.6 Proportionality heorems ssential uestion hat proportionality relationships eist in a triangle intersected by an angle bisector or by a line parallel to one of the sides? iscovering a Proportionality elationship ork with a partner. se dynamic geometry software to draw any. a. onstruct parallel to with endpoints on and, respectively. LOOIG O o be proficient in math, you need to look closely to discern a pattern or structure. b. ompare the ratios of to and to. c. ove to other locations parallel to with endpoints on and, and repeat part (b). d. hange and repeat parts (a) (c) several times. rite a conjecture that summarizes your results. iscovering a Proportionality elationship ork with a partner. se dynamic geometry software to draw any. a. isect and plot point at the intersection of the angle bisector and. b. ompare the ratios of to and to. c. hange and repeat parts (a) and (b) several times. rite a conjecture that summarizes your results. ommunicate our nswer 3. hat proportionality relationships eist in a triangle intersected by an angle bisector or by a line parallel to one of the sides?. se the figure at the right to write a proportion. ection.6 Proportionality heorems 99
.6 Lesson hat ou ill Learn ore ocabulary directed line segment, p. 50 Previous corresponding angles ratio proportion se proportionality theorems. Partition directed line segments. sing Proportionality heorems heorems riangle Proportionality heorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Proof. 7, p. 505 If, then =. onverse of the riangle Proportionality heorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. Proof., p. 505 If =, then. inding the Length of a egment In the diagram,, =, = 6, and = 9. hat is the length of? OLIO = riangle Proportionality heorem 9 6 9 = 6 ubstitute. = 6 ultiply each side by 9 and simplify. he length of is 6 units. 35 36 onitoring Progress 1. ind the length of. Help in nglish and panish at igideasath.com he theorems above also imply the following: ontrapositive of the riangle Inverse of the riangle Proportionality heorem Proportionality heorem If, then. If, then. 500 hapter imilarity
olving a eal-life Problem On the shoe rack shown, = 33 centimeters, = 7 centimeters, = centimeters, and = 5 centimeters. plain why the shelf is not parallel to the floor. OLIO ind and simplify the ratios of the lengths. = 5 ecause 5 9 = 7 33 = 9 11 11, is not parallel to. o, the shelf is not parallel to the floor. onitoring Progress. etermine whether P. Help in nglish and panish at igideasath.com P 50 90 7 0 heorem hree Parallel Lines heorem If three parallel lines intersect two transversals, then they divide the transversals proportionally. r s t m Proof. 3, p. 505 = G H ain t. 1 10 yd 150 yd econd t. 300 yd 3 outh ain t. sing the hree Parallel Lines heorem In the diagram, 1,, and 3 are all congruent, G = 10 yards, = 150 yards, and = 300 yards. ind the distance H between ain treet and outh ain treet. OLIO orresponding angles are congruent, so, G, and H are parallel. se the hree Parallel Lines heorem to set up a proportion. HG G = HG 10 = 300 150 HG = 0 hree Parallel Lines heorem ubstitute. ultiply each side by 10 and simplify. y the egment ddition Postulate, H = HG + G = 0 + 10 = 360. he distance between ain treet and outh ain treet is 360 yards. ection.6 Proportionality heorems 501
heorem riangle ngle isector heorem If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. Proof. 35, p. 506 = sing the riangle ngle isector heorem In the diagram, P P. se the given side lengths to find the length of. P 7 13 15 OLIO ecause P is an angle bisector of P, you can apply the riangle ngle isector heorem. Let =. hen = 15. = P P riangle ngle isector heorem 15 = 7 13 ubstitute. 195 13 = 7 ross Products Property 9.75 = olve for. he length of is 9.75 units. onitoring Progress Help in nglish and panish at igideasath.com ind the length of the given line segment. 3.. 0 3 16 1 30 G 16 15 1 H ind the value of the variable. 5. 1 6. y Partitioning a irected Line egment directed line segment is a segment that represents moving from point to point. he following eample shows how to use slope to find a point on a directed line segment that partitions the segment in a given ratio. 50 hapter imilarity
Partitioning a irected Line egment y (6, ) 6 (3, ) 6 6 y 3 (6, ) P(., 5.6) 6 (3, ) 1. 3.6 6 ind the coordinates of point P along the directed line segment so that the ratio of P to P is 3 to. OLIO In order to divide the segment in the ratio 3 to, think of dividing, or partitioning, the segment into 3 +, or 5 congruent pieces. Point P is the point that is 3 of the way 5 from point to point. ind the rise and run from point to point. Leave the slope in terms of rise and run and do not simplify. slope of : m = 6 3 = 6 3 = rise run o find the coordinates of point P, add 3 5 of the run to the -coordinate of, and add 3 5 of the rise to the y-coordinate of. run: 3 5 of 3 = 5 3 3 = 1. rise: 3 5 of 6 = 5 3 6 = 3.6 o, the coordinates of P are (3 + 1., + 3.6) = (., 5.6). he ratio of P to P is 3 to. onitoring Progress Help in nglish and panish at igideasath.com ind the coordinates of point P along the directed line segment so that P to P is the given ratio. 7. (l, 3), (, ); to 1. (, 1), (, 5); 3 to 7 ou can apply the riangle Proportionality heorem to construct a point along a directed line segment that partitions the segment in a given ratio. onstructing a Point along a irected Line egment onstruct the point L on so that the ratio of L to L is 3 to 1. OLIO tep 1 tep tep 3 G G raw a segment and a ray raw of any length. hoose any point not on. raw. raw arcs Place the point of a compass at and make an arc of any radius intersecting. Label the point of intersection. sing the same compass setting, make three more arcs on, as shown. Label the points of intersection,, and G and note that = = = G. raw a segment raw G. opy G and construct congruent angles at,, and with sides that intersect at,, and L. ides,, and L are all parallel, and they divide equally. o, = = L = L. Point L divides directed line segment in the ratio 3 to 1. L ection.6 Proportionality heorems 503
.6 ercises ynamic olutions available at igideasath.com ocabulary and ore oncept heck 1. OPL H If a line divides two sides of a triangle proportionally, then it is to the third side. his theorem is known as the.. OL In, point lies on and bisects. rite the proportionality statement for the triangle that is based on the riangle ngle isector heorem. onitoring Progress and odeling with athematics In ercises 3 and, find the length of. (ee ample 1.) 3. 3 1 In ercises 5, determine whether. (ee ample.) 5. L 5 1 7.5. 6. 1 1.5 5 L 1 1 0 In ercises 11 and 1, find the length of the indicated line segment. (ee ample 3.) 11. 1. 0 15 P 1 10 In ercises 13 16, find the value of the variable. (ee ample.) 13. y 6 1. 3.5 z 1.5 7. L. 15 10 1 35 16 L 15 In ercises 9 and 10, use the diagram to complete the proportion. 3 15. 16.5 11 9 p 16. q 16 In ercises 17 0, find the coordinates of point P along the directed line segment so that P to P is the given ratio. (ee ample 5.) 17. (, 0), (3, ); 1 to 1. (, ), (6, 1); 3 to 36 19. (1, 6), (, 3); 5 to 1 G 0. ( 3, ), (5, ); to 6 9. = G 10. G = OIO In ercises 1 and, draw a segment with the given length. onstruct the point that divides the segment in the given ratio. 1. 3 in.; 1 to. in.; to 3 50 hapter imilarity
3. O LI escribe and correct the error in solving for. 10 = 1 16 10 16 = 1 10 = =.. O LI escribe and correct the error in the student s reasoning. ecause = and =, it follows that =. HIL OIO In ercises 5 and 6, find the value of for which P. 5. P + 5 3 + 5 7 6. P 1 7. POIG HO Prove the riangle Proportionality heorem. Given Prove = 1 3 1. POIG HO Prove the onverse of the riangle Proportionality heorem. Given = Prove 9. OLIG IH HI he real estate term lake frontage refers to the distance along the edge of a piece of property that touches a lake. Lot 17 yd Lot yd 55 yd Lakeshore r. Lot 61 yd lake a. ind the lake frontage (to the nearest tenth) of each lot shown. b. In general, the more lake frontage a lot has, the higher its selling price. hich lot(s) should be listed for the highest price? c. uppose that lot prices are in the same ratio as lake frontages. If the least epensive lot is $50,000, what are the prices of the other lots? plain your reasoning. 30. OLIG IH HI our school lies directly between your house and the movie theater. he distance from your house to the school is one-fourth of the distance from the school to the movie theater. hat point on the graph represents your school? (, ) y (5, ) 31. OIG In the construction on page 503, eplain why you can apply the riangle Proportionality heorem in tep 3. 3. POIG HO se the diagram with the auiliary line drawn to write a paragraph proof of the hree Parallel Lines heorem. Given k 1 k k 3 Prove = t 1 t auiliary line k 1 k k 3 ection.6 Proportionality heorems 505
33. IIL HIIG In L, the angle bisector of also bisects L. lassify L as specifically as possible. ustify your answer. 3. HO O O I? uring a football game, the quarterback throws the ball to the receiver. he receiver is between two defensive players, as shown. If Player 1 is closer to the quarterback when the ball is thrown and both defensive players move at the same speed, which player will reach the receiver first? plain your reasoning. 3. IG G wo people leave points and at the same time. hey intend to meet at point at the same time. he person who leaves point walks at a speed of 3 miles per hour. ou and a friend are trying to determine how fast the person who leaves point must walk. our friend claims you need to know the length of. Is your friend correct? plain your reasoning. 0.6 mi 0.9 mi 39. OIO Given segments with lengths r, s, and t, construct a segment of length, such that r s = t. r 35. POIG HO se the diagram with the auiliary lines drawn to write a paragraph proof of the riangle ngle isector heorem. Given Prove = auiliary lines 36. HOGH POOIG rite the converse of the riangle ngle isector heorem. Is the converse true? ustify your answer. 0. POO Prove eva s heorem: If P is any point inside, then = 1. (Hint: raw segments parallel to through and, as shown. pply the riangle Proportionality heorem to. how that P P, P, and P.) s t P 37. OIG How is the riangle idsegment heorem related to the riangle Proportionality heorem? plain your reasoning. aintaining athematical Proficiency eviewing what you learned in previous grades and lessons se the triangle. (kills eview Handbook) 1. hich sides are the legs?. hich side is the hypotenuse? a c b olve the equation. (ection.3) 3. = 11. + 16 = 5 5. 36 + = 5 506 hapter imilarity