October 8, 2014
Any calculator permitted on N.Y.S. Regents examinations may be used.
The word “compute” calls for an exact answer in simplest form.
3 - Areas of Polygons
1. In rhombus RHOM , the diagonals intersect at B. Suppose RH = 10 and RO = 12.
Compute the area of RHOM .
2. Regular pentagon W AY N E is centered at O. If OY = 10, find the area of W AY N E,
rounded to the nearest thousandth.
3. Trapezoid ABCD is the union of square ABF E and right triangles BF C and AED as
shown in the figure below. The perimeter of ABCD is 152 units. Compute the area of
square ABF E in square units.
A
x+y+9
D
x+y+1
x
B
x+1
E
F y C
October 2014
3 - Areas of Polygons
1. In rhombus RHOM , the diagonals intersect at B. Suppose RH = 10 and RO = 12.
Compute the area of RHOM .
each other, RB = 12/2 = 6.
SOLUTION: 96 Since the diagonals of a rhombus bisect √
Since the diagonals of a rhombus are perpendicular, HB = 102 − 62 = 8. The area of the
rhombus is four times the area of 4RBH, or 4 · 12 · 6 · 8 = 96.
2. Regular pentagon W AY N E is centered at O. If OY = 10, find the area of W AY N E,
rounded to the nearest thousandth.
SOLUTION: 237.764 The area of W AY N E is five times the area of 4OY N , or
5 · 12 · 10 · 10 · sin 72◦ , which is approximately 237.764. Note: For those students who have
not yet studied A2Trig, students could also use right triangle trig on isosceles triangle
OY N to find the base and height of the triangle.
3. Trapezoid ABCD is the union of square ABF E and right triangles BF C and AED as
shown in the figure below. The perimeter of ABCD is 152 units. Compute the area of
square ABF E in square units.
A
x+y+9
D
x+y+1
x
B
x+1
E
F y C
SOLUTION: 576 The perimeter of ABCD is 5x + 3y + 11 = 152 → 5x = 141 − 3y. In
4BCF , x2 + y 2 = (x + 1)2 → 2x = y 2 − 1. Substituting,
5 2
(y − 1) = 141 − 3y → 5y 2 − 5 = 282 − 6y → 5y 2 + 6y − 287 = 0. Factoring,
2
(5y + 41)(y − 7) = 0, so y = 7 and x = 24. The area of ABF E is x2 = 242 = 576.
October 2, 2013
Any calculator permitted on N.Y.S. Regents examinations may be used.
The word “compute” calls for an exact answer in simplest form.
3 - Areas of Polygons
1. Square REUT has an area of 36. Let point A be on EU such that EA = 2(AU ).
Compute the area of triangle RAT .
2. Parallelogram MATH has a perimeter of 40. Altitude AD is drawn and AD = 4. If
M D = 3, compute the area of the parallelogram.
3. In quadrilateral ABCD, AB ⊥ BC, DA ⊥ AC, AC = 45, AD : AB = 7 : 9, and the
ratio of the areas of 4ABC and 4ADC is 27 : 35. Compute the area of ABCD. Do NOT
assume that ABCD is a trapezoid.
A
D
B
C
October 2013
3 - Areas of Polygons
1. Square REUT has an area of 36. Let point A be on EU such that EA = 2(AU ).
Compute the area of triangle RAT .
√
SOLUTION: 18 Each side of REUT measures 36 = 6. It does not matter where A is
on the segment EU . The area of RAT is 21 · 6 · 6 = 18.
2. Parallelogram MATH has a perimeter of 40. Altitude AD is drawn and AD = 4. If
M D = 3, compute the area of the parallelogram.
√
SOLUTION: 60 The length of M A is 32 + 42 = 5, so AT = 40−2·5
= 15. The area we
2
want is 15 · 4 = 60.
3. In quadrilateral ABCD, AB ⊥ BC, DA ⊥ AC, AC = 45, AD : AB = 7 : 9, and the
ratio of the areas of 4ABC and 4ADC is 27 : 35. Compute the area of ABCD. Do NOT
assume that ABCD is a trapezoid.
A
B
D
C
SOLUTION: 1116 We know that
9k
A
7k
D
1
(7k)(45)
2
=
27
→ x = 27.
35
B
45
y
1
x(9k)
2
x
C
In 4ABC, (27, 9k, 45) = 9(3, k, 5) → k = 4 → AD = 28. Thus, the area of ABCD is
1
· 36 · 27 + 12 · 28 · 45 = 1116.
2
October 3, 2012
Any calculator permitted on N.Y.S. Regents examinations may be used.
The word “compute” calls for an exact answer in simplest form.
3 - Areas of Polygons
1. In rectangle P ARK, P A = 6 and P K = 4. T is on KR such that KT = 4. Compute
the area of T RAP .
2. A rectangular piece of paper 9 inches long is folded in half, creating a smaller rectangle
that is similar in shape to the original piece of paper. Find the area of the original piece of
paper, to three decimal places.
3. If the sides of a square are increased by 25%, its area equals that of a rectangle whose
sides have lengths in a 2 : 5 ratio and whose perimeter is 28 units. Compute the original
length of a side of the square.
October 2012
3 - Areas of Polygons
1. In rectangle P ARK, P A = 6 and P K = 4. T is on KR such that KT = 4. Compute
the area of T RAP .
SOLUTION: 16 We compute [P ARK] − [P T K], which is 4 · 6 − 12 · 4 · 4 = 16.
2. A rectangular piece of paper 9 inches long is folded in half, creating a smaller rectangle
that is similar in shape to the original piece of paper. Find the area of the original piece of
paper, to three decimal places.
SOLUTION: 57.276 Since the two rectangles are similar, their sides are in proportion.
Further, the width of the original piece of paper becomes the length of the new smaller
w
to obtain w = 6.364, and the area of the original
rectangle. Therefore, we solve w9 = 4.5
rectangle is 9 · 6.364 = 57.276.
3. If the sides of a square are increased by 25%, its area equals that of a rectangle whose
sides have lengths in a 2 : 5 ratio and whose perimeter is 28 units. Compute the original
length of a side of the square.
√
SOLUTION: 85 10 Call the original side of the square x; then the sides of the new
square are 54 x, and the area of the new square is 25
x2 . If the length and width of a
16
rectangle are in the ratio 2 : 5 but the half-perimeter is 14, then the width and length are
found by solving 2y + 5y = 14 → y = √
2, so the rectangle is 4 by 10 and has area 40. We
25 2
40·16
8
2
solve 16 x = 40 → x = 25 → x = 5 10.
December 13, 2011
Calculators are not permitted on this contest.
The word “compute” calls for an exact answer in simplest form.
3 - Areas of Polygons
1. If the length of each side of a triangle is multiplied by 2, the area is multiplied by k.
Compute k.
2. In trapezoid ABCD, ABkDC, AB = 12, and DC = 20. P Q is the median of trapezoid
ABCD, with P on AD and Q on BC. Compute the ratio of the area of trapezoid ABQP
to the area of trapezoid P QCD.
3. ABCD is a square, AB is extended to F , DF intersects BC at E, BE : EC = 1 : 2,
and the area of 4BEF is 24. Compute the area of ABCD.
B
A
E
D
C
F
December 2011
3 - Areas of Polygons
1. If the length of each side of a triangle is multiplied by 2, the area is multiplied by k. Compute
k.
SOLUTION: 4 Let the base and height be b and h. After the dilation by 2, the base has
length 2b and the height has length 2h. Thus, the area becomes 12 · (2b) · (2h) = 4 · 12 · b · h, which
is 4 times the area of the original triangle. Our answer is k = 4.
2. In trapezoid ABCD, ABkDC, AB = 12, and DC = 20. P Q is the median of trapezoid
ABCD, with P on AD and Q on BC. Compute the ratio of the area of trapezoid ABQP to the
area of trapezoid P QCD.
SOLUTION: 79 As a median, P Q = 12+20
= 16 and the altitudes of trapezoids ABQP and
2
P QCD are equal in length. Thus, the required ratio is
7
9.
1
2 h(12
1
2 h(16
+ 16)
, which is
+ 20)
28
36 ,
which reduces to
3. ABCD is a square, AB is extended to F , DF intersects BC at E, BE : EC = 1 : 2, and the
area of 4BEF is 24. Compute the area of ABCD.
B
A
F
E
D
C
SOLUTION: 288 We first note that 4BEF ∼ 4CED and BE : CE = 1 : 2. Let BE = x and
CE = 2x. Then CD = 3x, which implies BF = 23 x. Therefore, 21 x · 32 x = 24 → x2 = 32. The area
of ABCD = (3x)(3x) = 9x2 = 9(32), or 288.
DECEMBER 14, 2010
NO CALCULATORS ON THESE TOPICS
#3 AREA OF POLYGONS
1. The area of ∆PQR is 45 square units.
The areas of ∆PQS and ∆PSR are unequal.
Determine the number of square units in
the smaller of the two areas.
2. Rectangle ABCD has an area of 500 inches2.
E and F are midpoints of two adjacent sides.
Determine the number of square inches in the
larger of the two regions inside ABCD created by EF .
3. If square ABCD with AB = 2′ is rotated 45° about
its center T, a new square PQRS is generated.
Compute the number of square feet in the area of
the overlap, i.e. the area of the shaded region.
ANSWERS:
1.
21
2.
437.5
3.
8 2 −8
2010
DECEMBER 15, 2009
NO CALCULATORS ON THESE TOPICS
#3 AREA OF POLYGONS
ANSWERS:
1.
4.5
2.
12
3.
16 2
1. A segment connects the midpoints of the legs of a right triangle with sides
of length 3″, 4″, and 5″, dividing the triangle into a triangle and a trapezoid.
Find the number of square inches in the area of the trapezoid.
2. In rectangle ABCD, AB = 12 cm and AD = 4 cm.
Point G is on AB, point H is on DC , and point E
is the intersection of the diagonals. Find the number
of square cm’s in the area of the shaded region.
3. In a regular octagon ABCDEFGH, AF = 8 units.
Compute the number of square units in the area of ∆AFC.
DECEMBER 9, 2008
NO CALCULATORS ON THESE TOPICS
#3 AREA OF POLYGONS
ANSWERS:
1.
2 65
2.
192
3.
61
1. A right triangle has an area of 60 and its legs have lengths in a 2:3 ratio.
Compute the length of the hypotenuse.
2. A line parallel to the short sides of a 12 X 25 rectangle subdivides the rectangle into two
similar, noncongruent rectangles. Determine the area of the larger of these two rectangles.
 y = x −1 + x − 2 + x − 4

x = 0
3. Compute the area of the region bounded by 
x = 8
y = 0

N{ONROE C@UNTY
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ANSWERS:
#3 AREA OF POLYGONS
2^,165
l.
r92
61
3.
1. A right triangle has an areaof 60 and its legs have lengths in a 2:3 ratio.
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DECEMBER
11,2OO7
NO CALC(TLATORSOATTHIS TOPIC
#3
AREA OF POLYGONS
ANSWERS:
120
4/
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t4,4
3
1. The side of a rhombus is 13" and one of its diagonalsis 24"
Find the number of squareinches in its area.
&*1 = tu
r ar t
A:
d. d ' o
..€h
T
2. A six-pointed star is formed by taking equilateralaABC, flipping it over a horizontal line to
form aDEF and placing it on top of the IABC so that all of its sides are trisectedby the
intersection points. Express (in simplest form) the ratio of the areaof the entire star to the
areaof the orieinal aABC.
D
A bo €*u, ta*e"a[B
easA t l-1ft1e-
sD 1-A -F=fE
q "\-t
qA
3. Thebasesof trapezoidABCD areAp : 12 andCD:8. The altitudeis 6.
The diiagonalsAC and BD intersectat E. Whatis the areaof aDEA?
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DECEM BER
12,2006
]YO CALCULATORSON THIS TOPIC
#3,
AREA OF POLYGONS
ANSWERS:
2 0 + 2 0 j;
|
816
18+ 6J3
l. SquareABCD hasarea400. Its diagorlals
intersectat E.
Computethe exactperimeterof triangleABE.
6
rt= {o o
sa ncc
2o
S = zo
ZS
:;
20
P= ton6.*ro{2.{
2-c{- uo Jz
2. RectangleSROHhasSrR= 40 andSl/:30. PointE is on RIl ro that SE -&*^(&
f RII.
penta
gon
SHORE.
Find the areaof concave
or.
,fJ
G
zrir" .lo
Cr.Lofro ) = 5c(;<)
ABC andBCD withmlA:45o
3. Givenrighttriangles
mlD: 30o,andDE: 12,computetheur"uof t.i*g'I. BCE.
cbStr.{,', -13- =
tr^.r^,
S frU
r>t{) = b 1..
g=
G.tc
?:-6 .
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r
9.tri3)= iAtb.fr)
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fL
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aJi'o ;
tl
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