3-3. Perpendicular Lines Going Deeper EXAMPLE. Constructing a Perpendicular Bisector REFLECT. Name Class Date

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1 Nae lass ate 3-3 erpendicular Lines Going eeper Essential question: How can you construct perpendicular lines and prove theores about perpendicular bisectors? erpendicular lines are lines that intersect at right angles. In the figure, line l is perpendicular to line and you write l. The right angle ark in the figure indicates that the lines are perpendicular. l Video Tutor 1 The perpendicular bisector of a line segent is a line perpendicular to the segent at the segent s idpoint. M9 12.G.O.12 EXMLE onstructing a erpendicular isector l onstruct the perpendicular bisector of. Work directly on the figure below. lace the point of your copass at. Using a copass setting that is greater than half the length of, draw an arc. Houghton Mifflin Harcourt ublishing opany Without adjusting the copass, place the point of the copass at and draw an arc intersecting the first arc at and. Use a straightedge to draw. is the perpendicular bisector of. REFLET 1a. How can you use a ruler and protractor to check the construction? Module 3 59 Lesson 3

2 You can use reflections and their properties to prove a theore about perpendicular bisectors. Refer to the diagra in the roof below as you read this definition of reflection. reflection across line aps a point to its iage as follows. Line is the perpendicular bisector of if and only if is not on line. The iage of is if and only if is on line. The notation r () = eans that the iage of point after a reflection across line is point. The notation r () = eans that the iage of point is point, which iplies that is on line. 2 M9 12.G.O.9 ROOF erpendicular isector Theore If a point is on the perpendicular bisector of a segent, then it is equidistant fro the endpoints of the segent. Given: is on the perpendicular bisector of. rove: = oplete the following proof. onsider the reflection across line. Then r () = because lso, r () = by the definition of reflection. Therefore, = because REFLET 2a. Suppose you use a copass and straightedge to construct the perpendicular bisector of a segent,. If you choose a point on the perpendicular bisector, how can you use your copass to check that is equidistant fro and? K M L 2b. What conclusion can you ake about KLJ in the figure? Explain. Houghton Mifflin Harcourt ublishing opany 2c. escribe the point on the perpendicular bisector of a segent that is closest to the endpoints of the segent. J Module 3 60 Lesson 3

3 The converse of the erpendicular isector Theore is also true. In order to prove the converse, you will use the ythagorean Theore. Recall that the ythagorean Theore states that in a right triangle with legs of length a and b and hypotenuse of length c, a 2 + b 2 = c 2. a c b a 2 + b 2 = c 2 3 M9 12.G.O.9 ROOF onverse of the erpendicular isector Theore If a point is equidistant fro the endpoints of a segent, then it lies on the perpendicular bisector of the segent. Given: = rove: is on the perpendicular bisector of. Use the ethod of indirect proof. ssue the opposite of what you want to prove and show this leads to a contradiction. ssue that point is not on the perpendicular bisector of. Then when you draw a perpendicular fro to the line containing and, the perpendicular intersects this line at a point Q, which is not the idpoint of. Q oplete the following to show that this assuption leads to a contradiction. Q fors two right triangles, Q and Q. Q 2 + Q 2 = 2 and Q 2 + Q 2 = 2 by Houghton Mifflin Harcourt ublishing opany Subtract these equations: However, 2-2 = 0 because Q 2 + Q 2 = 2 Q 2 + Q 2 = 2 Q 2 - Q 2 = 2-2 Therefore, Q 2 - Q 2 = 0. This eans Q 2 = Q 2 and Q = Q. This contradicts the fact that Q is not the idpoint of. Thus, the initial assuption ust be incorrect, and ust lie on the perpendicular bisector of. REFLET 3a. In the proof, once you know Q 2 = Q 2, why can you conclude Q = Q? 3b. Explain how the converse of the erpendicular isector Theore justifies the copass-and-straightedge construction of the perpendicular bisector of a segent. Module 3 61 Lesson 3

4 The perpendicular bisector construction can be used as part of the ethod for drawing the perpendicular to a line through a given point not on the line. 4 M9 12.G.O.12 EXMLE onstructing a erpendicular to a Line onstruct a line perpendicular to line that passes through point. Work directly on the figure at right. lace the point of your copass at. raw an arc that intersects line at two points, and. onstruct the perpendicular bisector of. This line will pass through and be perpendicular to line. REFLET 4a. oes the construction still work if point is on line? Why or why not? practice 1. onstruct the perpendicular bisector of the segent shown below. 2. onstruct a line perpendicular to line that passes through point. Houghton Mifflin Harcourt ublishing opany Module 3 62 Lesson 3

5 Nae lass ate dditional ractice onstruct a line perpendicular to line r. r 2. onstruct the perpendicular bisector of. 3. onstruct a line perpendicular to through. Then, using your two perpendicular lines, construct a right triangle that has as a vertex and a hypotenuse with length XY. Houghton Mifflin Harcourt ublishing opany Use the diagra to find the given quantity c 5. 6 c 1.5 ft G E X F 2 ft 2 ft Y = GE = Module 3 63 Lesson 3

6 roble Solving 1. Use geoetric constructions to find a single point that is equidistant fro and, and also equidistant fro and. (Note: The distance fro the point to does not have to be the sae as the distance fro the point to. It only atters that the point is equidistant fro each pair.) 2. If the two segents fro Exercise 1 were arranged so that they were both part of the sae line, as shown below, could you still find a point that is equidistant fro and, and also equidistant fro and? Explain why or why not. Select the best answer. 3. The road fro Westtown to Easttown is the perpendicular bisector of the road fro Northtown to Southtown. Given that fact and the distances arked on the ap, how far is Westtown fro Easttown? 3 i 4 i 5 i 6 i Westtown 5 i Northtown Southtown Easttown 3 i Houghton Mifflin Harcourt ublishing opany Module 3 64 Lesson 3