3. roofs with erpendiculr Lines Essentil Question Wht conjectures cn ou ke out perpendiculr lines? Writing onjectures Work with prtner. Fold piece of pper in hlf twice. Lel points on the two creses, s shown.. Write conjecture out nd. Justif our conjecture.. Write conjecture out O nd O. Justif our conjecture. O Eploring Segent isector Work with prtner. Fold nd crese piece of pper, s shown. Lel the ends of the crese s nd.. Fold the pper gin so tht point coincides with point. rese the pper on tht fold.. Unfold the pper nd eine the four ngles fored the two creses. Wht cn ou conclude out the four ngles? Writing onjecture ONSTRUTING VILE RGUMENTS To e proficient in th, ou need to ke conjectures nd uild logicl progression of stteents to eplore the truth of our conjectures. Work with prtner.. rw, s shown.. rw n rc with center on ech side of. Using the se copss setting, drw n rc with center on ech side of. Lel the intersections of the rcs nd. c. rw. Lel its intersection with s O. Write conjecture out the resulting digr. Justif our conjecture. O ounicte Your nswer. Wht conjectures cn ou ke out perpendiculr lines? 5. In Eplortion 3, find O nd O when = units. Section 3. roofs with erpendiculr Lines 7
3. Lesson Wht You Will Lern ore Voculr distnce fro point to line, p. 8 perpendiculr isector, p. 9 Find the distnce fro point to line. onstruct perpendiculr lines. rove theores out perpendiculr lines. Solve rel-life proles involving perpendiculr lines. Finding the istnce fro oint to Line The distnce fro point to line is the length of the perpendiculr segent fro the point to the line. This perpendiculr segent is the shortest distnce etween the point nd the line. For eple, the distnce etween point nd line k is. k distnce fro point to line Find the distnce fro point to. Finding the istnce fro oint to Line ( 3, 3) (2, 0) REMEMER Recll tht if (, ) nd ( 2, 2 ) re points in coordinte plne, then the distnce etween nd is = ( 2 ) 2 + ( 2 ) 2. (, 3) (, ) ecuse, the distnce fro point to is. Use the istnce Forul. = ( 3 ) 2 + [3 ( )] 2 = ( ) 2 + 2 = 32 5.7 So, the distnce fro point to is out 5.7 units. Monitoring rogress. Find the distnce fro point E to FH. Help in English nd Spnish t igidesmth.co F(0, 3) G(, 2) H(2, ) 2 2 E(, 3) 8 hpter 3 rllel nd erpendiculr Lines
onstructing erpendiculr Lines Use copss nd strightedge to construct line perpendiculr to line through point, which is not on line. onstructing erpendiculr Line Step Step 2 Step 3 Q Q rw rc with center lce the copss t point nd drw n rc tht intersects the line twice. Lel the intersections nd. rw intersecting rcs rw n rc with center. Using the se rdius, drw n rc with center. Lel the intersection of the rcs Q. rw perpendiculr line rw Q. This line is perpendiculr to line. n M Q The perpendiculr isector of line segent Q is the line n with the following two properties. n Q n psses through the idpoint M of Q. onstructing erpendiculr isector Use copss nd strightedge to construct the perpendiculr isector of. Step Step 2 Step 3 c 2 3 5 6 7 8 9 0 2 3 5 in. 2 3 5 6 M rw n rc lce the copss t. Use copss setting tht is greter thn hlf the length of. rw n rc. rw second rc Keep the se copss setting. lce the copss t. rw n rc. It should intersect the other rc t two points. isect segent rw line through the two points of intersection. This line is the perpendiculr isector of. It psses through M, the idpoint of. So, M = M. Section 3. roofs with erpendiculr Lines 9
roving Theores out erpendiculr Lines Theores Theore 3.0 Liner ir erpendiculr Theore If two lines intersect to for liner pir of congruent ngles, then the lines re perpendiculr. If l 2, then g h. roof E. 3, p. 53 g 2 h Theore 3. erpendiculr Trnsversl Theore In plne, if trnsversl is perpendiculr to one of two prllel lines, then it is perpendiculr to the other line. If h k nd j h, then j k. roof Eple 2, p. 50; Question 2, p. 50 j h k Theore 3.2 Lines erpendiculr to Trnsversl Theore In plne, if two lines re perpendiculr to the se line, then the re prllel to ech other. If p nd n p, then n. roof E., p. 53; E. 7, p. 62 n p roving the erpendiculr Trnsversl Theore Use the digr to prove the erpendiculr Trnsversl Theore. Given h k, j h rove j k j 2 3 5 6 7 8 h k STTEMENTS RESONS. h k, j h. Given 2. 2 = 90 2. efinition of perpendiculr lines 3. 2 6 3. orresponding ngles Theore (Theore 3.). 2 = 6. efinition of congruent ngles 5. 6 = 90 5. Trnsitive ropert of Equlit 6. j k 6. efinition of perpendiculr lines Monitoring rogress Help in English nd Spnish t igidesmth.co 2. rove the erpendiculr Trnsversl Theore using the digr in Eple 2 nd the lternte Eterior ngles Theore (Theore 3.3). 50 hpter 3 rllel nd erpendiculr Lines
Solving Rel-Life roles roving Lines re rllel The photo shows the lout of neighorhood. eterine which lines, if n, ust e prllel in the digr. Eplin our resoning. s t u p q Lines p nd q re oth perpendiculr to s, so the Lines erpendiculr to Trnsversl Theore, p q. lso, lines s nd t re oth perpendiculr to q, so the Lines erpendiculr to Trnsversl Theore, s t. So, fro the digr ou cn conclude p q nd s t. Monitoring rogress Use the lines rked in the photo. Help in English nd Spnish t igidesmth.co c d 3. Is? Eplin our resoning.. Is c? Eplin our resoning. Section 3. roofs with erpendiculr Lines 5
3. Eercises nic Solutions ville t igidesmth.co Voculr nd ore oncept heck. OMLETE THE SENTENE The perpendiculr isector of segent is the line tht psses through the of the segent t ngle. 2. IFFERENT WORS, SME QUESTION Which is different? Find oth nswers. Find the distnce fro point X to line WZ. X( 3, 3) Z(, ) Find XZ. Y(3, ) Find the length of XY. 2 2 2 W(2, 2) Find the distnce fro line to point X. Monitoring rogress nd Modeling with Mthetics In Eercises 3 nd, find the distnce fro point to XZ. (See Eple.) 3. 6 Z(2, 7) ONSTRUTION In Eercises 5 8, trce line nd point. Then use copss nd strightedge to construct line perpendiculr to line through point. 5. 6. Y(0, ) 2 (3, 0) X(, 2) 7. 8.. 3 (3, 3) 3 3 X(, 3) Z(, ) Y(2,.5) ONSTRUTION In Eercises 9 nd 0, trce. Then use copss nd strightedge to construct the perpendiculr isector of. 9. 0. 52 hpter 3 rllel nd erpendiculr Lines
ERROR NLYSIS In Eercises nd 2, descrie nd correct the error in the stteent out the digr. In Eercises 7 22, deterine which lines, if n, ust e prllel. Eplin our resoning. (See Eple 3.). 7. z v 8. w c Lines nd z re prllel. 9. 2. 20. n c p 2 c d q 8 c The distnce fro point to "" is 2 centieters. n 2. p z 22. v w ROVING THEOREM In Eercises 3 nd, prove the theore. (See Eple 2.) 3. Liner ir erpendiculr Theore (Th. 3.0) k. Lines erpendiculr to Trnsversl Theore (Th. 3.2) 23. USING STRUTURE Find ll the unknown ngle ROOF In Eercises 5 nd 6, use the digr to write proof of the stteent. esures in the digr. Justif our nswer for ech ngle esure. 5. If two intersecting lines re perpendiculr, then the intersect to for four right ngles. Given rove, 2, 3, nd re right ngles. 2 3 0 5 30 2 3 6. If two sides of two djcent cute ngles re perpendiculr, then the ngles re copleentr. """ """ Given rove nd 2 re copleentr. 2. MKING N RGUMENT Your friend clis tht ecuse ou cn find the distnce fro point to line, ou should e le to find the distnce etween n two lines. Is our friend correct? Eplin our resoning. 25. MTHEMTIL ONNETIONS Find the vlue of when nd ( c. (9 + 8) c [5( + 7) + 5] 2 Section 3. roofs with erpendiculr Lines 53
26. HOW O YOU SEE IT? You re tring to cross stre fro point. Which point should ou jup to in order to jup the shortest distnce? Eplin our resoning. 29. ONSTRUTION onstruct squre of side length. E 30. NLYZING RELTIONSHIS The pinted line segents tht for the pth of crosswlk re usull perpendiculr to the crosswlk. Sketch wht the segents in the photo would look like if the were perpendiculr to the crosswlk. Which tpe of line segent requires less pint? Eplin our resoning. 27. TTENING TO REISION In which of the following digrs is nd? Select ll tht ppl. E 3. STRT RESONING Two lines, nd, re perpendiculr to line c. Line d is prllel to line c. The distnce etween lines nd is eters. The distnce etween lines c nd d is eters. Wht shpe is fored the intersections of the four lines? 32. MTHEMTIL ONNETIONS Find the distnce etween the lines with the equtions = 3 2 + nd 3 + 2 =. 28. THOUGHT ROVOKING The postultes nd theores in this ook represent Eucliden geoetr. In sphericl geoetr, ll points re points on the surfce of sphere. line is circle on the sphere whose dieter is equl to the dieter of the sphere. In sphericl geoetr, how n right ngles re fored two perpendiculr lines? Justif our nswer. 33. WRITING escrie how ou would find the distnce fro point to plne. n ou find the distnce fro line to plne? Eplin our resoning. Mintining Mtheticl roficienc Siplif the rtio. 3. 6 ( ) 8 3 (Skills Review Hndook) 35. 3 5 36. Reviewing wht ou lerned in previous grdes nd lessons 8 ( 3) 7 ( 2) 37. 3 2 ( ) Identif the slope nd the -intercept of the line. (Skills Review Hndook) 38. = 3 + 9 39. = 2 + 7 0. = 8. = 8 6 6 5 hpter 3 rllel nd erpendiculr Lines