Algebra Area of Parallelograms

Transcription

1 Lesson 10.1 Reteach Algebra Area of Parallelograms The formula for the area of a parallelogram is the product of the base and height. The formula for the area of a square is the square of one of its sides. height 5 h side 5 s base 5 b A 5 bh A 5 s Find the area. yd.7 m 5 1 yd Step 1 Identify the figure. The figure is a parallelogram, so use the formula A 5 bh. Step Substitute 5 1_ for b and for h. A 5 5 1_ 3 Step 3 Multiply. A 5 5 1_ _ So, the area of the parallelogram is 11 yd. Find the area. 1. Figure: Formula: A 5 13 m A m mi 3 yd 1 mi yd mi yd 10-5 Reteach

2 Lesson 10.1 Enrich All Aboard Parallelogram and Square Express Find the areas of the train cars below. Write the shape of the car and the formula you used to find the area. (Do not include the wheels as part of the area.) Car D Car C Car B 0.6 ft Car A 61 ft 73 yd 13 1 yd 36. ft 3.3 ft 0.6 ft 38 ft Train Car A Train Car B Figure: Formula for the Area: Area: Train Car C Figure: Formula for the Area: Area: Train Car D Figure: Formula for the Area: Area: Figure: Formula for the Area: Area: 1. Stretch Your Thinking Add another car to the train. Draw and label the car. Then find the area.. Stretch Your Thinking If the unshaded triangles in Train Car C are removed, the figure is a square. What is the area of the shaded section of Train Car C? 10-6 Enrich

3 Lesson 10. Reteach Explore Area of Triangles You can use grid paper to find a relationship between the areas of triangles and rectangles. Step 1 on grid paper, draw a rectangle with a base of 8 units and a height of 6 units. Find and record the area of the rectangle. A 5 8 square units Step Cut out the rectangle. Step 3 Draw a diagonal from the bottom left corner up to the top right corner. Step Cut the rectangle along the diagonal. You have made triangles. Are the triangles congruent? yes How does the area of one triangle compare to the area of the rectangle? The area of the triangle is half the area of the rectangle. If l is the length and w is the width, you can use a rectangle to find the area of a triangle. Find the area of the triangle. Area of rectangle: A 5 lw m Area of triangle: A 5 1_ 3 area of rectangle 5 1_ m So, the area is 1 square meters. 7 m m Find the area of the triangle in. ft 6 m 10 in. 10 ft 9 m 10-7 Reteach

4 Lesson 10. Enrich Tricky Triangles Find the unknown base or height of the first triangle in each problem. Draw the triangle on the grid to help you visualize your answer. Then, on the other grid, draw another triangle with the same area but a different base and height. 1. b 5 10 cm h 5 A 5 15 sq cm b 5 h 5 A 5 15 sq cm. b 5 h 5 cm A 5 6 sq cm b 5 h 5 A 5 6 sq cm 3. b 5 h 5 cm A 5 8 sq cm b 5 h 5 A 5 8 sq cm. b 5 8 cm h 5 A 5 1 sq cm b 5 h 5 A 5 1 sq cm 10-8 Enrich

5 Lesson 10.3 Reteach Algebra Area of Triangles To find the area of a triangle, use the formula A 5 1_ 3 base 3 height. height h base b Find the area of the triangle. 3 cm 7 cm Step 1 Write the formula. A 5 1_ bh Step Rewrite the formula. Substitute the base and height measurements for b and h. A 5 1_ Step 3 Simplify by multiplying. A 5 1_ 3 1 A Step Use the appropriate units. A cm Find the area of the triangle. 1. Write the formula. A 5 1_ 3 Substitute for b and h. A 5 1_ ft 6 ft Simplify. A = 1_ 3 A = ft in. 6.5 m m 11 in. A = A = 10-9 Reteach

6 Lesson 10.3 Enrich Wordy Triangles Make a drawing to show the situation. Label the dimensions. Then use the drawing to find the area. 1. A model ship has a mast that is 7 inches tall. A right triangular sail goes from the top of the mast to 1 inch from the bottom of the mast. The length of the base of the sail is inches. The height of the sail is along the mast.. Two ropes extend from the top of a flagpole to the ground 18 feet away from the flagpole s base on either side. The flagpole is 30 feet high. The ropes each create a right triangle with the flagpole, and each right triangle is half of a larger triangle with the flagpole at its center. Area of sail: 3. Andrew is helping the first-grade class make paper trees. For each student, he folds a rectangular piece of construction paper in half lengthwise. Then, he cuts the folded paper into a triangular shape with the same base and height as the rectangle. Then he unfolds the paper and throws away the scraps. The original piece of paper is 11 inches by 8.5 inches. Area of larger triangle:. Lucy s rectangular garden is 15 feet long and 1 feet wide. She adds a right triangle to her garden plan, so that the triangle s height extends out 7 feet in line with the shorter side of the rectangle, and the base of the triangle runs from the corner of the rectangle to the center of the longer side of the rectangle. Area of paper thrown away per student: Total area of Lucy s garden: Enrich

7 Lesson 10. Reteach Explore Area of Trapezoids Show the relationship between the areas of trapezoids and parallelograms. Step 1 On grid paper, draw two copies of the trapezoid. Count the grid squares to make your trapezoid match this one. Step Cut out the trapezoids. 7 3 Step 3 Turn one trapezoid until the two trapezoids form a parallelogram. Step Find the length of the base of the parallelogram. Add the lengths of one shorter trapezoid base and one longer trapezoid base. Step 5 Find the area of the parallelogram. Use the formula A 5 bh. Step 6 The parallelogram is made of two congruent trapezoids. So, divide by to find the area of one trapezoid units A square units square units Find the area of the trapezoid. 1. Trace and cut out two copies of the trapezoid. Arrange them to form a parallelogram. a. Find the base of the parallelogram b. Find the area of the parallelogram, using A 5 bh. A square units c. Find the area of the trapezoid. 5 square units in. 3. ft. 9 mm 8 in. 5 ft 7 mm 6 in. 10 ft 3 mm in. ft mm Reteach

8 Lesson 10. Enrich Trapezoid Patterns Use the grid to draw each polygon. Label the measurements and give the areas. 1. Hexagon made from two congruent trapezoids with bases and 6 and height. Rectangle made from two congruent trapezoids with bases 5 and 7 and height 3 Area of one trapezoid: Area of hexagon: 3. Parallelogram made from two congruent trapezoids with bases 3 and 8 and height 5 Area of one trapezoid: Area of rectangle:. Square made from four congruent trapezoids with bases 5 and 3 and height Area of one trapezoid: Area of parallelogram: 5. Stretch Your Thinking Daniel made a rectangle from two congruent trapezoids with bases 11 and 8 and a height of 6. Give the length, width, and area of the rectangle. Area of one trapezoid: Area of square: 6. What is special about the trapezoids in Problem? Why did you draw them that way? 10-1 Enrich

9 Algebra Area of Trapezoids Lesson 10.5 Reteach base 1 b 1 To find the area of a trapezoid, use the formula Area ( base 1 1 base ) 3 height. height h base b Find the area of the trapezoid. b 1 5 mm height 5 5 mm b 5 1 mm Step 1 Write the formula to find the area. A 5 1_ ( b 1 1 b )h Step Replace the variable b 1 with, b with 1, and h with 5. Step 3 Use the order of operations to simplify. A ( 1 1) 3 5 A A A 5 50 Step Use the appropriate units. A 5 50 mm Find the area cm Write the formula. A 5 8. cm 1 cm Replace the variables. Simplify.. 5 in. 3. A ( 1 ) 3 A A 5 7 ft 17 in. 8 ft 3 in. 1 ft Reteach

10 Lesson 10.5 Enrich Split Into Two A trapezoid can be divided into two triangles. Label each triangle with its height and base. Then find the area of the triangles and the area of the trapezoid ft Areas 10 ft 18 ft A B Triangle A: Triangle B: Trapezoid:. Areas 10 cm 8 cm 0 cm A B Triangle A: Triangle B: Trapezoid: 3. Areas 5 m Triangle A: 1 m 15 m A B Triangle B: Trapezoid:. The formula for the area of a trapezoid is A 5 1_ (b 1 b )h. Rewrite the formula as the sum of the areas 1 of two triangles. Which properties did you use to rewrite the formula? 10-1 Enrich

11 Lesson 10.6 Reteach Area of Regular Polygons In a regular polygon, all sides have the same length and all angles have the same measure. To find the area of a regular polygon, divide it into triangles. Step 1 Draw line segments from each vertex to the center of the regular polygon. Step Examine the figure. The line segments divide the polygon into congruent triangles. This polygon is a hexagon. A hexagon has 6 sides, so there are 6 triangles. 1 in. Step 3 Find the area of one triangle. Use the formula A 5 1_ bh. The base of the triangle (or one side of the hexagon) is 1 in. The height of the triangle is 1.1 in. 1.1 in. A 5 1_ _ in. 1 in. Step Multiply by 6, because there are 6 triangles So, the area of the regular hexagon is 508. square inches. Find the area of the regular polygon. 1. Number of congruent triangles inside the pentagon: Area of each triangle: 5.5 mm A mm Area of the pentagon: 3 5 mm 8 mm m 6. ft 16.5 cm 8.3 m ft 19 cm m ft cm Reteach

12 Lesson 10.6 Enrich Off on a Tangent You can find the approximate height of a triangle inside a regular polygon by using the tangent of the bisected triangle, which is a right triangle. The tangent can be expressed as the ratio of the triangle s height, h, to half the polygon s side length, s: h s. Find the approximate height and area for the triangle, as well as the area of the polygon. Use the table of tangents and the formula h 5 tan 3 (s ). Round each value to the nearest hundredth. Polygon Tangent (tan) pentagon 1.38 hexagon 1.73 octagon.1 decagon Triangle height:. Triangle height: h h Triangle area: 6 Triangle area: 10 Name of polygon: Polygon area: Name of polygon: Polygon area: 3. Triangle height:. Triangle height: h Triangle area: h Triangle area: 1 Name of polygon: Polygon area: Name of polygon: Polygon area: 5. Stretch Your Thinking Bethany says she can draw a regular hexagon with side length 16 and triangle height 11. Is this possible? How can you tell? Enrich

13 Lesson 10.7 Reteach Composite Figures A composite figure is made up of two or more simpler figures, such as triangles and quadrilaterals. The composite figure shows the front view of a bird house. Complete Steps 1 to find the area of the shaded region. Step 1 Find the area of the rectangle. 8 cm A 5 lw cm Step Find the area of the triangle. A 5 1_ _ bh 5 1_ 3 3 Step 3 Find the area of the square. 3 5 cm 0 cm 10 cm 10 cm A 5 s 5 ( ) 16 cm 5 cm Step Add the areas of the rectangle and triangle. Then subtract the area of the square. Shaded area cm So, the area of the shaded region is cm. Find the area of the shaded region cm in. 16 in. 0 in. 8 in. 6 cm 8 cm 1 cm 1 cm Reteach

14 Lesson 10.7 Enrich Find Areas of Composite Figures Solve. 1. A skateboard ramp is made of the figures as shown. What is the area of the front of the ramp? 9 ft 9 ft ft ft 9 ft 9 ft 7 ft 7 ft. A bean bag game wall is shown. Each hole in the wall is in the shape of a square with side length 3 inches. What is the area of the wall, without counting the holes? in. 63 in. 3. Stretch Your Thinking Mr. Siers is building an entrance to the front of his ranch, as shown. What is the area of the front of the entrance? 0.5 yd 0.5 yd 3 yd. yd. How is finding the area of a parallelogram like finding the area of two congruent triangles? Explain your answer. 0.5 yd 0.5 yd Enrich

15 Lesson 10.8 Reteach Problem Solving Changing Dimensions Amy is sewing a quilt out of fabric pieces shaped like parallelograms. The smallest of the parallelograms is shown at the right. The dimensions of another parallelogram she is using can be found by multiplying the dimensions of the smallest parallelogram by 3. How do the areas of the parallelograms compare? cm 6 cm 5 cm Read the Problem What do I need to find? I need to find compare. 1 cm how the areas of the different parallelograms Sketch Multiplier Area 18 cm 15 cm What information do I need to use? the dimensions of the smallest parallelogram and the number by which the dimensions are multiplied I need to use. Solve the Problem none 3 How will I use the information? I can draw a sketch of parallelogram each and calculate the areas. Then I can look for patterns in my results A _ 5 _ cm A 5 _ 3 _ 5 cm When the dimensions are multiplied by 3, the area is multiplied by _. 1. Sunni drew a parallelogram with area 0 in.. If she doubles the dimensions, what is the area of the new parallelogram?. Abe drew a square with side length 0 mm. If he draws a new square with dimensions that are half that of the previous square, what is the area of the new square? Reteach

16 Lesson 10.8 Enrich Fixing Perimeter and Area You have explored how changing the dimensions of a polygon affects the area. Now you will explore what happens when you fix the perimeter or area and allow the other measurements to change. 1. Complete the table to find the dimensions of the rectangle with the greatest area whose perimeter is 0 cm. area: dimensions: Length (cm) Width (cm) Perimeter (cm) Area ( cm ) Complete the table to find the dimensions of the rectangle with the least perimeter whose area is 36 cm. perimeter: dimensions: Length (cm) Width (cm) Perimeter (cm) Area ( cm ) Stretch Your Thinking What can you conclude about the dimensions of a rectangle with a fixed perimeter?. Stretch Your Thinking What can you conclude about the dimensions of a rectangle with a fixed area? 10-0 Enrich

17 Lesson 10.9 Reteach Figures on the Coordinate Plane The vertices of a parallelogram are A(, ), B( 3, 5), C(, 5), and D(5, ). Graph the parallelogram and find the length of side AD. Step 1 Draw the parallelogram on the coordinate plane. Plot the points and then connect the points with straight lines. Step Find the length of side AD. Horizontal distance of A from 0: 5 Horizontal distance of D from 0: Points A and D are in different quadrants, so add to find the distance from A to D units y B 6 5 C 3 A D units 5 units x So, the length of side AD is 7 units. Graph the figure and find the length of the given side. 1. Triangle JKL. Trapezoid WXYZ J( 3, 3), K( 3, 5), L(5, ) W(, 3), X(, 3), Y(3, 5), Z(3, 3) y x y x length of JK 5 length of WZ Reteach

18 Lesson 10.9 Enrich Hidden Picture Use the clues to find the unknown vertices. Plot the unknown vertices on the coordinate grid and draw each quadrilateral to complete the picture below. 0 y 18 A E F D C J K G 10 O M S 8 6 N Q x CLUES: 1. ABCD is a rectangle. Point B :. EFGH is a square. Point H: 3. JKLM is a trapezoid, and LM is 6 units long Point L:. NOPQ and PQRS are parallelograms. Point P : Point R: 10- Enrich