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M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW The Perpendicular Bisector of a Segment Check off the steps as you complete them: ______ Open a Geometry window in Geogebra. ______ Use the segment tool of draw AB . ______ Under the point tool, select Midpoint of Center, and create the midpoint of AB . (This should be point C.) ______ Select the perpendicular line tool, and click on point C and AB . ______ Use the point tool to make point D anywhere on the perpendicular line. Your diagram should look something like the one at right. ______ Select Distance or Length (in the Angle menu), and measure the distances from D to A and from D to B by clicking on the points. ______ Drag point D anywhere along the line, and observe the measurements DA and DB. 1. What do you notice about DA and DB? 2. DC is called the perpendicular bisector of AB because it is perpendicular to AB and passes through the midpoint of AB . Complete this statement: Any point on the perpendicular bisector of a segment is: 1 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW Follow the directions to find the equation of the perpendicular bisector of the line segment through the given points. 3. Points: 2, 6 and 2, 4 a. Graph the segment with endpoints 2, 6 and 2, 4 . b. Find the midpoint of the segment. c. Find the slope of the segment. d. Find the slope of a line perpendicular to this segment. e. Use slope-intercept form y mx b or point-slope form y k m x h , as appropriate to find the equation of a line with slope from part (d) and point from part (b). f. Graph the perpendicular bisector. 2 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW Follow the directions to find the equation of the perpendicular bisector of the line segment through the given points. 4. Points: 1, 6 and 4,3 a. Graph the segment with endpoints 1, 6 and 4,3 . b. Find the midpoint of the segment. c. Find the slope of the segment. d. Find the slope of a line perpendicular to this segment. e. Use slope-intercept form y mx b or point-slope form y k m x h , as appropriate to find the equation of a line with slope from part (d) and point from part (b). f. Graph the perpendicular bisector. 3 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW ASSIGNMENT 2M Follow the directions to find the equation of the perpendicular bisector of the line segment through the given points. 1. Points: 4,7 and 1, 3 a. Graph the segment with endpoints 4,7 and 1, 3 . b. Find the midpoint of the segment. c. Find the slope of the segment. d. Find the slope of a line perpendicular to this segment. e. Use slope-intercept form y mx b or point-slope form y k m x h , as appropriate to find the equation of the perpendicular bisector. f. Graph the perpendicular bisector. 4 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW Follow the directions to find the equation of the perpendicular bisector of the line segment through the given points. 2. Points: 2,5 and 4,1 a. Graph the segment with endpoints 2,5 and 4,1 . b. Find the midpoint of the segment. c. Find the slope of the segment. d. Find the slope of a line perpendicular to this segment. e. Use slope-intercept form y mx b or point-slope form y k m x h , as appropriate to find the equation of the perpendicular bisector. f. Graph the perpendicular bisector. 5 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW Practice for Section 3-6 Follow the directions for each: 1. Points: 4, 2 and 8, 4 a. Graph the segment with endpoints 4, 2 and 8, 4 . b. Find the midpoint of the segment. c. Find the slope of the segment. d. Find the slope of a line perpendicular to this segment. e. Find the equation of the perpendicular bisector. f. Graph the perpendicular bisector. g. Use algebra to rewrite your equation from part (e) in slope-intercept form y mx b . h. Use algebra to rewrite your equation from part (g) in standard form (x and y on the left side, no decimals or fractions). 6 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW 2. Points: 4, 6 and 6, 2 a. Graph the segment with endpoints 4, 6 and 6, 2 . b. Find the midpoint of the segment. c. Find the slope of the segment. d. Find the slope of a line perpendicular to this segment. e. Find the equation of the perpendicular bisector. f. Graph the perpendicular bisector. g. Use algebra to rewrite your equation from part (e) in slope-intercept form y mx b . h. Use algebra to rewrite your equation from part (g) in standard form (x and y on the left side, no decimals or fractions). 7 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW Review of 3-3 to 3-6 Figures in this review may not be drawn to scale. Multiple Choice: ____ 1. In the diagram at right, TY MV , and mHRY 100 . Which of the following statements does not have to be true? (A) (B) (C) (D) mFHM 100 mMHR 80 TRS and FHM are alternate exterior angles YRS and VHR are alternate interior angles ____2. In the diagram at right, which statement is the reason that mPIR mYMS ? When two parallel lines are cut by a transversal, then… (A) (B) (C) (D) Alternate exterior angles are congruent Alternate interior angles are congruent Consecutive interior angles are supplementary Corresponding angles are congruent _____3. In the diagram at right, 2 3 . Which of the following must be true? (A) (B) (C) (D) r t m8 m6 m4 m6 m5 m3 ____4. In the diagram at right, m6 m7 180 . Which of the following does not have to be true? (A) (B) (C) (D) m1 m4 180 m5 m4 180 r s m2 m7 8 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW ____5. Annie wanted to solve for x, so she set up the equation 5x 2 3x 12 180 . What is her reason? If two parallel lines are cut by a transversal, then… (A) (B) (C) (D) Linear pairs are supplementary Corresponding angles are supplementary Alternate interior anlges are congruent Consecutive interior angles are supplementary ____6. To solve for x in the diagram at right, Ben used the equation 9 x 5 10 x 5 . The reason that justifies Ben’s equation is: If two parallel lines are cut by a transversal, then… (A) (B) (C) (D) Alternate interior angles are congruent Alternate exterior angles are congruent Corresponding angles are congruent Consecutive interior angles are supplementary 7. Complete the proof: Given: m n , r s , m1 130 Prove: m3 50 Statements Reasons 1. 1. 2. 2. Corresponding s are 3. m2 130 3. s have = measures 4. m2 m3 180 4. 5. 130 m3 180 5. 6. 6. 9 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW 8. Complete the proof: Given: 1 2 , m3 88 Prove: m4 92 Statements Reasons 1. 1. 2. c d 2. 3. m3 m4 180 3. 4. 4. Substitution 5. 5. Subtraction Find the slope of each line. 9. 10. 11. 12. Find the slopes of PQ and RS , and determine whether the lines are parallel, perpendicular, or neither. 13. P 1,0 , Q 5,3 , R 6, 1 , S 0,2 14. P 5,1 , Q 1, 1 , R 2,1 , S 3, 2 10 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW Graph each line, and write its equation. 15. the horizontal line through 3,1 16. the line with slope Equation: 8 passing through 1, 5 5 Equation: 17. the line through 1, 3 and 2, 4 18. the line with x-intercept 2 and y-intercept 1 Equation: Equation: 11 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW Graph each line. 19. y 3 3 x 1 4 20. y 4 x2 3 Find the slope of each line. 21. y x 1 4 23. y 4 22. 3x 2 y 6 24. (a) Graph MN with M 2,1 and N 2, 4 . (b) Graph line parallel to MN with y-intercept 4 . (c) Write an equation for line . (d) Graph line k perpendicular to MN and passing through 1, 2 . (e) Write an equation for line k. 12 2 x 2 3 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW 25. Write an equation of the line through the point 4,5 and parallel to the line 2 x y 6 . 26. Write an equation of the line through the point 3, 1 and perpendicular to x 3 y 2 . 27. (a) Graph AB with A 1, 5 and B 3,5 . (b) Find the midpoint of AB . (c) Find the slope of AB . (d) Find the slope of a line perpendicular to AB . (e) Find an equation of the perpendicular bisector of AB in point-slope form. (f) Use algebra to rewrite your equation from (e) in slope-intercept form. (g) Use algebra to rewrite your equation from (g) in standard form. 13 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW 28. (a) Graph AB with A 5, 2 and B 1, 4 . (b) Find the midpoint of AB . (c) Find the slope of AB . (d) Find the slope of a line perpendicular to AB . (e) Find an equation of the perpendicular bisector of AB in point-slope form. (f) Use algebra to rewrite your equation from (e) in slope-intercept form. (g) Use algebra to rewrite your equation from (g) in standard form. 14 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW 29. (a) Graph AB with A 6,0 and B 4, 5 . (b) Find the midpoint of AB . (c) Find the slope of AB . (d) Find the slope of a line perpendicular to AB . (e) Find an equation of the perpendicular bisector of AB in point-slope form. (f) Use algebra to rewrite your equation from (e) in slope-intercept form. (g) Use algebra to rewrite your equation from (g) in standard form. 15 M2 GEOMETRY PACKET 5 FOR UNIT 2 – SECTION 3-6 & TEST REVIEW 30. (a) Graph AB with A 7,1 and B 3,5 . (b) Find the midpoint of AB . (c) Find the slope of AB . (d) Find the slope of a line perpendicular to AB . (e) Find an equation of the perpendicular bisector of AB in point-slope form. (f) Use algebra to rewrite your equation from (e) in slope-intercept form. (g) Use algebra to rewrite your equation from (g) in standard form. 16