Proportions in Triangles

Transcription

1 - roportions in Triangles ontent tandard G.RT. rove theorems about triangles...a line parallel to one side of a triangle divides the other two proportionally... Objective To use the ide-plitter Theorem and the Triangle-ngle-isector Theorem n artist uses perspective to draw parallel lampposts along a city street, as shown in the diagram. What is the value of? Justify your answer. Use what you know about similar triangles to plan a pathway to a solution. MTHEMTIL RTIE. in.. in.. in.. in.. in. The olve It involves parallel lines cut by two transversals that intersect. In this lesson, you will learn how to use proportions to find lengths of segments formed by parallel lines that intersect two or more transversals. Essential Understanding When two or more parallel lines intersect other lines, proportional segments are formed. Theorem - ide-plitter Theorem Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. If... R R Then... R R Lesson - roportions in Triangles

2 roof roof of Theorem -: ide-plitter Theorem Given: with R rove: R R tatements Reasons ) R ) Given ), ) If lines are, then corresponding are. ) R ) ostulate ) R ) orresponding sides of are proportional. ) R R, ) egment ddition ostulate R R ) R ) ubstitution roperty ) R R ) roperty of roportions () R How can you use the parallel lines in the diagram? KL is parallel to one side of MN. Use the ide-plitter Theorem to set up a proportion. roblem Using the ide-plitter Theorem What is the value of in the diagram at the right? K NL KM LM ide-plitter Theorem ubstitute. ross roducts roperty K ubtract from each side. Divide each side by. Grid in the number. N L M Got It?. a. What is the value of a in the diagram at the right? b. Reasoning In Z, R joins and Z with R on and on Z, and R Z. If R R Z, what must be true about R? Justify your reasoning. a a hapter imilarity

3 orollary orollary to the ide-plitter Theorem orollary If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. If... a b c a b c W Then... W ou will prove the orollary to Theorem - in Eercise. roblem Finding a Length What information does the diagram give you? The lines separating the campsites are parallel. Think of the river and the edge of the road as transversals. Then the boundaries along the road and river for each campsite are proportional. amping Three campsites are shown in the diagram. What is the length of ite along the river? Let be the length of ite along the river.. orollary to the ide-plitter Theorem. ross roducts roperty Divide each side by.. The length of ite along the river is yd. yd ite ite yd. yd. yd ite Got It?. What is the length of ite along the road? Essential Understanding The bisector of an angle of a triangle divides the opposite side into two segments with lengths proportional to the sides of the triangle that form the angle. Theorem - Triangle-ngle-isector Theorem Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. If... D bisects D Then... D D ou will prove the Triangle-ngle-isector Theorem in Eercise. Lesson - roportions in Triangles

4 roblem Using the Triangle-ngle-isector Theorem lgebra What is the value of in the diagram at the right? R bisects R. Use the Triangle-ngle- isector Theorem to write a proportion. R R ubstitute corresponding side lengths in the proportion. Use the ross roducts roperty. Divide each side by.. Got It?. What is the value of y in the diagram at the right? y. Lesson heck Do you know HOW? Use the figure to complete each proportion.. a b. b e e a b c f. a b c e f What is the value of in each figure?.. d e f Do you UNDERTND? MTHEMTIL RTIE. ompare and ontrast How is the orollary to the ide-plitter Theorem related to Theorem -: If three (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal?. ompare and ontrast How are the Triangle-ngle- isector Theorem and orollary to Theorem - alike? How are they different?. Error nalysis classmate says you can use the ide-plitter Theorem to find both and y in the diagram. Eplain what is wrong with your classmate s statement. y. hapter imilarity

5 ractice and roblem-olving Eercises ractice lgebra olve for. ee roblem..... Marine iology Use the information shown on the auger shell.. What is the value of?. What is the value of y? ee roblem. lgebra olve for.... mm mm y. mm.. lgebra olve for..... ee roblem.. Writing The size of an oil spill on the open ocean is difficult to measure directly. Use the figure at the right to describe how you could find the length of the oil spill indirectly. What measurements and calculations would you use? D. The lengths of the sides of a triangle are cm, cm, and cm. Find the lengths, to the nearest tenth, of the segments into which the bisector of each angle divides the opposite side. E Lesson - roportions in Triangles

6 pply Use the figure at the right to complete each proportion. Justify your answer.. R JR KJ. L M. KL LW MW. KJ K J. T. K T K L K K J M R T W L TEM Urban Design In Washington, D.., E. apitol treet, Independence venue, treet, and D treet are parallel streets that intersect Kentucky venue and th treet.. How long (to the nearest foot) is Kentucky venue between treet and D treet?. How long (to the nearest foot) is Kentucky venue between E. apitol treet and Independence venue? Lincoln ark ft ft th t. E apitol t. Independence ve. Walter t. t. outh arolina ve. Massachusetts ve. Kentucky ve. ft t. th t. ft th t. D t. lgebra olve for..... Think bout a lan The perimeter of the triangular lot at the right is m. The surveyor s tape bisects an angle. Find the lengths and y. How can you use the perimeter to write an equation in and y? What other relationship do you know between and y?. rove the onverse of the ide-plitter Theorem: If a line divides two roof sides of a triangle proportionally, then it is parallel to the third side. Given: R R rove: R R m m y hapter imilarity

7 Determine whether the red segments are parallel. Eplain each answer. ou can use the theorem proved in Eercise..... n angle bisector of a triangle divides the opposite side of the triangle into segments cm and cm long. second side of the triangle is. cm long. Find all possible lengths for the third side of the triangle.. Open-Ended In a triangle, the bisector of an angle divides the opposite side into two segments with lengths cm and cm. How long could the other two sides of the triangle be? (Hint: Make sure the three sides satisfy the Triangle Inequality Theorem.). Reasoning In, the bisector of bisects the opposite side. What type of triangle is? Eplain your reasoning. lgebra olve for.. R. T E.. D F hallenge. rove the orollary to the ide-plitter Theorem. In the diagram from roof page, draw the auiliary line W and label its intersection with line b as point. Given: a b c rove: W. rove the Triangle-ngle-isector Theorem. In the diagram from page, roof draw the auiliary line E so that E D. Etend to meet E at point F. Given: D bisects. rove: D D. Use the definition in part (a) to prove the statements in parts (b) and (c). a. Write a definition for a midsegment of a parallelogram. b. parallelogram midsegment is parallel to two sides of the parallelogram. c. parallelogram midsegment bisects the diagonals of a parallelogram. a b c D W E F Lesson - roportions in Triangles

8 . tate the converse of the Triangle-ngle-isector Theorem. Give a convincing argument that the converse is true or a countereample to prove that it is false.. In, the bisectors of,, and cut the opposite sides into lengths a and a, b and b, and c and c, respectively, labeled in order counterclockwise around. Find the perimeter of for each set of values. a. b, b, c b. a, a, b tandardized Test rep T/T. What is the value of in the figure at the right?. uppose VL. If mv and ml, what is m?. In the diagram at the right, R R. For what value of is T parallel to?. Leah is playing basketball on an outdoor basketball court. The -ft pole supporting the basketball net casts a -ft shadow. t the same time, the length of Leah s shadow is ft in. What is Leah s height in inches? ou can assume both Leah and the pole supporting the net are perpendicular to the ground. T R Mied Review Use the figure to complete each proportion. a ee Lesson -.. n h h. b b c b h m n c. n a a. m h n Find the center of the circle that you can circumscribe about each.. (, ). (, ). (, ) (, ) (, ) (, ) (, ) (, ) (, ) ee Lesson -. Get Ready! To prepare for Lesson -, do Eercises. quare the lengths of the sides of each triangle. ee p..... m in. in. m in. m m m m hapter imilarity