1999
State High School Mathematics Contest
Geometry Test
1.)
2.)
3.)
4.)
If the hypotenuse of a 30  60  90 right triangle has length 4, then
the area of the triangle is
a.)
4 3
b.)
2 3
d.)
6
e.)
15
2
c.)
12
The area of a first circle is 3 times that of a second circle. The ratio of
the radius of the first circle to the radius of the second circle is
a.)
3 :1
b.)
9:1
d.)
1: 3
e.)
1:9
c.)
3 :1
Among the sets of integers A  2,5,8 , B  3, 4,5 , C  5,12,13 , and
D  10,10,10 , which cannot be the lengths of sides of a triangle?
a.)
A only
b.)
B and C
c.)
d.)
A and D
e.)
none of A, B, C, or D
C only
A circle is inscribed in an equilateral triangle with side length 2
inches. The area of the circle is
a.)
 square inches
b.)
c.)

square inches
3
d.)

square inches
2

square inches
4
e.)
5.)
6.)

square inches
6
In a rectangular solid with side lengths 1, 2, and 3, the greatest
distance between any two corners is
a.)
14
b.)
d.)
5
e.)
c.)
13
10
6
The figure below shows a shaded sector of a circle of radius 5 inches.
If the length of the circular arc of the boundary of the sector is 10
inches, then the area of the sector is
5
7.)
a.)
25
square inches
36
b.)
25
square inches
3
c.)
15 square inches
d.)
25 square inches
e.)
50 square inches
In the figure below, lines L1 and L2 are parallel, the measure of
angle 1 is 37 , and the measure of angle 2 is 85 . The measure of
angle 3 is
1
3 3
3
2
L1
L2
8.)
9.)
a.)
58
b.)
53
c.)
d.)
43
e.)
none of the above
48
If the hypotenuse of a right triangle has length 7 meters and the sum
of the lengths of the legs is 8 meters, then the area of the triangle is
a.)
7
square meters
2
b.)
10
square meters
3
c.)
4 square meters
d.)
15
square meters
2
e.)
15
square meters
4
A rectangle is inscribed in a circle of diameter 12 inches. If one of the
sides of the rectangle has length 6 inches, then the area of the
rectangle is
a.)
36 3 square inches
b.)
36   1 square inches
c.)
72 square inches
d.)
6 3 square inches
e.)
24 1  3 square inches


10.) The sum of the areas of the shaded triangles within the 6 inch by inch
rectangle shown below is
3
a.)
9 square inches
b.)
10 square inches
c.)
11 square inches
d.)
12 square inches
e.)
There is not enough information to decide
11.) The area of a triangle with side lengths 5, 10, and 13 is
a.)
30
b.)
6 14
d.)
25
e.)
28
c.)
12
12.) Suppose that n  3 points lie in a plane and no three of them lie on the
same line. The number of distinct triangles formed by the points –
taken three at a time – is
a.)
n
3
b.)
3n - 8
d.)
n 2  11n  36
6
e.)
n3  3n 2  2n
6
c.)
n!
6
13.) If K is a positive constant and the line x + 2y = K is tangent to the
circle x 2  y 2  1 at a point in the first quadrant, then K equals
a.)
d.)
2
1
2
b.)
e.)
3
c.)
5
1
5
14.) The figure below is formed by placing the 6 inch diameter of a
semicircle along the 6 inch side of a 5 in. – 5 in. – 6 in. isosceles
triangle. The area of the figure, in square inches, is
a.)
24  9
2
b.)
d.)
36  18
2
e.)
24  36
2
c.)
24  18
2
12  9
2
15.) A line segment drawn from a vertex of a triangle to the midpoint of
the opposite side of a triangle always divides the triangle into two
triangles which are
a.)
right triangles
b.)
similar triangles
c.)
congruent triangles
d.)
triangles with equal area
e.)
triangles with at least one equal angle
16.) The sum of all interior angles of the (non-convex) polygon shown
below is
a.)
1350
d.)
1800
b.)
e.)
c.)
1440
1620
there is not enough information to decide
17.) Two straight lines L and M lie in the x-y plane. L passes through
the points (7, -3) and (2, 1), and line M is perpendicular to line L.
The slope of line M is
a.)
- 0.80
b.)
1.25
d.)
0.80
e.)
1.20
c.)
- 1.25
18.) A spherical balloon of diameter 10 inches contains water, which is 2
inches deep at the deepest point. The area of the top surface of the
water is
a.)
25 
b.)
21 
d.)
9
e.)
4
c.)
16 
19.) Three points A, B, and C lie on the circumference of a circle of radius
2 as shown below. If the length of arc AB is 2 and the length of arc
AC is 4, then the radian measure of angle BAC is
B
A

2
C
a.)
3
2
d.)

3
2
b.)

2
e.)
 2
 1
c.)
20.) The three medians of a triangle meet at a point which always
a.)
is the center of the inscribed circle of the triangle
b.)
is the center of the circumscribed circle of the triangle
c.)
bisects each median
d.)
trisects each median
e.)
none of the above
21.) In the right triangle ABC shown below, the length of AD is 6, the
length of BC is 5, and the length of DE is 2. The area of triangle
ABC is
B
D
5
6
2
A
C
E
a.)
10 2
b.)
10 3
d.)
35
e.)
30 2
c.)
25 2
22.) A 3 by 20 rectangle intersects a circle of radius 5 as shown below.
The center O of the circle lies on AC . The product of the lengths of
AB and BC is
B
C
O
A
a.)
27
b.)
30
d.)
36
e.)
39
c.)
33
23.) Let f  x  denote the volume of a cube with surface area x. If
f  x   x, then x equals
a.)
1
b.)
6
6
d.)
6
e.)
216
c.)
3
2
24.) If H is the point where the three altitudes of triangle ABC meet, then
it is always true that
a.)
the perpendicular bisectors of the sides of triangle HBC meet at
A
b.)
the internal angle bisectors of the angles of triangle HBC meet
at A
c.)
the medians of triangle HBC meet at A
d.)
the altitudes of triangle HBC meet at A
e.)
H is always inside triangle ABC
25.) A quadrilateral ABCD is inscribed in a circle of diameter 5, with AC
passing through the center 0 of the circle, as shown below. Given that
the length of BC is 1 and the length of CD is 2, the perimeter of the
quadrilateral is closest to
B
C
A
0
D
a.)
11
b.)
12
d.)
20
e.)
23
c.)
13
26.) A Mobius strip of width 1 inch is folded and flattened on the plane so
as to frame an equilateral triangle as shown below. The longer
outside line segments on the border of the flattened figure have length
3 inches. If the strip is cut along a width as shown and unfolded, the
resulting rectangular strip will have length (in inches) equal to
a.)
9
b.)
9 3
d.)
21
2
e.)
12
c.)
92 3
27.) A circle C in the x-y plane has center (-1,2) and radius 3. The number
of points with integer coordinates and which lie on or inside of C is
a.)
25
b.)
26
d.)
28
e.)
29
c.)
27
28.) The area of shaded rectangular portion of the regular hexagon shown
below accounts for what proportion of the total area of the hexagon?
a.)
7
12
b.)
2
3
d.)
3
4
e.)
5
6
2 3
5
c.)
2
29.) A 5in. by 5 in. square target is
colored black and white along 45
angle stripes as shown. The
probability that a dart thrown
randomly into the target lands in
black is
1
2
1
2
2
a.)
1
2
b.)
3
5
d.)
12
25
e.)
13
25
c.)
9
25
30.) For two points P and Q, let m(PQ) denote the length of PQ . In the
triangle shown below AE , BF , and CD all intersect at 0. Moreover, 16
m(AD) = 15 m(DB) and 5 m(BE) = 2 m(EC).
C
F
0
A
E
B
a.)
3
b.)
5
2
d.)
2
e.)
3
2
c.)
8
3
31.) For two points P and Q, let m(PQ) denote the length of PQ . Suppose
that triangle ABC is inscribed in a circle of radius 3, with AB lying
along a diameter of the circle. Then m(AC) + m(BC) must be
a.)
strictly less than 6 2
b.)
less than or equal to 6 2
c.)
equal to 6 2
d.)
greater than or equal to 6 2
e.)
strictly greater than 6 2
32.) Suppose that the length of the diagonal of a square is a rational
number. Then the perimeter of the square must always be
a.)
an integer
b.)
a rational number
c.)
an integral multiple of 2
d.)
an irrational number
e.)
none of the above
33.) A circular race track consists of the ring enclosed by two circles with
the same center. If the track is 8 feet wide, then the difference
between the outer circumference and the inner circumference is
closest to
a.)
20 feet
b.)
30 feet
c.)
40 feet
d.)
50 feet
e.)
60 feet
34.) Consider an isosceles trapezoid, and suppose that the longer base is
equal in length to a diagonal, and the shorter base is equal in length to
an altitude. Then the ratio of larger base to the smaller base is
a.)
5:3
b.)
5:2
c.)
4:3
d.) 3:2
e.)
2:1
35.) In a right triangle, let the length of the hypotenuse be c and the lengths
of the two legs be a and b. If a and c differ by 1, then b 2 equals
a.)
ac
b.)
c
a
d.)
c–a
e.)
2a
c.)
a+c
36.) Let A denote the set of points (x,y) in the x-y plane satisfying
3 x  y  5 , and let B denote the set of points (x,y) in the x-y plane
satisfying x2  y 2  25. Then the intersection of A and B consists of
a.)
a single point
b.)
two points c.)
an arc of a circle
d.)
a straight line segment including its endpoints
e.)
a straight line segment excluding its endpoints