A(0, 4), B(-4, 2), C(6, -3). Give the exact values. 15 9. A circle has center (4, 4) and radius 3 2 . Suppose the points (7, y) and (1, z) are on the circle. Find all possible values of y + z. 10. At which point(s) does the circle in problem #9 intersect the y-axis? 11. Suppose A(-3, -2), B(-7, 2), C(0, 4), and D(t, r) are vertices of a quadrilateral. Find point D such that CD is parallel to AB and D is the midpoint of some segment CG of length 12. 12. Suppose A(-2, 5), B(3, y), C(2, 7), and D(x, -4). Find the values of x and y if AB and CD intersect at the point E(-1, 4). What else can you say about AB and CD? 13.