GEOMETRY INDIVIDUAL TEST January 2015

Transcription

1 The abbreviatin NOTA means "Nne f the Abve." Diagrams may nt be drawn t scale. All angle measures are in degrees. 1. Cnsider the cnditinal statement "If a quadrilateral is a rectangle then it has cngruent diagnals." Which is the cnverse f the cntrapsitive f the statement? A. If a quadrilateral has cngruent diagnals then it is a rectangle. B. If a quadrilateral is nt a rectangle then it des nt have cngruent diagnals. C. If a quadrilateral des nt have cngruent diagnals then it is nt a rectangle. D. If a quadrilateral is a rectangle then it has cngruent diagnals.. m ABC = 4x + 4 and m CBD = 50 x. If BD BA then find m ABC in degrees. A. 9. B. 1 C D A B C D E F AC is parallel t DF, with transversal BE. A, B and C are cllinear, as are D, E and F. m ABE = 3x + 5 and m DEB = x Find m CBE in degrees. A. 10 B. 3 C. 35 D. 79 D B C A 4. The cmplement f the supplement f btuse angle A has measure x. Which is the degree measure f angle A? A. ( x 90) B. (180 x) 5. C. ( x + 90) D. (180 x) (x+94) S, T and U are cllinear, and in TUR, angles R, U and exterir angle RTS have measures given in the diagram. Find the measure f RTS in degrees. A. 15 B. 14 C. 115 D RST ~ DEF. RS=1, ST=8, DE=5, m R = x + 10 and m D = 40. Give the value f x. A. 53 B. 43 C. 5 D A right triangle has ne side with length 1 and all sides are integers. Find the sum f the lengths f all f the pssible hyptenuses f the triangles. A. 8 B. 48 C. 85 D. 97 R 3x S T U 3

2 8. RST is issceles with tw f its angles measuring ( x + 10) and (x 10). Which culd NOT be a degree measure f ne angle f RST? A. 0 B. 30 C.48 D AB bisects RAT. If m BAT = x + 30 and m RAT = 3x 10 then find m BAR in degrees. A. 100 B. 80 C. 70 D P U G Right PUT has altitude t the hyptenuse UG. If PG= x + and GT= x + 7 and GU=3x fr x > 0, then find the length GU. A. 7 8 B. C. 3 D. 6 T 10. P A flag ple is 15 feet in height and is x feet hrizntally frm a pint P n the grund. Angle P is the angle f elevatin frm P t the tp f the flag ple and sin P = 0.6. Give the value f x in feet. A. 18 B. 0 C. D d P 6 Tw vertical ples are 4 feet and 6 feet frm pint P as shwn. The heights f the ples are 3 and 8 feet as shwn. Find the distance between the tps f the ples (d) abve, in feet. A. 5 5 B. 5 3 C. 5 D Which three numbers culd be the lengths f the sides f an btuse triangle? A. 3, 4, 5 B. 6, 8, 9.5 C. 7, 4, 6 D. 9, 40, In RST, RS=5 and ST=6. In ABC, AB=5 and BC=6. Which culd NOT necessarily be used t prve that RST ABC? A. RST ABC B. AC RT C. R, A are right angles D. RST, ABC are right angles

3 E, R, T and N are cllinear. UR BE and BT UN. A is the intersectin f UR and BT. EB=8, ER=3, RT=, TN=4. Give the sum f the lengths f AU and RU. 1 B 8 A E 3 R T 4 N A. 16 B C. 6 D T 14 R Z P Trapezid TRPZ has bases TR and PZ as shwn. m Z = 45 and m P = 60. Find the perimeter f TRPZ. A B C D A regular plygn has each interir angle 170 degrees. If the plygn has n sides, then give the value f 1 10 n. A. 9 B. 8 C. 7 D. 6 U T P Q R S In right triangle PTS, PT=8 and TS=6. TQ is the median t the hyptenuse PS and TR is the altitude t PS, fr Q and R bth n PS. Give the length f segment QR. A. 1.5 B. 1.4 C. 1.3 D. 1. D A regular ctagn and a regular pentagn share a side CD as shwn. B and A are vertices f their respective plygns as shwn. Give the degree measure f CAB. A. 1.5 B. C. 30 D B S R C 40 9 A U and S lie n line p, R and T lie n line q and p q. m SAR = 9 and m USA = 40. Find m ART in degrees. A. 40 B.50 C. 5 D.54 A U T p q

4 1. A B D ABCD is a parallelgram, with AB=10 and BC=6. Line EF is the perpendicular bisectr f AB, with E and F (nt shwn) n sides AB and DC respectively. If EF=5 then give the length f segment FC. A. 11 B C. 6 D An equilateral triangle has sides f length 6 1 each. Give the length f the altitude f the triangle. A. 9 7 B. 3 7 C. 3 1 D A rhmbus has diagnals f lengths 10 cm and 4 cm. Diagnals RM R H and BH S intersect at pint S. Give the C perimeter f pentagn RHMSB. A. 44 B. 53 C. 54 D. 56 B M 4. TRY has side lengths TR=1 and RY=14. Which can NOT be the length f side TY? A. 3 B. 11 C. 0 D Tw bugs start at the same pint and g in ppsite directins (ne east and ne west) mving 3 feet each. One then turns nrth, mves 4 feet and stps. The secnd turns suth, mves 6 feet and stps. Find the distance in feet between the tw bugs when they are at the end f the described "trips." A. 16 B C. 34 D R Ralph's cane drifted away n the water. Ralph is in his cabin n the shre and sees the cane at an angle f depressin f 1. Ralph is at a height f 10 feet abve the cane. Which equatin can be used t find the distance (RC) the cane is frm Ralph's eye, in feet? A. cs1 = B. sin1 = RC RC RC C. tan1 = D. 10 cs1 C RC = 10

5 7. Given that p, q and r are lines (nt necessarily cplanar) and p q and q r, which must be true? I. p r II. p r 30. The vertices f a triangle are A(0, 4), B(5, 8) and C(9, 3). The triangle is A. acute B. btuse C. right D. nt pssible A. I nly B. II nly C. I and II D. neither I nr II 8. 6' 8' 5' 4' Sharn is standing in frnt f a wallmirrr. Her reflectin and her bdy are parallel (cnsider Sharn's bdy t have n depth and linear, as the side f the triangle). Her reflectin is 4 feet tall. If the sides f the "mirrr triangle" shwn abve are 6, 4 and 5 feet (including the reflectin) then hw tall is Sharn? A. 5 ft 4 inches B. 5 ft 3 inches C. 5 ft inches D. 5 ft 1 inch 9. What is the psitive difference between the number f diagnals f a cnvex hexagn and the number f diagnals f a cnvex pentagn? A. 1 B. 4 C. 5 D. 6

6 1. B 11. C 1. B. D 1. D. A 3. D 13. A 3. D 4. C 14. E 4. E 5. A 15. A 5. C 6. D 16. D 6. B 7. C 17. B 7. D 8. A 18. B 8. A 9. A 19. D 9. B 10. B 0. C 30. C 1. B. The cnverse switches rder. The cntrapsitive switches rder and negates bth parts. S we are back t the riginal rder, with the negatin f each part.. D. 4x x = 90. 3x + 54 = 90. x=1. m ABC = 4(1) + 4 =5. 3. D. ABE, DEB are supplementary if the lines are parallel. 3x x + 15 = x+0=180. 5x=160. x=3. m CBE = m DEB = x + 15 = (3)+15 = C. (90-(180-A))=x A=x. -90+A=x s A=x A. The exterir angle has measure equal t the sum f the measures f the remte interir angles. 3x+3=x+94. x=6. x=31. m STR = x + 94= D. Crrespnding angles f similar triangles are cngruent. x+10=40. x=15 7. C. Using sme f the cmmn Pythagrean Triples, we can immediately include 4(3-4-5)=1-16-0, and 3(3-4-5)= and Otherwise, t get ther pssible triples, we use the frmulas uv, u v, u + v which generate Pythagrean Triples. S case 1: uv = 1 means uv=6 s we generate using u=6,v=1 and u=3, v= t get and which we already had. Case : u v = 1 (u-v)(u+v)=1. {u+v=1 and u-v=1} r {u+v=6 and u-v=} r {u+v=4 and u-v=3}. The first gives decimals since u=13. The secnd gives u=8, u=4, v=. Triple which we had already. The last gives decimals. Case 3: u + v = 1 we slve by trial and errr. v=1 gives decimals; v= and v=3 same. Abve that, we have n slutins. S the list is , , , Sum f the hyptenuses is =85. Answer C. 8. A. Case 1: x+10=x-10 gives x=0 fr angles 30, 30 and 10. Case gives x+10 is a base angle fr (x+10)+x-10=180 fr x=85/ and angles are 5.5, 5.5, 75. Case 3: x-10 is the base angle. (x-10)+x+10=180. x=38 and angles are 48, 66, 66. Chice A, 0 degrees, is nt an angle measure. 9. A. (x+30)= 3x-10. x=70. m BAR = m BAT = x + 30 = B. sine is the rati f ppsite t hyptenuse, and if it is 0.6=3/5 then we knw the ppsite and hyptenuse are 3a and 5a. Since the flag is 15, we say 3a=15 and the sides are 3a, 4a, 5a = 15, 0, 5. The value f x, adjacent side is 4a which is C. We use the Pythagrean Th. with the tw least numbers and cmpare the greatest number t the result. Knwledge f Pythagrean Triples makes this prblem easier. In chice A, = 5 s it is a right triangle. In chice B, we get 10, and since 9.5 is smaller, chice B is an acute triangle. Chice C gives 5 s this is btuse, since 6 is greater. Chice D is acute, as 40.5 is less than D. ( x + 7)( x + ) = (3 x) by a gemetric mean frmula (similarity). x + 9x + 14 = 9x 8x 9x 14 = 0 (8x + 7)( x ) = 0. x= because x is psitive. s GU=3x= A. The tw triangles with the ples as a leg are similar and since the crrespnding sides are reversed, the tw angles by pint P are cmplementary. S the

7 "dtted" triangle is als a right triangle. The hyptenuses f the "ple triangles" are 5 and 10 s = E. All can prve the triangles cngruent. Chice A wuld use SAS, chice B wuld use SSS, chice C HL and chice D SAS. 15. A RU RA 5 6 Separate the three similar triangles. 8 RA 8 RU = and = gives RA= 16/5 and RU= 48/5, s AU=48/5-16/5 = 3/5 and AU+RU= 3/5+48/5 = 80/5=16. T 14 R Z 6 14 P P= = = Chice D. 17. B. If the interir angle is 170 then the exterir angle is 10, and 360/10= 36 sides. 1 n 10 =18-10 = B The hyptenuse is 10 s the median will divide PS, s PQ=5 and QS=5. Use gemetric mean t get 8 = 10( PR) and PR=6.4. QR=6.4-5 = D. An ctagn has angles (180)(8 ) / 8 = 135 and pentagns 108 by the same frmula =117. The base angles f the issceles triangle is ( )/= S R 40 9 A 40 Extend SA t create a transversal. Angle T is 40 degrees, and angle R is 9-40=5, using the exterir angle therem. Chice C E 11 B 5 6 F C Use the Pythagrean Th. t get EB=5, and FC= Chice B.. A. Since each angle is 60 degrees, the height is half the side, multiplied by i 3 = D. Diagnals are 13 perpendicular and R H bisect each ther 5 1 s each quarter S f the rhmbus 1 5 is a B M triangle. Perimeter is =56 4. E. The third side must be between 14-1 and 14+1 by the Triangle Inequality therem. S all f the chices are pssible. 5. C. The bugs are apart n the N/S line by 10 feet and n the E/W line by 6 feet. Use the Pythagrean Therem: 136 = Angles f R depressin and 10 x elevatin are 1 alternate interir Cane angles and between tw parallel lines are 10 cngruent, s sin1 =. Chice B. RC U T

8 7. D. Since the lines need nt all be cplanar then neither statement must be true. 8. A = Shann. S Shannn = 16 = feet, and since 1 (1) = 4 inches then 3 Shannn is 5 ft, 4 inches. 9. B. Hexagn diagnals = 1 (6)(6 3) = 9 and pentagn diagnals = 1 (5)(5 3) = 5. The difference is C. Slpes are AB: 4 5, AC: 1, and 9 5 BC:. Since we have negative 4 reciprcals, tw lines are perpendicular. Right triangle! GEOMETRY INDIVIDUAL TEST