Download Meet 3 Cat 2 SG

Park Forest Math Team
Meet #3
Self-study Packet
Problem Categories for this Meet (in addition to topics of earlier meets):
1.
2.
3.
4.
5.
Mystery: Problem solving
Geometry: Properties of Polygons, Pythagorean Theorem
Number Theory: Bases, Scientific Notation
Arithmetic: Integral Powers (positive, negative, and zero), roots up to the sixth
Algebra: Absolute Value, Inequalities in one variable including interpreting line graphs
Important Information you need to know about GEOMETRY…
Properties of Polygons, Pythagorean Theorem
Formulas for Polygons where n means the number of sides:
• Exterior Angle Measurement of a Regular Polygon: 360÷n
• Sum of Interior Angles: 180(n – 2)
• Interior Angle Measurement of a regular polygon:
• An interior angle and an exterior angle of a regular polygon
always add up to 180°
Interior
angle
Exterior
angle
Diagonals of a Polygon where n stands for the number of vertices (which is equal
to the number of sides):
•
• A diagonal is a segment that connects one vertex of a polygon to
another vertex that is not directly next to it. The dashed lines represent
some of the diagonals of this pentagon.
Pythagorean Theorem
• a2 + b2 = c2
• a and b are the legs of the triangle and c is the hypotenuse (the side
opposite the right angle)
c
a
b
•
Common Right triangles are ones with sides 3, 4, 5, with sides 5, 12, 13,
with sides 7, 24, 25, and multiples thereof—Memorize these!
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Category 2 ± Geometry
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Answers
1. _______________
/!
2. _______________
3. _______________
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Answers
1. 50
2. 12
3. 30
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Category 2
Geometry
Meet #3, January 2009
1.
How many degrees are in the sum of the interior angles of a convex
decagon?
2.
Let the number of diagonals in a regular octagon be !, and the number of
diagonals in a regular hexagon be ". What is the value of ! # " ?
3.
Quadrilateral ABCD has right angles at A and C. The lengths of CD, BC,
and AB are 7 cm, 11cm, and 1cm respectively. How many centimeters long
is AD?
C
B
A
D
Answers
1. _______________
2. _______________
3. _______________
Solutions to Category 2
Geometry
Meet #3, January 2009
Answers
1. 1440
2.
11
3.
13
1. For an !-sided convex polygon, the expression
"#$%! & '( will give the number of degrees in the sum of the
interior angles of the polygon. So a decagon %! ) "$( would
have "#$%"$ & '( ) "#$%#( ) "**$ degrees.
+%+,-(
2. For an !-sided convex polygon, the expression
will
.
give the number of diagonals in the polygon. So an octagon has
/%0(
1%-(
) '$ diagonals and a hexagon has
) 2 diagonals. So
.
.
3 ) '$ and 4 ) 2. Therefore 3 & 4 ) '$ & 2 ) "".
3. First draw in segment 56. Since 56 is the hypotenuse of triangle 576, we
can use the Pythagorean Theorem to find the length of drawn in segment 56.
56. ) 8. 9 "". ) *2 9 "'" ) "8$. However BD is also the hypotenuse of
triangle :56 and therefore 56. ) ". 9 :6. ) " 9 :6. ) "8$. So :6. ) ";2
and :6< ) <"=.
11
C
B
7
1
A
?
D
Category 2
Geometry
Meet #3, January 2007
1. How many degrees are in the measure of an exterior angle of a regular
decagon?
exterior angle
2. A Pythagorean Triple is a set of three natural numbers that satisfy the
Pythagorean Theorem a 2 + b 2 = c 2 , where a and b are legs on a right triangle and c
is the hypotenuse. One way to find Pythagorean Triples is with the three
equations a = m 2 − n 2 , b = 2mn , and c = m 2 + n 2 , where the values of m and n are
natural numbers with m > n. How many units are in the perimeter of the right
triangle that it is produced when m = 7 and n = 4?
3. An isosceles triangle has sides measuring 34 units, 34 units, and 32 units. If the
32-unit side is considered the base, how many units are in the height (or the
altitude) of this triangle?
Answers
1. _______________
2. _______________
3. _______________
34 units
34 units
32 units
www.imlem.org
Solutions to Category 2
Geometry
Meet #3, January 2007
Answers
1. 36
2. 154
1. Imagine taking a walk counter-clockwise around the
regular decagon. You will make ten left-hand turns of
the same measure. When you get back to where you
started, you will have turned a total of 360 degrees.
Therefore, each exterior angle must be 360 ÷ 10 = 36
degrees.
3. 30
2. Using m = 7 and n = 4, we find that
a = 7 2 − 4 2 = 49 −16 = 33 , b = 2 ⋅ 7 ⋅ 4 = 56 , and
c = 7 2 + 4 2 = 49 + 16 = 65 . The perimeter of a right
triangle with sides of 33, 56, and 65 units is 33 + 56 + 65
= 154 units.
3. First we draw the height we wish to find. An altitude
line is always perpendicular to the base. On an isosceles
triangle this line also meets the base at the midpoint.
Now we can use the Pythagorean Theorem to find the
missing leg of a right triangle with an hypotenuse of 34
units and a leg of 16 units. 342 = 1156 and 162 = 256.
1156 – 256 = 900 and 900 = 30 , so the height of the
isosceles triangle must be 30 units.
34 units
34 units
16 units and 16 units
www.imlem.org
Category 2
Geometry
Meet #3, January 2005
1. A certain polygon has twice as many diagonals as sides. How many sides are
there on this polygon?
Note: A diagonal in a polygon is any line segment that connects two vertices and
is not a side.
2. The figure below shows the design of a raft which floats in the middle of a
lake. The raft is made of a number of square sections that are linked together.
If the perimeter of the raft is 220 feet, how many square feet are in the area of the
raft?
3. In the figure below, triangles ABC, ACD, and ADE are right triangles, and
sides AB, BC, CD, and DE have the same measure. If the measure of side AB is 2
centimeters, how many centimeters are there in the measure of side AE?
E
D
C
Answers
1. _______________
2. _______________
3. _______________
B
www.Imlem.org
A
Solutions to Category 2
Geometry
Meet #3, January 2005
Average team got 18.63 points, or 1.55 questions correct
1. The table below shows the number of diagonals for
several polygons. The heptagon with 7 sides has 14
diagonals, which is twice the number of sides.
Answers
1. 7
Polygon
Triangle
Square
Pentagon
Hexagon
Heptagon
Octagon
2. 625
3. 4
Sides
3
4
5
6
7
8
Diagonals
0
2
5
9
14
20
2. There are 44 side lengths of the square sections in the
perimeter of the raft, so each square must have a side
length of 220 feet ÷ 44 = 5 feet. The area of each square
is thus 5 feet × 5 feet = 25 square feet.
The raft is made of 25 sections, so the area of the raft is
25 × 25 square feet = 625 square feet.
E
D
C
3. We will have to use the Pythagorean Theorem three
times to calculate the length of each hypotenuse. Let the
length of AC = x, the length of AD = y, and the length of
AE = z. Then we find x as follows:
x2 = 22 + 22  x2 = 8  x = 8  x = 2 2
We find y as follows:
y2 = 22 +
B
A
( 8)
2
 y 2 = 12  y = 12  y = 2 3
Finally, we find z as follows:
z 2 = 22 +
( 12 )
2
 z 2 = 16  z = 4 .
AE is 4 centimeters.
www.Imlem.org
So the length of