The Perpendicular Bisector of a Segment
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- Edwin Wood
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1 Check off the steps as you complete them: The Perpendicular Bisector of a Segment 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
2 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 2, 4. and 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
3 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 4,3. and 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
4 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 equation of the perpendicular bisector. or point-slope form y k m x h, as appropriate to find the f. Graph the perpendicular bisector. 4
5 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 equation of the perpendicular bisector. or point-slope form y k m x h, as appropriate to find the f. Graph the perpendicular bisector. 5
6 Follow the directions for each: 1. Points: 4, 2 and 8,4 M2 GEOMETRY PACKET 5 FOR UNIT 2 SECTION 3-6 & TEST REVIEW Practice for Section 3-6 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
7 2. Points: 4,6 and 6,2 M2 GEOMETRY PACKET 5 FOR UNIT 2 SECTION 3-6 & TEST REVIEW 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
8 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 m HRY 100. Which of the following statements does not have to be true? (A) m FHM 100 (B) m MHR 80 (C) TRS and FHM are alternate exterior angles (D) YRS and VHR are alternate interior angles 2. In the diagram at right, which statement is the reason that m PIR m YMS? When two parallel lines are cut by a transversal, then (A) Alternate exterior angles are congruent (B) Alternate interior angles are congruent (C) Consecutive interior angles are supplementary (D) Corresponding angles are congruent 3. In the diagram at right, 2 3. Which of the following must be true? (A) r t (B) m 8 m 6 (C) m 4 m 6 (D) m 5 m 3 4. In the diagram at right, m 6 m Which of the following does not have to be true? (A) m 1 m (B) m 5 m (C) r s (D) m 2 m 7 8
9 5. Annie wanted to solve for x, so she set up the equation 5x 2 3x What is her reason? If two parallel lines are cut by a transversal, then (A) Linear pairs are supplementary (B) Corresponding angles are supplementary (C) Alternate interior anlges are congruent (D) Consecutive interior angles are supplementary 6. To solve for x in the diagram at right, Ben used the equation 9x 5 10x 5. The reason that justifies Ben s equation is: If two parallel lines are cut by a transversal, then (A) Alternate interior angles are congruent (B) Alternate exterior angles are congruent (C) Corresponding angles are congruent (D) Consecutive interior angles are supplementary 7. Complete the proof: Given: m n, r s, m Prove: m 3 50 Statements Reasons Corresponding s are 3. m s have = measures 4. m 2 m m
10 8. Complete the proof: Given: 1 2, m 3 88 Prove: m 4 92 M2 GEOMETRY PACKET 5 FOR UNIT 2 SECTION 3-6 & TEST REVIEW Statements Reasons c d m 3 m Substitution Subtraction Find the slope of each line 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
11 Graph each line, and write its equation. M2 GEOMETRY PACKET 5 FOR UNIT 2 SECTION 3-6 & TEST REVIEW 15. the horizontal line through 3,1 16. the line with slope passing through 1, Equation: Equation: 17. the line through 1, 3 and 2,4 18. the line with x-intercept 2 and y-intercept 1 Equation: Equation: 11
12 Graph each line y 3 x 1 4 y x 2 3 Find the slope of each line. x 21. y x 2y y 4 x (a) Graph MN with M 2,1 and 2,4 N. (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
13 25. Write an equation of the line through the point 4,5 and parallel to the line 2x y Write an equation of the line through the point 3, 1 and perpendicular to x 3y (a) Graph AB with A 1, 5 and 3,5 B. (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
14 28. (a) Graph AB with A 5,2 and 1,4 M2 GEOMETRY PACKET 5 FOR UNIT 2 SECTION 3-6 & TEST REVIEW B. (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
15 29. (a) Graph AB with A 6,0 and 4, 5 B. (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
16 30. (a) Graph AB with A 7,1 and 3,5 M2 GEOMETRY PACKET 5 FOR UNIT 2 SECTION 3-6 & TEST REVIEW B. (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
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