Geometry Contest 2013

Transcription

1 eomety ontet One pizza ha a diamete twice the diamete of a malle pizza. What i the atio of the aea of the lage pizza to the aea of the malle pizza? ) to 1 ) to 1 ) to 1 ) 1 to ) to 1. In ectangle diagonal i pependicula to at. If = 3 and =, find the aea of the ectangle. ) 50/3 ) 16/3 ) 5 ) 1 ) 100/3 3. In the figue ae thee imila ight tiangle with meaue of leg a indicated. ind the length of dotted egment. ume all meauement ae in centimete (cm). ) 87.5 cm ) 1 cm ) 1 cm ) 15 cm ) 10.5 cm The figue i compoed of multiple non-conguent and non-paallel line egment which inteect each othe with non-conguent angle a hown. ind the um of the meaue of the angle P, Q, R, S, T, and U. ) 180 degee ) 70 degee ) 360 degee ) 50 degee ) 70 degee U P T Q S R 5. If m = 80 degee, =, and =, find m in the figue. ) 0 degee ) 50 degee ) 100 degee ) 130 degee ) 10 degee

2 6. If the aea of i 3 quae unit, and the length ae given on the figue, then find the aea of. ) 6 ) 8 ) 1 ) 16 ) mall quae with a ide of length a i placed at the cone of a lage quae with a ide of length a+b. If the aea of the mall quae i 1/9 of the lage quae, then find the atio of a to b. ) 1 to ) 1 to 3 ) 1 to ) 1 to 8 ) 1 to 9 a b a b 8. Pentagon i fomed by adjoining quae and equilateal. iagonal and inteect at. ind m. ) 60 degee ) 67.5 degee ) 7 degee ) 75 degee ) 108 degee 9. iamete of length 10 i pependicula to chod of length 6 at point. ind the length of the longet chod joining thei endpoint. ) 10 unit ) 3 unit ) 8 unit ) 3 10 unit ) 8 unit 10. ectangle ha ide of length 6 and 8 unit. ind the ditance fom a vetex to the diagonal though two othe vetice. ) 3.6 unit ).8 unit ) 10 unit ) 7 unit ) 10 unit

3 11. onguent tiangle ae contucted on oppoite ide of a quae a hown below. If m = 5 degee, m = 65 degee, and i paallel to, then find m. ) 0 degee ) 5 degee ) 70 degee ) 110 degee ) 160 degee 1. iagonal and of egula pentagon inteect at. ind m. ) 36 degee ) 7 degee ) 90 degee ) 108 degee ) 1 degee 13. quae and a egula pentagon hae a ide a hown. ind m (not dawn in figue). ) 90 degee ) 99 degee ) 100 degee ) 108 degee ) 16 degee 1. Thee cicle with adii of 3, 5, and x unit ae mutually tangent in pai whee x i the adiu of the laget cicle. If the line of cente of thee cicle ae alo ide of a ight tiangle, then find the adiu of the laget cicle. ) unit ) 8 unit ) 10 unit ) 1 unit ) 8 unit 15. Two tangent cicle with cente and ae alo tangent to the ide of a 0 degee angle (i. e., ) a hown. If egment i extended to meet the cicle again at H, find m H. ) 0 degee ) 0 degee ) 70 degee ) 100 degee ) 10 degee H

4 16. Quadilateal i a tapezoid with cicle P incibed in ight and cicle Q incibed in quae. If the two iocele ide of the tapezoid ae 5 unit each and the aea of the quae i 16 quae unit, find the peimete of the tapezoid. ) 16 unit ) 18 unit ) 0 unit ) unit ) unit P Q 17. ind the length of tangent fo the cicle with cente if = 1, = 8, =1, and = 10. ) 6 ) 0 ) 105 ) 3 ) The ide of an equilateal tiangle inide a cicle have it ide extended to meet the cicle with the extended ide having meauement a hown. ind x. ) 1.5 ) 13 ) 13.5 ) 1 ) 15 1 x In the figue, ac = 0 degee, ac = 50 degee, ac = 180 degee, and egment i tangent to the cicle at. Segment inteect the cicle at and inteect at. ind m. ) 5 degee ) 35 degee ) 0 degee ) 5 degee ) 55 degee 0. Segment,, and ae all paallel to each othe, egment and inteect at, and i a point on. If = 6 and = 3, find. ) 1 ) ) 1.5 ) ).5

5 1. In calene, M i the midpoint of, N i the midpoint of M, and i the inteection of ay N and. ind the atio of N to N. ) to 1 ).5 to 1 ) 3 to 1 ) 3.5 to 1 ) to 1 N M. point P i 13 unit fom the cente of a cicle. If the diamete of the cicle i 10 unit, find the length of a tangent egment fom P to the cicle. ) 10 unit ) 65 unit ) 1 unit ) 18 unit ) 19 unit 3. If two tiangle ae imila, then what i the maximum numbe of pai of pat of the tiangle which can be conguent and yet the tiangle ae not conguent? ) ) 3 ) ) 5 ) 6. Let be the adiu of a cicle incibed in a ight tiangle with hypotenue of length c and leg of length a and b. ind a fomula fo in tem of a, b, and c. ) a b - c ) c = ) = a b ) a b ) a b - c 5. pentagon i fomed by an adjoining quae and an equilateal tiangle a hown. If each ide of the quae ha length, find the aea of. ) ) 3 ) ) ) In ome quadilateal, diagonal and inteect at and i paallel to. If = 1, = 10, and = 8, then find. ) 8 ) 9.6 ) 10 ) 15 ) 0

6 7. If i fomed by extending the ide of outwad uch that = = 5, = = 3, and = =, then find the atio of the aea of to the aea of. ) 3 to 1 ) to 1 ) 5 to 1 ) 6 to 1 ) 7 to 1 8. In the figue two cicle ae tangent to each othe and ae incibed in quadilateal with the meauement a hown. ind the length of tangent egment x. (The figue i not to cale.) ) 6 unit ) 7 unit ) 8 unit ) 9 unit ) 10 unit x iven: and ae imila with m = m = 90 degee. If m = 60 degee and = 1 unit, then find. ) 3 unit ) unit ) 3 unit ) 3 unit ) 7 unit 30. Let be an iocele ight tiangle with hypotenue. Let be a median and be an angle biecto which meet at P. If ay P inteect at, then find /. ) ) ) ) 1 1 ) 1 P

7 31. Two unequal quae in quadant II and IV hae a common vetex at the oigin of a ectangula coodinate ytem. Segment (dotted) fom the emote vetex of one quae join with the emaining vetice of the othe quae and meet the coodinate axe at,,, and a hown. What kind of quadilateal i? ) a quae ) a ectangle which i not a quae ) a kite which i not a quae ) a hombu which i not a quae ) an iocele tapezoid which i not a quae 3. In quadilateal ide and ae conguent. The diagonal meet at. Point,, H, I, and J ae the epective midpoint of,,,, and. Which of the following tatement ae not alway tue fo thee condition? (The figue i not neceaily to cale.) ) = ) H, J, and I ae collinea ) HI i pependicula to ) I = H ) J i the midpoint of HI H J I 33. Let P be a point in the inteio of uch that m = ½ m P, m = ½ m P, and m = ½ m P, then which tatement mut follow? ) Point P i the centoid of the tiangle. ) Point P i the cicumcente of the tiangle. ) Point P i the incente of the tiangle. ) Point P i the othocente of the tiangle. ) No uch point exit.

8 3. In the figue thee ae two pai of concentic cicle with cente and. The egment i a common chod fo the two lage cicle. The egment i alo tangent to the two malle cicle. If adii S = 15, R = 9, and Q = 13, find P. ) 5 unit ) 6 unit ) 9 unit ) 1 unit ) unit P Q R 35. In cicle c 1 with adiu a malle cicle c i contucted uing one of the quadant of cicle c 1 and foming O. ind the quae unit fo the aea of the cecent inide the malle cicle but outide the lage cicle. S ) ) ) ) ) 1 c 1 O M c 36. djacent ide of a paallelogam ae one unit and two unit. If they meet at 60 degee, find the diffeence of the length of thei diagonal. ) 1 unit ) unit ) 3 unit ) 7 unit ) 7 3 unit 37. n equilateal tiangle i contucted inwad fom each vetex of a quae whoe ide meet at the vetice of a malle quae. ind the atio of the aea of the oiginal quae to the aea of the malle quae. ) 3 to 1 ) 3 to 1 ) 3 to 1 ) 3 to 1 ) 3 1 to 1

9 38. In = 10, = 8, = 9. If i the midpoint of X, Z i 3 time, and X, Y, and Z ae collinea, find Y. ) 1 ) 13 ) 1 ) 15 ) 16 Z X Y 39. In the figue the dotted egment i a chod of length 16 which i tangent to the malle emicicle and alo paallel to the diamete of the lage emicicle. ind the aea of the egion inide the lage emicicle that i outide the malle emicicle. ) ) 8 ) 16 ) 0 ) 3 0. ope i tetched aound two pulley wheel a hown. If the ditance between the cente of the wheel i 10 inche and the adii of the wheel ae inche and 7 inche epectively. ind the length of the ope. ) 9 10 inche ) 9 0 inche ) 3 10 inche ) 10 inche ) inche

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