Park Forest Math Team. Meet #3. Self-study Packet

Transcription

1 Park Forest Math Team Meet #3 Self-study Packet Problem Categories for this Meet (in addition to topics of earlier meets): 1. Mystery: Problem solving 2. : Properties of Polygons, Pythagorean Theorem 3. Number Theory: Bases, Scientific Notation 4. Arithmetic: Integral Powers (positive, negative, and zero), roots up to the sixth 5. Algebra: Absolute Value, Inequalities in one variable including interpreting line graphs

2 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 a 2 + b 2 = c 2 a and b are the legs of the triangle and c is the hypotenuse (the side opposite the right angle) c a 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! b

3 Category 2 Meet #3 - January, 2015 Figures are not necessarily drawn to scale. 1) Angle ADC is a right angle. AB = 4 cm and BC = 9 cm. DB is perpendicular to AC. How many cm long is DB? D C A B 2) DEHJ is a square with an area of 64 square meters. Diagonal HF = 17 meters. How many square meters are in rectangle DFGJ? D E F J H G 3) Polygon PENTAGONAL is a pentagram (star) consisting of a regular pentagon with five isosceles triangles attached at its five edges. How many degrees are in one of the exterior A N angles (for example, angle PEN)? 1) E T G 2) P L N O 3) A

4 Solutions to Category 2 Meet #3 - January, ) A student who knows the Pythagorean Theorem should also know that, at its foundation, is the notion of similar triangles. In this diagram are three similar 1) 6 triangles. Using triangles DAB and DBC, we can say that ratios of corresponding sides are proportional: 2) 184. So, and cross products are 3) 108 equal, so and DB = 6. A few students may recognize this diagram as representing this theorem: "The altitude to the hypotenuse of a right triangle is the geometric mean (or mean proportional) to the two segments of the hypotenuse into which it is divided." 2) One side of square DEHJ is 8 meters because its area is 64 square meters. For one of the right triangles of rectangle EFGH, using the Pythagorean Theorem,. So,, and, so, and EF = 15. So, rectangle DFGJ now measures 8 by (15 + 8), or 8 by 23, so its area is (8)(23), or 184 square meters. 3) Each interior angle of the regular pentagon measures (3)(180)/5, or 108 degrees. Any one of the exterior angles of the pentagram is vertical to one of these interior 108 degree angles and, therefore, is equal to 108 degrees.

5 Category 2 Meet #3, January Mia drew regular hexagons on each side of a pentagon. If she draws all the diagonals in all six shapes, how many diagonals will she have to draw? Note: A diagonal is a line segment that connects two vertices that are not already connected by a side. 2. Right triangle ABC below has legs of length 7 units and 11 units. How many square units are there in the area of square ACDE which is constructed on the hypotenuse of this triangle? 3. A small rectangular box has sides of lengths 6 cm, 6 cm, and 7 cm. How many centimeters are there in the space diagonal of the box? Note: A space diagonal is a line that connect two opposite vertices of the box and goes through the interior space of the box. 1. diagonals 2. sq. units 3. cm

6 Solutions to Category 2 Meet #3, January Three diagonals can be drawn from each of the six vertices on a hexagon, but this would count each diagonal at both ends. So there are = 9 diagonals in each hexagon. Similarly, there are = 5 diagonals in the pentagon. Mia will have to draw = 50 diagonals diagonals sq. units cm 2. According to the Pythagorean theorem, the sum of the areas of the squares on the legs of a right triangle is equal to the area of the square on the hypotenuse. If we were to construct a square on leg AB, it would have an area of = 121 square units. A square on leg BC would have an area of 7 7 = 49 square units. Their sum is = 170 square units and this is the area of square ACDE. 3. We can calculate the length of the space diagonal of the box by using the Pythagorean theorem twice. First we can find the length of the diagonal of the bottom face, which is = 72 = 6 2 cm. Then we find the length of the space diagonal using this length and the height of the box as follows: ( ) 2 = = 121 = 11 cm. Alternatively, we can use a 3- dimensional version of the Pythagorean theorem as follows: = = 121 = 11 cm.

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9 Category 2 Meet #3, January 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? B C A D

10 Solutions to Category 2 Meet #3, January 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)'$&2)"".. 3. First draw in segment 56. Since 56 is the hypotenuse of triangle 576, we can use the Pythagorean Theorem to find the length of drawn in segment )8. 9"". )*29"'")"8$. However BD is also the hypotenuse of triangle :56 and therefore 56. )". 9:6. )"9:6. )"8$. So :6. )";2 and :6< ) <"=. 1 B A 11? C 7 D

11 Category 2 Meet #3, January 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? units 34 units 32 units

12 Solutions to Category 2 Meet #3, January 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 = 36 degrees. 2. Using m = 7 and n = 4, we find that a = = = 33, b = = 56, and c = = = 65. The perimeter of a right triangle with sides of 33, 56, and 65 units is = 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 = 1156 and 16 2 = = 900 and 900 = 30, so the height of the isosceles triangle must be 30 units. 34 units 34 units 16 units and 16 units

13 Category 2 Meet #3, January 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 B A

14 Solutions to Category 2 Average team got points, or 1.55 questions correct Meet #3, January 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. Polygon Sides Diagonals Triangle 3 0 Square 4 2 Pentagon 5 5 Hexagon 6 9 Heptagon 7 14 Octagon 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 square feet = 625 square feet. C D E 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: x 2 = x 2 = 8 x = 8 x = 2 2 We find y as follows: B A y 2 = ( ) 2 y 2 = 12 y = 12 y = 2 3 Finally, we find z as follows: z 2 = ( ) 2 z 2 = 16 z = 4. So the length of AE is 4 centimeters.