Do Now Take a ruler from the bookshelf. Take out a compass. Draw an obtuse angle. Construct (using only a compass and straightedge) a duplicate angle. PERPENDICULAR BISECTORS Perpendicular Bisectors—Terms A segment bisector—a line, ray, or segment that passes through the midpoint of a segment. Cuts the line segment in half Perpendicular lines—intersect at a right angle. Perpendicular bisector—passes through the midpoint of a segment at a right angle. Equidistant—the same distance Constructing Perpendicular Bisectors Step 1: Draw a line segment. Set your compass to more than half the distance between the two endpoints. Step 2: Using one endpoint as center, swing an arc on both sides of the segment. Step 3: Using the same compass setting, swing an arc from the other endpoint to intersect each arc. Step 4: Mark your two intersection points and connect them. Perpendicular Bisector Conjecture If a point is on the perpendicular bisector of a segment, then it is _________ from the endpoints. equidistant Converse of Perpendicular Bisector Conjecture If a point is equidistant from the endpoints of a segment, then it is on the _______________of the segment. perpendicular bisector Also true! Practice Draw and label AB. Construct the perpendicular bisector of AB. Practice Draw and label QD. Construct perpendicular bisectors to divide QD into four congruent segments. Hint: To divide it into two congruent segments, you need a perpendicular bisector. Can you divide each of those segments again? Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line. Exploring Slopes What do you notice about the slopes of parallel lines? What do you notice about the slopes of perpendicular lines? Slopes of Parallel Lines Find the slope of each line. What do you notice? Parallel Lines have equal slopes Equations of Parallel Lines Are these lines parallel? y=3x +8 y=3x – 4 How do you know? Slopes of Perpendicular Lines Find the slope of each of these lines. What do you notice? Perpendicular lines have opposite reciprocal slopes. (Both opposite AND reciprocal) Equations of Perpendicular Lines Are these lines perpendicular? y= 5x + 7 y= 5x – 2 NO! y= ½ x – 3 y= - ½ x – 9 NO! y= ¼ x y= 4x + 7 NO! y= -⅓x + 2 y= 3x – 4 Derive the Expression for Slopes of Perpendicular Lines If this were the slope of a line, what would be the slope of a line perpendicular to it? 3 1/6 -8 -1/2 3/4 -t a/b m Slope and Midpoint To find slope: To find midpoint: Practice Line segment AB starts at A (-4, 1) and ends at B (0, 3). Line segment CD starts at C (-1, 5) and ends at D (1, 1). 1. Determine if these lines are perpendicular bisectors. A. B. Hint: They would have to be perpendicular (opposite reciprocal slopes) Hint: They would also have to be bisectors (have the same midpoint) Before the Exit Slip If you finish early, you may take some time to finish the 3.1 worksheet or the 3.2 worksheet. Save about 12 minutes for the exit slip. Today’s Objectives Duplicate a line segment, an angle and a polygon Construct perpendicular bisectors and midpoints Make conjectures about perpendicular bisectors Use Problem Solving skills Exit Slip For all exercises, do not erase your construction marks. For #1-2, Determine if these two lines are parallel, perpendicular, or intersecting. How do you know? 1. y=7x+3, y=-1/7x – 6 2. y=1/3x – 8, y=-1/3x – 3 3. Draw a line segment. Label it PQ, then construct its perpendicular bisector. 4. Line segment AB starts at A (1, 2) and ends at B (4, 0). Line segment CD starts at C (.5, -2) and ends at D (4.5, 4). A. B. Determine if these lines are perpendicular bisectors. Explain your reasoning.