In a chemical mixture, 2 g of chemical A are used for each Given that , are the following proportions

21. In , is trisected

by and so that . Write two different proportions that follow from this information. ∠1 ⬵ ∠2 ⬵ ∠3 CE CD ∠ACB 䉭ABC

33. Use Theorem 5.6.1 and the drawing to complete the proof

of this theorem: “If a line is parallel to one side of a triangle and passes through the midpoint of a second side, then it will pass through the midpoint of the third side.” Given: with M the midpoint of ; Prove: N is the midpoint of RT Í MN 7 ST RS 䉭RST

28. In shown in Exercise 27, suppose that

, and are medians. Find the value of: a b

29. Given point D in the interior of , suppose that

, , , , and . Find RK.

30. Given point D in the interior of , suppose that

, , , and . Find

31. Complete the proof of this property:

If , then and PROOF Statements Reasons 1. 1. ? 2. 2. ? 3. 3. ? 4. 4. ? 5. 5. Means-Extremes Property symmetric form 6. 6. ?

32. Given:

, with , Prove: RX XS = ZT RZ Í YZ 7 RS Í XY 7 RT 䉭RST a + c b + d = c d a + c b + d = a b b a + c = ab + d ab + bc = ab + ad b c = a d a b = c d a + c b + d = c d a + c b + d = a b a b = c d KT KR . HT = 4 SH = 3 GS = 3 RG = 2 䉭RST KT = 3 HT = 5 SH = 4 GS = 4 RG = 3 䉭RST TH HS RK KT SK TG , RH 䉭RST

27. Given point D in the interior of

, which statements is are true? a b TK KR RG GS SH HT = 1 RK KT TH HS GS RG = 1 䉭RST

25. Given: bisects ,

, , , and Find: x HINT: You will need to apply the Quadratic Formula.

26. Given: bisects , , ,

, and Find: x MP = 3x - 1 RP = x + 1 NR = x MN = 2x ∠NMP MR VT = x + 2 RT = 2 - x SV = 3 RS = x - 6 ∠SRT RV

23. In right not shown with right

, bisects so that V lies on side . If , , and , find SV and VT.

24. Given: AC

is the geometric mean between AD and AB. , and Find: AC DB = 6 AD = 4 RT = 12 ST = 6 13 RS = 6 ST ∠SRT RV ∠S 䉭RST

22. In , ,

, and . With the angle bisectors as shown, which line segment is longer? a or ? b or ? c or ? FB AF DB CD EC AE m ∠ABC = 40° m ∠ACB = 60° m ∠CAB = 80° 䉭ABC A B E D C 1 2 3 B F A C E D D A B C V S T R 3 x – 6 x + 2 2 – x M P R 2 x x + 1 3 x – 1 x N T S R G K H D Exercises 27–30 Z S T R X Y S R M N T