Download geometrical gems

REMT 2016
GEOMETRICAL
GEMS
ANSWERS
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LAST NAME
FIRST NAME
GRADE
2016E2
1.
In this figure, DE is parallel to AB. If AB has
length 5, BE has length 3, and EC has length 7,
what is the length of DE?
A
D
B
C
E
2.
In a certain triangle ABC, the length of AB is equal to the length of BC. If angle B
measures 40 degrees, what is the degree measure of angle C?
3.
In a regular 5-pointed star, what is the degree measure of
each of the angles at the star's points?
4.
In the given triangle ABC, the length of AB is 13, the
length of AC is 15, and the length of the altitude AD is
12. What is the area of this triangle?
A
B
D
C
2016E3
5.
From a certain point P, the two tangents to a circle of radius 5 form an angle of 60
degrees. What is the distance from this point P to the center of the circle?
6.
Triangle ABC is an isosceles right triangle with sides AB
and BC of length 10. BD and BE are circular arcs with
centers at A and C respectively. What is the exact area of
the shaded region BDE?
A
E
D
B
C
7.
A circle is inscribed in a right triangle whose legs are 5 and 12 units long, respectively.
What is the exact value of the area of this circle?
8.
The sides of a 10 by 10 square are cut by various points into
pieces of length 2 and 8, 3 and 7, 4 and 6, and 5 and 5, as
shown. What is the area of the quadrilateral determined by
these four points?
2
8
5
3
5
7
6
4
2016E4
9.
In a certain triangle ABC, point D is one-third of the way from A to B, and point E is
two-thirds of the way from A to C. What is the ratio of the area of triangle ADE to the
area of triangle ABC?
10.
AB and CD are two chords of a circle. When
extended through A and C, these chords
meet at a point P outside the circle. If the
length of PA is 4, the length of AB is 6, and
the length of PC is 5, what is the length of
CD?
B
A
P
C
D
11.
Two right circular cones are constructed of cardboard. One has radius 7 inches and
height 9 inches; the other has radius 5 inches and height 14 inches. The two cones
are places base down on a table with their bases touching. How many inches apart
are the vertices of these cones?
12.
A water tank is made in the shape of a horizontal
circular cylinder with hemispherical caps at both
ends. The length of the tank, including the caps, is
16 feet and the height is 6 feet. What is the volume
of this tank, to the nearest cubic foot?