18. Given: S is equidistant f rom E and D;V is equidistant f rom E and D.

Prove: SV is the perpendicular bisector of ED.

19. a. A t own wants to build a beach house onthe lake front equidistant from the recreationcenter and the school. Co p y the diagramand show the point B where the beach houseshould be located.

b. The town also wants to build a boat-launchingsite that is equidistant f rom Elm Road andMain Street. Find the point L where it shouldbe built .

c. On your diagram, locate the spot F for aflagpole that is to be the same distance fromthe recreation center, the school, and thecourthouse.

20. Given: A L M N A R S T ;_LX and RY are altitudes.

Prove: L X R Y

21. a. Given: AB A C ; BD I AC; CE I ABProve: B D C E

b. The result you proved in part (a) can be stated as a theoremabout certain alt itudes. S t a t e t his theorem i n y our o wnwords.

courthouse•

Congruent Triangles I 157

school• •

recreation center

Main St.

22. Prove that the medians drawn t o the legs o f an isosceles t riangle arecongruent. Wr i t e the proof in two-column form.

For Exercises 23-27 write proofs in paragraph form. (Hi nt: You can usetheorems fr om this section to wri te fair ly short proofs for Exercises 23and 24.)

23. Given: SR is the I bisector of QT;QR is the I bisector of SP.

Prove: P Q = TS

24. Given: DP bisects L A DE ;EP bisects L DEC.

Prove: B P bisects L ABC.

EL 1 1x M Y S

A

158 / Chapter 4

25. Given: Plane M is the perpendicular bisecting plane of AB.(That is , AB I plane M and 0 is the midpoint of AB.)Prove: a . A D B D

b. A C B Cc. L CAD -=----- L CB D

26. Given: m LRTS = 90;MN is the I bisector of TS.

Prove: T M is a median.

27. Given: E H and F.1 are medians of scalene AEFG; P is on EH such thatEH H P ; Q is on F l such that F.I

Prove: a . GQ G Pb. GQ and GP are both parallel to EF.c. P, G, and Q are collinear.

Write paragraph proofs. (I n this book a star designates an exercise that isunusually difficult.)

28. Given: AE 11 BD; BC 11 AD;AE B C ; AD B D

Prove: a . A C B Eb. ECI1 AB

* 29. Given: A M is the I bis. o f BC;AE I BD; AF I DF;L l 12Prove: B E C F

Exp,These exploratory exercises can be done using a computer with a programthat draws and measures geometric figures.

Decide i f the following statements are true or false. I f you think the statementis true, give a convincing argument to support your belief. I f you think thestatement is false, make a sketch and give all the measurements of the trianglethat you f ind as your counterexample. Fo r each false statement, also discoverif there are types of triangles for which the statement is true.1. A n angle bisector bisects the side opposite the bisected angle.2. A median bisects the angle at the vertex f rom which it is drawn.3. The length of a median is equal to half of the length of the side it bisects.

m ExercisesComplete.

1. I f K is the midpoint of ST, then RK is called a(n) ? o f ARST.2. I f RK I ST, then RK is called a(n) 9 o f ARST.3. I f K is the midpoint of ST and RK I ST, then RK is called a(n)

? o f ST.

4. I f RK is both an alt itude and a median of ARST, then:a. A RS K A R T K by ? b . A RS T is a(n) ? t riangle.

5. I f R is on the perpendicular bisector of ST, then R is equidistantfrom ? and ? T h u s ? = ?

6. Refer to AABC and name each of the following.a. a median of AABCb. an alt itude of AABCc. a bisector of an angle of AABC

7. Draw XY. Label its midpoint Q. A

11. Plane M is the perpendicular bisecting plane of AB at 0(that is, M is the plane that is perpendicular to AB at itsmidpoint, 0) . Points C and D also lie in plane M. L i s tthree pairs of congruent triangles and tell which congruencemethod can be used to prove each pair congruent.

Congruent Triangles I 155

a. Select a point P equidistant f rom X and Y. Dr a w PX, PY, and PQ.b. What postulate justif ies the statement APQX A P Q Y ?c. What reason justif ies the statement I PQX Z _ PQY?d. What reason justif ies the statement PQ I XY?e. What name for PQ best describes the relationship between PQ and XY?

8. Given: A D E F is isosceles wit h DF = EF;FX bisects L DFE.

a. Would the median drawn f rom F to DE be the samesegment as FX?

b. Would the alt itude drawn f rom F to DE be the samesegment as FX?

9. What k ind of triangle has three angle bisectors that are also altitudes andmedians?

10. Given: NO bisects I N .What can you conclude f rom each of the following addit ional statements?a. P lies on NO.b. The distance f rom a point Q to each side of L N is 13.

156 / Chapter 4

iritten Exercises

1. a. Draw a large scalene t riangle ABC. Caref ul ly draw the bisector o fL A , the alt itude f rom A, and the median f rom A. Thes e three shouldall be different.

b. Draw a large isosceles t riangle ABC wit h vertex angle A. Caref ul lydraw the bisector of LA , t h e a l t i t u d e f r o m A , a nd t he m e di a n f ro m A.

Are these three different?

2. Draw a large obtuse triangle. Then draw its three altitudes in color.3. Draw a right triangle. Then draw its three altitudes in color.4. Draw a large acute scalene triangle. Then draw the perpendicular bisectors

of its three sides.

5. Draw a large scalene right triangle. Then draw the perpendicular bisectorsof its three sides and tell whether they appear to meet in a point. I f so,where is this point?

6. Cut out any large triangle. Fo l d the two sides of one angle of the triangletogether to f orm the angle bisector. Us e the same method to f orm thebisectors of the other two angles. Wh a t do you notice?

Complete each statement.

7. I f X is on the bisector of LSKN, then X is equidistantfrom ? and ?

8. I f X is on the bisector of L SNK, then X is equidistantfrom ? and ?

9. I f X is equidistant f rom SK and SN, then X lies on the

10. I f 0 is on the perpendicular bisector of LA, then 0 isequidistant f rom ? and ?

11. I f 0 is on the perpendicular bisector of AF, then 0 isequidistant f rom '? and ?

12. I f 0 is equidistant from L and F, then 0 lies on the ?

13. Given: P is on the perpendicular bisector of AB;P is on the perpendicular bisector of BC.

Prove: PA = PC

Use the diagrams on pages 153 and 154 to prove the following theorems.14. Theorem 4-5 1 5 . Theorem 4-616. Theorem 4-7 1 7 . Theorem 4-8