Altitudes and Perpendicular Bisectors

Transcription

1 6 2 hat ou ll Learn ou ll learn to identify and construct s and perpendicular bisectors in triangles. ltitudes and erpendicular isectors In geometry, an of a triangle is a perpendicular segment with one endpoint at a vertex and the other endpoint on the side opposite that is perpendicular to side vertex. he. hy It s Important onstruction arpenters use perpendicular bisectors and s when framing roofs. ee xercise 22. In the following activity, you will construct an of a triangle. onstruction aterials: tep 1 compass straightedge raw a triangle like. tep 2 lace the compass point at and draw an arc that intersects side in two points. Label the points of intersection and. tep 3 tep hapter 6 ore bout riangles lace the compass point at and. Using the draw an arc below same compass setting, place the compass point at and draw an arc to intersect the one drawn. Use a straightedge to align the vertex and the point where the two arcs intersect. raw a segment from vertex to side. Label the point of intersection.

2 ry hese 1. hat can you say about? 2. oes have any other s? If so, construct them. 3. ake a conjecture about the number of s in a triangle. n of a triangle may not always lie inside the triangle. ltitudes of riangles acute triangle right triangle obtuse triangle he is inside the triangle. he is a side of the triangle. xamples he is outside the triangle. ell whether each red segment is an of the triangle , is a vertex, and is on the side opposite. o, is an of the triangle. N N, is a vertex, and is on the side opposite. o, is an of the triangle. is not a perpendicular segment. o, is not an of the triangle. our urn a. b. c. In this text, we will assume that a perpendicular bisector can be a line or a segment contained in that line. nother special line in a triangle is a perpendicular bisector. perpendicular line or segment that bisects a side of a triangle is called the perpendicular bisector of that side. Line m is a perpendicular bisector of side. m I perpendicular bisector is the midpoint of. Lesson 6 2 ltitudes and erpendicular isectors 235

3 In some triangles, the perpendicular bisector and the are the same. If the perpendicular bisector of a side contains the opposite vertex, then the perpendicular bisector is also an. he line containing is the perpendicular bisector of. is an. is the midpoint of. xamples ell whether each red line or segment is a perpendicular bisector of a side of the triangle o, is a and is the midpoint of perpendicular bisector of side in. but is not the midpoint of. o, is not a perpendicular bisector of side in. 6 is a and is the midpoint of. o, perpendicular bisector of side in. our urn d. e. f. N 236 hapter 6 ore bout riangles U

4 e rld al o xample 7 usic Link balalaika is a stringed musical instrument that has a triangular body. alalaikas are commonly played when performing ussian songs and dance music. three-stringed balalaika is shown at the right. ell whether string is an, a perpendicular bisector, both, or neither. I tring contains the midpoint of I. In addition, string is perpendicular to I. ince it also contains the vertex,, opposite I, string is both a perpendicular bisector and an. tring heck for Understanding ommunicating athematics 1. raw a right triangle. hen construct all of the s of the triangle. 2. raw a triangle like. hen use a segment bisector construction to construct the perpendicular bisector of. 3. uided ractice xamples 1 6 ompare and contrast s and perpendicular bisectors. perpendicular bisector xercise 2 or each triangle, tell whether the red segment or line is an, a perpendicular bisector, both, or neither O 6. xample 7 N 7. amping he front of a pup tent is shaped like a triangle. ell whether the roof pole is an, a perpendicular bisector, both, or neither. Lesson 6 2 ltitudes and erpendicular isectors 237

5 xercises ractice or each triangle, tell whether the red segment or line is an, a perpendicular bisector, both, or neither N 10. omework elp or xercises 8 16, , 19, 20 ee xamples 1 6 I 4 6 xtra ractice 11. O 12. K U 13. ee page K V 17. Name a perpendicular bisector in. 18. ell whether is a perpendicular bisector, an, both, or neither. xercises In, is the perpendicular bisector. Is it possible to construct other of perpendicular bisectors in? ake a conjecture about the number of perpendicular bisectors of a triangle. pplications and roblem olving 20. rchitecture he ransamerica building in an rancisco is triangular in shape. opy the triangle onto a sheet of paper. hen construct the perpendicular bisector of each side. ransamerica uilding 238 hapter 6 ore bout riangles

6 ighway ramp 21. ransportation here are four major types of highway interchanges. One type, a cloverleaf interchange, is shown. Notice that each ramp along with sections of the highway form a triangle. ell whether highway is an, a perpendicular bisector, both, or neither. ighway 22. onstruction he most common type of design for a house roof is a gable roof. he illustration shows the structural elements of a gable roof. rafter collar tie wall stud top plate a. hich structural element is a perpendicular bisector? b. ell whether the top plate is an. xplain your reasoning. c. ell whether the collar tie is a perpendicular bisector. xplain your reasoning. 23. ritical hinking raw two types of triangles in which the is on the line that forms the perpendicular bisector. Identify the types of triangles drawn, and draw the and perpendicular bisector for each triangle. ixed eview 24. lgebra In,,, and are medians. If x 2 and 2x 10, find the value of x. (Lesson 6 1) 25. etermine whether and are congruent by,,, or. If it is not possible to prove that they are congruent, write not possible. (Lessons 5 5 & 5 6) 26. ind the slope of the line passing through points at (2, 3) and ( 2, 4). (Lesson 4 5) tandardized est ractice 27. hort esponse raw the next figure in the pattern. (Lesson 1 1) 28. ultiple hoice ndrew is buying a pair of sunglasses priced at $ hat is the total cost of the sunglasses if he needs to pay a sales tax of 6%? ound to the nearest cent. (ercent eview) $19.10 $19.94 $20.13 $ Lesson 6 2 ltitudes and erpendicular isectors 239