Sample: Do Not Reproduce GEO1 STUDENT PAGES. GEOMETRY AND MEASUREMENT Student Pages for Packet 1: Length and Area.

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1 Name Period Date GEOMETRY AND MEASUREMENT Student Pages for Packet 1: GEO1.1 Congruence Plot simple figures on coordinate graphs, and determine their lengths and areas. Make conjectures about perimeters of rectangles with fixed areas. Find lengths of horizontal and vertical line segments on a coordinate system. Identify congruent figures. Determine the effect of translations, reflections, and rotations on figures. GEO1.2 Polygons Derive formulas for the areas of parallelograms, triangles, and trapezoids. Find areas of irregular polygons. Use area and perimeter formulas to solve problems. GEO1 STUDENT PAGES GEO1.3 Vocabulary, Skill Builders, and Review GEO1 SP

2 area WORD BANK (GEO1) Word Definition Example or Picture congruent length perimeter polygon quadrilateral transformation vertex Learn more vocabulary in lesson 1 (SP5) and lesson 2 (SP12). GEO1 SP0

3 1.1 Congruence Ready (Summary) We will use coordinate graphs to plot simple figures, and we will determine their images under various rigid transformations. We will explore how the perimeters change for rectangles with fixed area. CONGRUENCE Go (Warmup) Set (Goals) Plot simple figures on coordinate graphs, and determine their lengths and areas. Make conjectures about perimeters of rectangles with fixed areas. Find lengths of horizontal and vertical line segments on a coordinate system. Identify congruent figures. Determine the effect of translations, reflections, and rotations on figures. Find the coordinates of points Q, R, S, and T. Then, find the perimeter and area of the rectangle. 1. Point Q: (, ) 5. Perimeter of rectangle QRST: 2. Point R: (, ) 3. Point S: (, ) 6. Area of rectangle QRST: 4. Point T: (, ) Q T y 5 R S 5 x GEO1 SP1

4 1.1 Congruence (-5,3) GRAPHING RECTANGLES Draw four different rectangles whose areas are 12 square units. A vertex for each rectangle is given. Different orientations of the same rectangle are permitted. Label the coordinates of each vertex. Find the length and width of each rectangle. Find the perimeter for each rectangle. (6,2) Length and width:, Area: Perimeter: Length and width:, Area: Perimeter: GEO1 SP2

5 1.1 Congruence (-6,0) GRAPHING RECTANGLES (continued) Draw four different rectangles whose areas are 12 square units. A vertex for each rectangle is given. Different orientations of the same rectangle are permitted. Label the coordinates of each vertex. Find the length and width of each rectangle. Find the perimeter for each rectangle. (-7,7) Length and width:, Area: Perimeter: Length and width:, Area: Perimeter: GEO1 SP3

6 1.1 Congruence FIXED AREA Refer to the rectangles you created on SP2 and SP3. 1. Were any of the rectangles you sketched congruent? How do you know? 2. Which of your rectangles has the greatest perimeter? What are its length and width? What is its perimeter? 3. Which of your rectangles has the least perimeter? What are its length and width? What is its perimeter? 4. Using a complete sentence, make a conjecture about the perimeter of rectangles with a fixed area. Consider the shape of the rectangles as the perimeters change. 5. Sketch and label a rectangle with a length of 24 units and an area of 12 square units in the space below. What are its length and width? What is its perimeter? Does your conjecture hold for this example? Explain. 6. What do you think will be the shape of a rectangle with a fixed area and the least perimeter? GEO1 SP4

7 1.1 Congruence RIGID TRANSFORMATIONS Explain each mathematical term in your own words: 1. congruent 2. translation 3. reflection 4. rotation 5. rigid transformation Determine whether each pair of figures is congruent. If congruent, state the rigid transformation(s) used to show that the figures are congruent Figures Congruent Figures? Rigid transformation(s) Used 11. GEO1 SP5

8 1.1 Congruence RIGID TRANSFORMATIONS (continued) A clothing company created the following diagram for a shirt. To show the other side of the shirt, the company will reflect the drawing across the y-axis. 12. Draw the reflection. 13. Label the coordinates of each vertex. 14. What do you notice about the coordinates that are reflected about the y-axis? Mathematicians often use superscripts to label related points. The symbol C ' (read C prime ) is used here to remind us that we are finding a point that is related to point C. Triangle CAT has coordinates C(-1,5), A(3, 5), and 15. Draw the triangle and label the coordinates of the vertices. 16. Then draw the translation of the triangle so that: C C' A A' T T ' and C ' = (-4,0) T(0,3). 17. Label the new triangle C' A' T '. y y x x GEO1 SP6

9 1.2 Polygons Ready (Summary) We will use logical reasoning and our knowledge about areas of rectangles to derive formulas for areas of parallelograms, triangles, and trapezoids. We will also find areas of complex figures. We will apply area and perimeter formulas to solve problems. Find the shaded area of each figure. 1 linear unit area = area = POLYGONS Go (Warmup) Set (Goals) Derive formulas for the areas of parallelograms, triangles, and trapezoids. Find areas of irregular polygons. Use area and perimeter formulas to solve problems. 1 square unit area = area = GEO1 SP7

10 1.2 Polygons AREAS OF PARALLELOGRAMS Directions: Use a ruler to draw the altitude (h) of each parallelogram. Rearrange the pieces to form a rectangle. Draw the rectangle next to the corresponding parallelogram. Find the area of each figure h 4. Write in words: The area of a parallelogram is equal to 5. Write in symbols: A = GEO1 SP8

11 1.2 Polygons AREAS OF TRIANGLES Directions: Duplicate each triangle. Rearrange the pieces to form a rectangle or a parallelogram. Draw the rectangle or parallelogram next to the corresponding triangle. Find the area of each triangle Write in words: The area of a triangle is equal to 5. Write in symbols: A = GEO1 SP9

12 1.2 Polygons FINDING AREAS OF POLYGONS For each problem: Identify the polygon. Write the corresponding area formula. Measure and label the relevant dimensions to the nearest centimeter. Substitute values into the formula. Solve for the area. Polygon #1: (Measure and label relevant dimensions in cm.) Area formula: Substitute and solve: Area of polygon: Polygon #2: (Measure and label relevant dimensions in cm.) Area formula: Substitute and solve: Area of polygon: GEO1 SP10

13 1.2 Polygons FINDING AREAS OF POLYGONS (continued) For each problem: Identify the polygon. Write the corresponding area formula. Measure and label the relevant dimensions to the nearest centimeter. Substitute values into the formula. Solve for the area. Polygon #3: (Measure and label relevant dimensions in cm.) Area formula: Substitute and solve: Area of polygon: Polygon #4: (Measure and label relevant dimensions in cm.) Area formula: Substitute and solve: Area of polygon: GEO1 SP11

14 1.2 Polygons Fill in the chart below and use for reference. SUMMARIZING FORMULAS Shape/Definition Sketch Perimeter Area 1. Rectangle a quadrilateral with four right angles 2. Square a rectangle with four congruent sides 3. Parallelogram a quadrilateral with parallel opposite sides 4. Rhombus a quadrilateral with four congruent sides 5. Triangle a three-sided polygon 6. Trapezoid a quadrilateral with a pair of parallel sides h b b h P = 2(b h) P = 2b 2h A = bh GEO1 SP12

15 1.2 Polygons USING FORMULAS 1 Draw a picture, write an appropriate formula, and substitute to solve each problem. 1. Find the height of a rectangle with an area of 20.4 cm 2 and a base of 3.4 cm. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question e. If this rectangular space were a tabletop, would it be big enough for a laptop computer? 2. Find the area of a parallelogram with a 1 base of 10 in. and a height of in. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question e. Do you think this parallelogram is as big as most kites you might see flying at the beach or in a park? GEO1 SP13

16 1.2 Polygons USING FORMULAS 1 (continued) Draw a picture, write an appropriate formula, and substitute to solve each problem. 3. Find the area of a triangle with a base of 235 cm and a height of 104 cm. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question e. Do you think this triangle would be big enough for the base of a cat s scratching pole? 4. Find the perimeter of a square tabletop whose area is 225 square inches. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question e. Do you think this tabletop would be enough space for a hamster cage? GEO1 SP14

17 1.2 Polygons USING FORMULAS 1 (continued) Draw a picture, write an appropriate formula, and substitute to solve each problem. 5. A parallelogram has adjacent sides that 3 1 measure 15 in and 5 in. Find it 8 4 perimeter. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question e. Do you think this parallelogram would be enough sitting room for a park bench? 6. Find the base of a triangular lawn with an area of 55 square meters and a height of 10 meters. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question e. Do you this space would be big enough for a child s swing? GEO1 SP15

18 1.2 Polygons AREAS OF TRAPEZOIDS Directions: Duplicate each trapezoid. Rearrange the pieces to form a rectangle or a parallelogram. Draw the rectangle or parallelogram next to the corresponding trapezoid. Find the area of each trapezoid Write in words: The area of a trapezoid is equal to 5. Write in symbols: A = GEO1 SP16

19 1.2 Polygons USING FORMULAS 2 Draw a picture, write an appropriate formula, and substitute to solve each problem. 1. The height of an isosceles trapezoid measures 2 inches and the bases measure 7 inches and 13 inches. Find its area. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question e. Do you think this trapezoid is a good size for a place mat on a table? 2. The area of an isosceles trapezoid is square meters, and the measures of its bases are 6.02 meters and 8.1 meters. Find the height of the trapezoid. a. Sketch the figure b. Write an appropriate formula c. Substitute and solve d. Answer the question e. Do you think this trapezoid could be a computer tabletop? GEO1 SP17

20 1.2 Polygons THE BALLROOM PROBLEM A rectangular ballroom has dimensions 30 feet by 50 feet. A stage is constructed inside the rectangular ballroom in the shape of an isosceles trapezoid with parallel edges that are 10 feet and 25 feet, and with a stage depth of 12 feet. 1. Draw and label a diagram that shows the ballroom with the stage. You may put the stage wherever you like. 2. Find the area of the ballroom. 3. Find the area of the stage. 4. Find the area of the remaining space in the ballroom after the stage is constructed. GEO1 SP18

21 1.3 Vocabulary, Skill Builders, and Review FOCUS ON VOCABULARY (GEO1) Unscramble the letters to reveal words from the word bank. Then, write their definitions or give examples. Example: eiepr darrod ordered pair (3,5) is an example of an ordered pair which labels the points on a coordinate plane with x-coordinate 3 and y-coordinate nhegtl 2. itemprree 3. eraa 4. evxrte 5. tuncnoerg 6. tqardraueiall 7. nlgypoo 8. onmtinaraoftrs dgiirtrs Word Bank polygon length perimeter quadrilateral rigid transformation area vertex congruent GEO1 SP19

22 1.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 1 Below is a chart that shows place value. Each place has a value ten times that of the place value to its right and one-tenth the value of the place to its left. ten thousands ten thousands thousands hundreds tens Round to the nearest hundredths Round to the nearest thousandths What does the 2 represent in ? 8. What does the 3 represent in ? Write the following numbers in words Compute ones tenths 13. hundredths 12.8 thousandths , ten-thousandths GEO1 SP20

23 1.3 Vocabulary, Skill Builders, and Review Compute. Simplify when possible. SKILL BUILDER GEO1 SP21

24 1.3 Vocabulary, Skill Builders, and Review SKILL BUILDER 3 Triangle MAN has coordinates M(0,-2), A(3,-2), N(0,3). 1. Draw the triangle and label the coordinates. 2. Draw the reflection of the triangle across the y-axis. 3. What do you notice about the coordinates that are reflected about the y-axis? 4. Sketch two different rectangles each with an area of 40 square units but with different perimeters. Label the dimensions of each rectangle and find the perimeter. Rectangle 1 Rectangle 2 Dimensions: Perimeter: Dimensions: Perimeter: y x GEO1 SP22

25 1.3 Vocabulary, Skill Builders, and Review TEST PREPARATION (GEO1) Show your work on a separate sheet of paper and choose the best answer. 1. Which of the following figures are NOT congruent? A. B. C. D. 2. Which rigid transformation is used to show congruence? A. translation B. reflection C. rotation D. dilation 3. A rectangular playground has a perimeter of 128 feet and a width of 30 feet. What is the area of the playground? A. 1,920 sq. ft B. 3,840 sq. ft C. 2,940 sq. ft D. 1,020 sq. ft 4. What is the area of a parallelogram with a height of 3.5 cm and a base of 8.01 cm? A cm 2 B cm 2 C cm 2 D cm 2 5. What is the formula for the area of a trapezoid? A. 1 a = ( b1 b2) h 2 B. 1 a = bh 2 2 C. a = bh D. a = r 6. The area of a trapezoid is 27 square meters, and the measures of its base are 5.5 meters and 3.5 meters. What is the height of the trapezoid? A. 3 meters B. 6 meters C. 10 meters D meters GEO1 SP23

26 1.3 Vocabulary, Skill Builders, and Review This page is left intentionally blank for notes. GEO1 SP24

27 1.3 Vocabulary, Skill Builders, and Review KNOWLEDGE CHECK (GEO1) Show your work on a separate sheet of paper and write your answers on this page. 1.1 Congruence Determine whether each pair of figures is congruent. If congruent, state the rigid transformation(s) used to show that the figures are congruent Polygons Figures Congruent Figures? Rigid transformation(s) Used Draw a picture, write an appropriate formula, and substitute to solve each problem. 3. Find the height of a rectangle whose base is 76 cm and area of 3,800 square centimeters. If this rectangular space were a dance floor, would it be big enough for 5 people to dance on? 4. Find the height of a trapezoid whose bases are 8 meters and 4.2 meters and whose area is square meters. GEO1 SP25

28 Home-School Connection (GEO1) Here are some questions to review with your young mathematician. 1. Are the figures congruent? If so, what rigid transformations can you use to show they are congruent? If not, why? 2. Find the area of a trapezoid whose bases are 6.1 in and 4.02 in and whose height is 3 in. 3. Sketch two different rectangles each with an area of 30 square units but with different perimeters. Label the dimensions of each rectangle and find the perimeter. Parent (or Guardian) signature Selected California Mathematics Content Standards AF Use variables in expressions describing geometric quantities (e.g. P = 2w 2l, A = 1 bh, C = d the 2 formulas for the perimeter of a rectangle, the area of a triangle, and the circumference of a circle, respectively). NS Convert fractions to decimals and percents and use these representations in estimations, computations, and applications. AF Use correct order of operations to evaluate algebraic expressions such as 3(2x 5) 2. MG Use formulas routinely for finding the perimeter and area of basic two-dimensional figures and the surface area and volume of basic three-dimensional figures, including rectangles, parallelograms, trapezoids, squares, triangles, circles, prisms, and cylinders. MG Estimate and compute the area of more complex or irregular two-and three-dimensional figures by breaking the figures down into more basic geometric objects. MG Understand and use coordinate graphs to plot simple figures, determine lengths and areas related to them, and determine their image under translations and reflections. MG Demonstrate an understanding of conditions that indicate two geometrical figures are congruent and what congruence means about the relationship between the sides and angles of two figures. FIRST PRINTING DO NOT DUPLICATE 2009 GEO1 SP26