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GR E Math 강좌 – Set 11
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Category 11 Quadrilateral & Other Polygons
1. What is the area of a square with perimeter P ?
(A)216P
(B) P4
(C)4
2P
(D)16
P
(E)
16
2P
2. If the area of a square region having sides of length 6 centimeters is equal to the area of a
rectangular region having width 2.5 centimeters, then the length of the rectangle, in
centimeters, is
(A) 8.5
(B) 9.5
(C) 9.6
(D) 10.5
(E) 14.4
3. In the figure above, if PQRS is a parallelogram, then =− x y
(A) 30 (B) 35 (C) 40 (D) 70 (E) 100
answer
answer
answer
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GR E Math 강좌 – Set 11
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4. A rectangular circuit board is designed to have width w inches, perimeter p inches, and
area k square inches. Which of the following equations must be true?
(A) 02=++ k pww
(B) 022 =+− k pww
(C) 022 2=++ k pww
(D) 022 2=−− k pww
(E) 022 2=+− k pww
5. The size of a television screen is given as the length of the screen’s diagonal. If the screenswere flat, then the area of a square 21-inch screen would be how many square inches greater
than the area of a square 19-inch screen?
(A) 2
(B) 4
(C) 16
(D) 38
(E) 40
6. If the figure above is a parallelogram, what i s the value of y in terms of x ?
(A)2
x
(B) x2
(C) x−90
(D)2
180x
−
(E)2
180 x−
answer
answer
answer
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GR E Math 강좌 – Set 11
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Note : Not drawn to scale.
7. The figure above represents a square plot measuring x feet on a side. The plot consists of a
rectangular garden, 48 square feet in area, surrounded by a walk that is 3 feet wide on two
opposite sides and 2 feet wide on the other two sides. What is the value of x ?
(A) 8
(B) 10(C) 12
(D) 16
(E) 18
8. What is the area of the region enclosed by the figure above?
(A) 116
(B) 144
(C) 176
(D) 179
(E) 284
9. The sum of the interior angles of any polygon with n sides is )2(180 −n degrees. If the
sum of the interior angles of polygon P is three times the sum of the interior angles of
quadrilateral Q , how many sides does P have?
(A) 6
(B) 8
(C) 10
(D) 12
(E) 14
answer
answer
answer
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GR E Math 강좌 – Set 11
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10. The perimeter of a rectangular flower bed is 100 feet. What is the area of the flower bed, in
square feet, if its length is 10 feet greater than its width?
(A) 400
(B) 500
(C) 600
(D) 900
(E) 2,400
11. In the figure above, the shaded rectangular portion of the square region has perimeter 20.
What is the perimeter of the unshaded portion?
(A) 28
(B) 34
(C) 36
(D) 40
(E) 68
12. In the figure above, = x
(A) 75(B) 90
(C) 100
(D) 105
(E) 150
answer
answer
answer
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GR E Math 강좌 – Set 11
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13. A square picture frame has an outer perimeter of 36 inches and is 1 inch wide on all sides.
What is the inner perimeter of the frame, in inches?
(A) 27
(B)2
127
(C) 28
(D)2
131
(E) 32
<High Level Questions>
14. The figure shown above has area A . If the length of each side were doubled, what would
then be the area in terms of A ?
(A) 2 A
(B) 4 A
(C) 6 A
(D) 8 A
(E) 20 A
15. The area of a rectangular garden would be increased by 150 square feet if either the length
were increased by 7.5 feet or the width were increased by 5 feet. What is the area of the
garden, in square feet?
(A) 600
(B) 525
(C) 375
(D) 300
(E) 225
answer
answer
answer
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16. The figure above represents a window, with the shaded regions representing the openings for
the glass. If all line segments in the figure are either horizontal or vertical ad the openings
are all the same size, what are the dimensions, in inches, of each opening? (1 foot = 12 inches)
(A) 12.0 by 18.0
(B) 10.5 by 16.5
(C) 9.0 by 15.0
(D) 8.0 by 10.0
(E) 7.5 by 13.5
Q R
P S
17. The figure above shows a rectangular parcel of undeveloped land partitioned into four regions,
P , Q , R , and S . In square meters, the area of square region Q is2 x , the area of
rectangular region R is x5 , and the area of rectangular region P is x4 . What is thearea, in square meters, of rectangular region S ?
(A) x x −2
(B) x x 92+
(C)220 x x −
(D) 9
(E) 20
answer
answer
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18. A rectangular-shaped carpet remnant that measures x feet by y feet is priced at $50.
What is the cost of the carpet, in dollars per square yard? (9 square feet = 1 square yard)
(A) xy50
(B) xy450
(C)9
xy
(D)50
xy
(E) xy
450
19. A rectangular circuit board is designed to have width w inches, perimeter p inches, and
area k square inches. Which of the following equations must be true?
(A) 02=++ k pww
(B) 022=+− k pww
(C) 022 2=++ k pww
(D) 022 2=−− k pww
(E) 022 2=+− k pww
20. The size of a television screen is given as the length of the screen’ s diagonal. If the screens
were flat, then the area of a square 21-inch screen would be how many square inches greater
than the area of a square 19-inch screen?
(A) 2
(B) 4
(C) 16
(D) 38
(E) 40
answer
answer
answer
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21. The dimensions of a rectangular floor are 16 feet by 20 feet. When a rectangular rug is
placed on the floor, a strip of floor 3 feet wide is exposed on all sides. What are the
dimensions of the rug, in fee?
(A) 10 by 14
(B) 10 by 17
(C) 13 by 14
(D) 13 by 17
(E) 14 by 16
22. A square board that has an area of 25 square inches is to be cut into pieces, each of which is a
square with sides of length 1, 2, or 3 inches. What is the least number of such square pieces
into which the board can be cut?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9
23. In rectangle QRST above, SU RU = . What is the ratio of the perimeter of QUT ∆ to
the perimeter of rectangle QRST ?
(A)8
3(B)
2
1(C)
8
5(D)
4
3(E)
11
9
answer
answer
answer
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24. The area of each of the 16 square regions in the figure above is T . What is the area of the
shaded region?
(A)3
13T
(B) T 5
(C)3
16T
(D)2
11T
(E) T 7
25. Rectangular region PQRS shown above is partitioned into ten identical smaller rectangular
regions, each of which has width x . What is the perimeter of PQRS in terms of x?
(A) x15
(B) x25
(C) x30
(D) x50
(E) It cannot be determined from the information given.
STOP
answer
answer
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수학해설 Category 11 Quadrilateral & Other Polygons
1. What is the area of a square with perimeter P ?
(A)216P
(B) P4
(C)4
2P
(D)16
P
(E)16
2
P
정사각형의 둘레가 P이면 한 변의 길이는 4
p, 따라서 면적은
16)
4(
22 p p=
. 정답은 (E)
2. If the area of a square region having sides of length 6 centimeters is equal to the area of a
rectangular region having width 2.5 centimeters, then the length of the rectangle, in
centimeters, is
(A) 8.5
(B) 9.5
(C) 9.6
(D) 10.5
(E) 14.4
한 변이 6인 정사각형이 한 변이 2.5 centimeters 인 직사각형과 면적이 같을 때, 이 직사각형
의 다른 한 변의 길이를 구하는 문제입니다. 36 = 2.5× P ⇒ P = 14.4
. 정답은 (E)
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GR E Math 강좌 – Set 11
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3. In the figure above, if PQRS is a parallelogram, then =− x y
(A) 30 (B) 35 (C) 40 (D) 70 (E) 100
평행사변형의 성질 중에 PQ 와 RS는 평행하며 길이도 같습니다. 2y와 x는 서로 보각(합쳐서
180도)의 관계이고 2y = 140, x = 40이죠. 다시 정리하면 y = 70이네요. =− x y 30
. 정답은 (A)
<Properties of Parallelograms(평행사변형의 성질)>
A B
EE
C D
l AB | |CD, AC | | BD
l AB = CD, AC = BD
l ∠ A =∠ D,∠ B =∠C
l AE = ED, CE = EB
l ∠ A +∠ B = 180, ∠ B +∠ D = 180, ∠ D + ∠C = 180, ∠C +∠ A = 180
4. A rectangular circuit board is designed to have width w inches, perimeter p inches, and
area k square inches. Which of the following equations must be true?
(A) 02=++ k pww
(B) 022
=+− k pww
(C) 022 2=++ k pww
(D) 022 2=−− k pww
(E) 022 2=+− k pww
j직사각형의 면적 = length × width(=w) = k,k직사각형의 둘레 = 2length + 2w
length을 L라 가정하면 L × w = k, 2L + 2w = P. 이 식을 L에 의해 정리하면
L = 022 2=+− k pww
. 정답은
(E)
E
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5. The size of a television screen is given as the length of the screen’s diagonal. If the screens
were flat, then the area of a square 21-inch screen would be how many square inches greater
than the area of a square 19-inch screen?
(A) 2 (B) 4 (C) 16 (D) 38 (E) 40
Television screen의 크기는 정사각형의 대각선의 길이로 정해진다고 할 때, 대각선의 길이가
21 inch인 정사각형의 면적이 대각선의 길이가 19 inch인 정사각형보다 면적이 얼마나 크냐
를 묻는 문제네요.
일단 21인치의 한 변의 길이를 x라 하면 면적(2 x )은 :
2
441212 222
=⇒= x x
19인치의 한 변의 길이를 y 라 하면 면적(2 y )은 :
2
361192
222=⇒= y y
두 정사학형의 면적의 차이 = 402
361
2
441=−
. 정답은 (E)
6. If the figure above is a parallelogram, what is the value of y in terms of x ?
(A)2
x
(B) x2
(C) x−90
(D)2
180x
−
(E)2
180 x−
평행사변형의 성질 중에 “맞각은 같다”라는 것이 기억 나시죠. 즉 x = 2y이죠. 이 것을 다시
y에 의해 정리하면 y =2
x
. 정답은 (A)
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Note : Not drawn to scale.
7. The figure above represents a square plot me asuring x feet on a side. The plot consists of a
rectangular garden, 48 square feet in area, surrounded by a walk that is 3 feet wide on two
opposite sides and 2 feet wide on the other two sides. What is the value of x ?
(A) 8
(B) 10(C) 12
(D) 16
(E) 18
한 직사각형의 작은 땅(plot) 의 면적은 48 square feet 이며, 그 작은 땅은 폭이 3 feet이고 또
다른 쪽은 2 feet 인 산책로가 둘러싸고 있을 때 x 값을 구하라는 문제네요. 일단 작을 땅
의 가로 a 세로를 b라 하면 ab = 48. 그 다음은 바깥쪽은 정사각형으로 가로로 6 , 세로로 4
만큼 증가했으며
정사각형이니까
변의
길이가
같죠
.다시
말하면
a + 6 = b + 4⇒
a−
b =− 2
j a × b = 48 k a − b = − 2 j과 k을 연립방정식으로 풀면 a = 6 , b = 8
다시 a나 b중에 하나를 a + 6 이나 b + 4 에 대입하면 x값을 구할 수 있죠.
. 정답은 (C)
<여러 가지 사각형>
Quadrilaterals(사각형) > Trapezoid(사다리꼴) > Parallelogram(평행사변형) >
Rectangle(직사각형) > Square(정사각형)
Rhombus(마름모)
직사각형 : 두 대각선의 길이가 같고, 서로 다른 것을 이등분한다.
마름모 : 네 변의 길이가 모두 같은 사각형
정사각형 : 두 대각선의 길이가 같다
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8. What is the area of the region enclosed by the figure above?
(A) 116 (B) 144 (C) 176 (D) 179 (E) 284
. 정답은 (C)
9. The sum of the interior angles of any polygon with n sides is )2(180 −n degrees. If the
sum of the interior angles of polygon P is three times the sum of the interior angles of
quadrilateral Q , how many sides does P have?
(A) 6 (B) 8 (C) 10 (D) 12 (E) 14
문제에서 )2(180 −n 는 한 사각형의 내각의 합의 3배라 했으니
)2(180 −n = 3(360) e n = 8
. 정답은 (B)
10. The perimeter of a rectangular flower bed is 100 feet. What is the area of the flower bed, in
square feet, if its length is 10 feet greater than its width?
(A) 400
(B) 500
(C) 600(D) 900
(E) 2,400
화단의 length = L, width = W라 가정하면 둘레는 2L + 2W = 100feet. 이번에는 L = W + 10의
조건을 주었죠. 이것을 위의 식 2L + 2W = 100feet에 대입하면 W = 20, L = 30. 당연히 면적은
600이네요.
. 정답은
(C)
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11. In the figure above, the shaded rectangular portion of the square region has perimeter 20.
What is the perimeter of the unshaded portion?
(A) 28
(B) 34
(C) 36
(D) 40
(E) 68
정사각형에서 shaded 부분의 높이는 2이고 둘레가 20일 때 밑변은 8 값이 나오죠. 밑변이 8
이라는 말은 이 도형이 정사각형이기 때문에 다른 변의 길이도 8이고 unshaded 부분의 높
이는 8 − 2 = 6. 따라서 unshaded의 둘레는 8+ 8+ 6+ 6 = 28
. 정답은 (A)
12. In the figure above, = x
(A) 75
(B) 90
(C) 100(D) 105
(E) 150
사다리꼴은 두 개의 직각삼각형과 하나의 직사각형으로 나눌 수 있죠. 이 삼각형에서 두 변
의 길이는 50과 40이므로 나머지 한 변은 222 4050 a+= . 이 식을 정리하면 a = 30.
= x 30 + 45 + 30 = 105
. 정답은
(D)
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13. A square picture frame has an outer perimeter of 36 inches and is 1 inch wide on all sides.
What is the inner perimeter of the frame, in inches?
(A) 27 (B)2
127 (C) 28 (D)
2
131 (E) 32
9
9
Inner frame의 둘레길이는 4×7 = 28
. 정답은 (C)
<High Level Questions>
14. The figure shown above has area A . If the length of each side were doubled, what would
then be the area in terms of A ?
(A) 2 A (B) 4 A (C) 6 A (D) 8 A (E) 20 A
위의 사각형은 아래와 같이 나눌 수 있고 면적은 cd + ab = A가 됩니다. 나눈 후에 각 변의
길이가 2배로 증가하므로 면적은 2d × 2c = 4dc가 되고 2a× 2b =4ab가 되므로 처음 면적 A에 4
배가 증가했습니다.
d + a
c b
c
ÿ 정답은 (B)
1 7
7
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15. The area of a rectangular garden would be increased by 150 square feet if either the length
were increased by 7.5 feet or the width were increased by 5 feet. What is the area of the
garden, in square feet?
(A) 600 (B) 525 (C) 375 (D) 300 (E) 225
증가하기 전 정원의 area(면적)을 A, length(세로)을 L,그리고 width(가로)을 W라 가정하면:
j (L + 7.5)(W) = A + 150, k L× (W + 5) = A + 150
위의 식에서 L×W = A와 동일하므로 L×W 대신에 A를 대입해서 풀면 L = 20, W=30을 구할 수
있고 면적은 600이 됩니다.
ÿ 정답은 (A)
16. The figure above represents a window, with the shaded regions representing the openings for
the glass. If all line segments in the figure are either horizontal or vertical ad the openings
are all the same size, what are the dimensions, in inches, of each opening? (1 foot = 12 inches)
(A) 12.0 by 18.0
(B) 10.5 by 16.5
(C) 9.0 by 15.0
(D) 8.0 by 10.0
(E) 7.5 by 13.5
일단 세로부터 계산하면 3ft = 36 inches가 되고 36 inches − (3+3+3) = 27 inches을 2로 나누면
13.5 inches가 나오네요. 문제 중에서 13.5가 있는 것은 (E) 뿐이므로 정답!
유리창의 가로의 길이도 위와 동일한 방법으로 풀면 됩니다.
ÿ 정답은 (E)
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Q R
P S
17. The figure above shows a rectangular parcel of undeveloped land partitioned into four regions,
P , Q , R , and S . In square meters, the area of square region Q is2 x , the area of
rectangular region R is x5 , and the area of rectangular region P is x4 . What is the
area, in square meters, of rectangular region S ?
(A) x x −2
(B) x x 92+
(C)220 x x −
(D) 9
(E) 20
S 의 면적을 구하려면 가로와 세로의 길이를 알아야 합니다. 그런데 S 의 가로는 R 과 같
고 세로는 P와 같습니다.
일단 Q의 한 변의 길이는 x이므로 R 의 세로도 x이고 R의 면적이 x5 이므로 가로는 5가 되죠. 동일한 방법으로 P의 세로의 길이는 4가 됩니다. 따라서 S 의 면적은 20.
ÿ 정답은 (E)
18. A rectangular-shaped carpet remnant that measures x feet by y feet is priced at $50.
What is the cost of the carpet, in dollars per square yard? (9 square feet = 1 square yard)
(A) xy50 (B) xy450 (C) 9
xy(D) 50
xy(E) xy
450
dollars yfeet x 50=× ⇒ 1 square feet =9
1square yard을 왼쪽에 대입하면
1 square yard = xy
450
ÿ 정답은 (E)
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GR E Math 강좌 – Set 11
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19. A rectangular circuit board is designed to have width w inches, perimeter p inches, and
area k square inches. Which of the following equations must be true?
(A) 02=++ k pww
(B) 022 =+− k pww
(C) 022 2=++ k pww
(D) 022 2=−− k pww
(E) 022 2=+− k pww
직사각형의 세로를 L이라 하면
j p LW =+ 22 k k W L =×
위의 두 식으로 L값을 구하면 022 2=+− k pww
ÿ 정답은 (E)
20. The size of a television screen is given as the length of the screen’ s diagonal. If the screens
were flat, then the area of a square 21-inch screen would be how many square inches greater
than the area of a square 19-inch screen?
(A) 2
(B) 4
(C) 16
(D) 38
(E) 40
TV 크기는 스크린의 대각선의 길이로 주어진다고 했고, 스크린이 평면이라고 했습니다. 한
정사각형 스크린이 21-inch 일 때 면적과, 또 다른 정사각형 모양의 스크린이 19-inch일 때
두 면적의 차이를 구하는 문제입니다.
일단 스크린 크기 21-inch는 한 변을 a라 가정하면 면적은 2a 을 구하면 됩니다.
22 )21(2 =a ⇒ 2
a = 441/2
2b 도 a를 구한 방법과 동일하게 구하면 2b = 361/2
2a −
2b = 80/2 = 40
ÿ 정답은 (E)
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GR E Math 강좌 – Set 11
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21. The dimensions of a rectangular floor are 16 feet by 20 feet. When a rectangular rug is
placed on the floor, a strip of floor 3 feet wide is exposed on all sides. What are the
dimensions of the rug, in fee?
(A) 10 by 14
(B) 10 by 17
(C) 13 by 14
(D) 13 by 17
(E) 14 by 16
가로 16 세로 20인 바닥에 rug을 깐 후 바닥 가장자리에 3 feet가 남을 때 이 rug의 치수를
구하려면 가로는 16 − 6 =10이고 세로는 20 − 6 = 14입니다.
ÿ 정답은 (A)
22. A square board that has an area of 25 square inches is to be cut into pieces, each of which is a
square with sides of length 1, 2, or 3 inches. What is the least number of such square pieces
into which the board can be cut?
(A) 5
(B) 6
(C) 7
(D) 8
(E) 9
위의 테이블에서 한 셀의 크기는 가로 1 inch 세로 1inch이고 각 변의 길이는 5 inches 일 때,
각 변이 1, 2, or 3 inches 중에서 한 수치를 선택해서 정사각형 모양으로 나눌 때 가장 작은
경우의 수는 8가지입니다. 위의 그림을 참고하세요!
ÿ 정답은 (D)
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23. In rectangle QRST above, SU RU = . What is the ratio of the perimeter of QUT ∆ to
the perimeter of rectangle QRST ?
(A)8
3(B)
2
1(C)
8
5(D)
4
3(E)
11
9
22222 534)()( =+== UT QU
QUT ∆ 의
둘레
= 8 + 5 + 5 = 18QRST 의 둘레 = 8 + 8 + 3 + 3 = 22
따라서 the ratio of QUT ∆ to QRST = 18 : 22 = 9 : 11 =11
9
ÿ 정답은 (E)
24. The area of each of the 16 square regions in the figure above is T . What is the area of the
shaded region?
(A)3
13T (B) T 5 (C)3
16T (D)2
11T (E) T 7
각각의 정사각형의 면적이 T 이므로 전체 면적은 16T 가 되고 16T 에서 unshaded region의
면적을 빼주면 shaded region의 면적을 구할 수 있습니다.
16T − ( 4T + 6T + T ) = 5T
ÿ 정답은 (B)
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25. Rectangular region PQRS shown above is partitioned into ten identical smaller rectangular
regions, each of which has width x . What is the perimeter of PQRS in terms of x?
(A) x15
(B) x25
(C) x30
(D) x50
(E) It cannot be determined from the information given.
PQ = x5 , RS = l이며 l x =5 이 성립합니다.
PS = l x +5 = x10
따라서 직사각형의 둘레는 x10 + x10 + x5 + x5 = x30
ÿ 정답은 (C)
<평행사변형이 되는 조건>
사각형에서 다음 중 어느 하나의 조건을 만족하면 평행사변형이 된다.
l 두 쌍의 대변이 각각 평행하다
l 두 쌍의 대변의 길이가 각각 같다.
l 두 쌍의 대각선의 크기가 각각 같다
l 두 대각선이 서로 다른 대각선을 이등분한다.
l 한 쌍의 대변이 평행하고 그 길이가 같다.
STOP