Meet #2. Park Forest Math Team. Self-study Packet

Transcription

1 Park Forest Math Team Meet #2 Self-study Packet Problem Categories for this Meet (in addition to topics of earlier meets): 1. Mystery: Problem solving 2. : rea and perimeter of polygons 3. Number Theory: Divisibility GCF, LCM, prime factorization 4. rithmetic: Fractions, terminating and repeating decimals, percents 5. lgebra: Word problems with 1 unknown; working with formulas; reasoning in number sentences

2 Important Information you need to know for Meet 2, Category 2 GEOMETRY: rea and Perimeter of Polygons Shape Perimeter rea Rectangle 2L + 2W LW Square 4s s 2 Triangle + B + C! Bh Parallelogram 2 + 2B Bh Trapezoid + C + B + b!h(b + b) Rectangle Square L W Triangle s Parallelogram H h C h B B Trapezoid b h B C To find the area of a more complex polygon, break the area into smaller parts and find the area of each part. Then add the areas together. If you memorize the formula for area of a rectangle and a triangle, you can find the area of virtually any polygon!

3 Category 2 Meet #2, November, If the trapezoid shown here has an area of square centimeters, how many centimeters are in the length of base one? base one height = 17 cm base two = 14 cm 2. rnold bought a rectangular sheet of plywood measuring 4 feet by 8 feet. From each of the four corners of the sheet of plywood, he cut rectangles for some shelves. Two of the rectangles measured 12 inches by 36 inches and the other two measured 10 inches by 30 inches. If we disregard any waste from the thickness of the cuts, how many feet are in the perimeter of the remaining piece of plywood? 3. If the side length of each square in the grid below measures 1 centimeter, how many square centimeters are there in the area of polygon BCDEFGHI? B D C I H G F E

4 Solutions to Category 2 Meet #2, November, The formula for the area of a trapezoid is ( ), where h is the height, b 1 is base one, = 1 2 h b 1 + b 2 and b 2 is base two. In our case, we know the area and need to find the value of b 1. Substituting the known ( ). values into the equation, we have 195.5= b If we had two such trapezoids, then we would have: = ( b ) 391= 17 b or ( ). Dividing both sides of the equation by 17, we get 23= ( b ), which means the sum of the bases is 23 and base one must be 9 cm. 2. Cutting rectangles from the corners of the sheet of plywood decreases the area but has no affect on the perimeter. The perimeter remains the same as the original 4 ft by 8 ft sheet, which is = 24 feet. 3. The interior of the polygon BCDEFGHI can be subdivided into triangles and rectangles in number of ways. One way is shown below. n alternative solution is to subtract the area of the regions that are outside the polygon from the total of 8 12 = 96 squares in the entire grid. This way is shown B as well. D Interior sum: = 70 square centimeters. Exterior sum: = 26 and = 70 square centimeters. I H C 3 G F E

5 Category 2 Meet #2, November How many different rectangles have a perimeter of 48 units, if both the length and the width of the rectangle are positive whole numbers? Note: rectangle with length a and width b is considered the same as a rectangle with length b and width a. 2. n irregular octagon is made on a grid as shown at right. Notice that some sides are longer than others. If the octagon has an area of 252 square inches, how many inches are in the length of a short side? 3. Three right triangles and two squares were cut out of a rectangle with length 20 centimeters and width 12 centimeters as shown below. The dimensions of the triangles are given in the picture and the squares are each 3.5 cm by 3.5 cm. How many square centimeters are in the area of the shaded region? 8 cm 3 cm cm 3 cm 7 cm 5 cm

6 Solutions to Category 2 verage team got points, or 1.1 questions correct Meet #2, November Since the perimeter of a rectangle is twice the sum of the length and the width, we will consider all whole number lengths and widths with a sum of 24 units. The 12 possible dimensions of the rectangle are: 1 23, 2 22, 3 21, 4 20, 5 19, 6 18, 7 17, 8 16, 9 15, 10 14, 11 13, and There are 5 full squares and 4 half squares in the shaded area, for a total of 7 squares the size of the grid. Since the area of the shaded region is 252 square inches, each square in the grid must be = 36 square inches. This means each grid square must be 6 inches on a side (6 6 = 36), and there are also 6 inches in the short side of the octagon. 3. Before the triangles and squares were cut out of the rectangle, it had an area of 12 cm 20 cm = 240 square centimeters. The areas of the triangles are = 15 cm 2, = 12 cm 2, and = 17.5 cm 2. The area of the two squares is = 24.5 cm 2. Subtracting these areas from the original 240 square centimeters, we get = 171 square centimeters.

7 Category 2 Meet #2, December The square at the far left in the figure below is a unit square. s you move to the right, the side length of each square increases by 1 unit. How many units are in the perimeter of the entire figure? 2. In the figure below, sides TR and P are parallel, but sides TP and R are not parallel. The length of side TR is 11 units and the length of side P is 17 units. If the area of TRP is 84 square units, how many units apart are the two parallel lines? T R P 3. If the area of the inner-most square EFGH is 9 square units, how many units are in the perimeter of the outer-most square BCD? B E H F G D C

8 Solutions to Category 2 Meet #2, December The distance across the bottom of the figure is = 28 units. The sum of the horizontal distances across the tops of the squares is also 28 units. The height of the square at the far right is 7 units. There are also seven 1 unit lengths on the left of each square. The total perimeter of the figure is thus = 70 units. 2. Quadrilateral TRP is a trapezoid. The formula for the area of a trapezoid is = 1 2 h ( b 1 + b 2 ), where h is the height and b 1 and b 2 are the two parallel bases. We know the area and the lengths of the two bases. Solving for h in the area formula as shown below, we find that the two parallel lines are 6 units apart. Students with correct answer in a cluster of 6 schools: 1. 28/ / /36! (Many got 144 for #3, which is the RE of the outer square, not the perimeter.) 84 = 1 h ( ) 84 = 1 2 h( 28) 84 = 14h h = = 6 3. Each larger square in the figure is twice the area of the one inside it. Since the area of square EFGH is 9 square units, the area of square BCD must be = 144 square units. The side length of square BCD must be 12 units, since = 144. The perimeter of square BCD is thus 4 12 = 48 units.

9 Category 2 Meet #2, December The four squares below have areas of 16, 9, 4, and 1 respectively. The squares are lined up one next to the other as shown below. What is the perimeter of the overall shape below? 2. rectangle with a perimeter of 60 inches is cut into 4 smaller congruent rectangles by cutting the rectangle in half both horizontally and vertically as shown to the right. How many inches are in the sum of the perimeters of the four new rectangles? 3. The trapezoid on the right has an area that is 16 more than the area of the trapezoid on the left. What is the value of h (the height) in the trapezoid on the right? h

10 Solutions to Category 2 Meet #2, December The total perimeter of the figure is: = Labeling the rectangles length and width with L and W we can call the perimeter of the original rectangle 2W + 2L. Each of the smaller rectangles would have a length of! and a width of # giving them each a perimeter of W + L and the " " four of them combined would be 4W + 4L. That s exactly twice the perimeter of the original, so the combined perimeter of the four rectangles must be 2(60) = 120 L L 2 W W W 2 W 2 3. The area of the trapezoid on the left is $ %&'()*+,-. / :: " L ". /782"9457: ". ;<<= So the area of the trapezoid on the right is = 216. Using the formula for area of a trapezoid we can find the missing height given the area. ;>?. /7@2"@456 " ;>?. 96 " ;>?. ;BC DDDDDE. DC L 2.

11 Category 2 Meet #2 December 2010 B 1. The perimeter of rectangle measures cm. How many centimeters in the measure of? E D C 2. In the drawing below, the area of trapezoid is four times the area of rectangle. [The drawing is not to scale]. B If measures inches, then how many inches are there in the measure of? D E C 3. The rectangle below is divided into congruent (identical) rectangles. The total area of all is square inches. How many inches in the perimeter of each one?

12 Meet #2 December 2010 Solutions to Category 2 Geometery 1. For the perimeter to measure cm, must measure cm. The area of triangle then is and this must equal the area of triangle which can be expressed as. When we plug in the known values we get. [Using similar triangles, you can also observe that ]. 2. If we call the trapezoid s height, then we know that the rectangle s area is, and the trapezoid s area is larger by the triangle s area. Knowing that, and naming, we can write: [Expressing the fact that the triangle s area is 3 times the rectangle s, and cancelling out ]. Solving, we get 3. Let s call a rectangle s width, and its height. Each reactangle s area is. In the drawing we see 4 rectangles in the top row and 3 in the bottom row, so we can conclude that or. When we subtitute this in the first equation we get or. So we get, and the perimeter is: [nother way to solve is to notice that the whole area is. Combined with this leads to ].

13 Category 2 Meet #2, November/December Find the number of inches in the perimeter of the figure at right. ll angles are right angles and all lengths are in inches. 2. Five squares of gold all have the same thickness, but they have edge lengths of 1 cm, 5 cm, 7 cm, 7 cm, and 11 cm. If the gold is melted down and recast with the same thickness as before into five identical squares, how many centimeters are there in the edge length of each square? 3. In trapezoid BCD, side D, which is 15 cm, is parallel to side BC, which is 21 cm. The area of trapezoid BCD is 234 square centimeters. If point E is on side D, how many square centimeters are there in the area of triangle BEC? Express your answer to the nearest tenth of a square centimeter. 1. inches 2. cm 3. sq. cm

14 Solutions to Category 2 Meet #2, November/December The total vertical rise on the right side of the figure must be equal to the 9 inches we see on the left side of the figure, so a b = 9. Similarly, the total of the horizontal lengths would equal the 12 inches we see on the bottom, but there is an extra = 6 inches because the figure turns into itself for 3 inches and then must come back 3 inches. The total perimeter is thus = 48 inches inches 2. 7 cm sq. cm 2. The total surface area of the 5 squares is square cm. If the gold is to be recast in 5 equal squares of the same thickness as before, they must each have a surface area of = 49 square cm. The side length of each square would be 49 = 7 cm. Notice that we squared the side lengths, then averaged these squared numbers, and finally took the square root of this average. This square root of the mean of the squares is called the root mean square or the quadratic mean. 3. Using the area formula for a trapezoid 1 2 h b 1 b 2, we substitute in the known values and solve for the unknown height h as follows: h h h 13. Triangle BEC has the same heigth, so it s 1 area is = square centimeters.