GRADE 8 PERFORMANCE TASK: GEOMETRY

Standards

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

7. G.4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Understand and apply the Pythagorean Theorem.

8. G.3 Describe the effect of dilations, translations, rotations, and reflectionson two-dimensional figures using coordinates.

8. G.7. Apply the Pythagorean Theorem to determine unknown side lengthsin right triangles in real-world and mathematical problems in two andthree dimensions.

8. G.8 Apply the Pythagorean Theorem to find the distance between twopoints in a coordinate system.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

8. G.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Skills

  • Plot points on a coordinate plane
  • Substitute into the Pythagorean Theorem and solve the equation to find the missing side of a right triangle.
  • Answer word problems using the Pythagorean Theorem by drawing a diagram.
  • Apply formulae for perimeter, area and volume.

Fencing and Grassing Your Property

You want to put a fence around your large yard. There are two companies that you have found to do the work. They have each given you a quote for how much the work will cost. Of course, you want to find out which company will be the cheapest.

The boundary of your yard is determined by five trees. The lines connecting them form the edge of your property. Shown below are the descriptions for the positions of the trees relative to your house.

Tree / Position (relative to your house)
A / 100 ft east
B / 40 ft east, 80 ft south
C / 40 ft west, 120 ft south
D / 90 ft west, 60 ft north
E / 20ft east, 110ft north

PART 1: On graph paper, mark the position of each of the trees on your land. Let each block of the graph paper represent a 10-foot by 10-foot square. Using a straightedge, connect Tree A to Tree B, Tree B to Tree C, Tree C to Tree D, and so on.

PART 2: a) Use the Pythagorean Theorem to find the length of each side of your property. Round each answer to the nearest hundredth, if necessary.

b) Determine the perimeter of your property.

c) Company 1 says that they will complete the job for $12 per foot of fencing. Company 2 says that they will charge you $250 for the first 100 feet of fencing and $13 for each additional foot. Determine the cost of fencing for both companies.

d) Figure out which company will complete the job for the least amount of money

PART 3: a) A circular swimming pool with a radius of 20 ft is positioned with a center at 40 ft west and 30 ft north. Mark the position of the pool on your graph. The depth of the swimming pool is 5ft. Calculate the volume of water needed to fill the pool.

b) A barn is being built with corners at (30 ft north, 30 ft east); (70 ft east, 30 ft north); (30 ft east, 20 ft south); (70 ft east, 20 ft south). Mark this barn on your graph.

c) The rest of the property is to be grassed at a cost of 20 cents per square foot. You have given the contract for the grassing to the company who was cheapest in Step 5. They will give you a discount of 8% on the cost of grassing. Calculate the cost of the grassing.

Name: ______Date: ______

Part 1: Graph of Your Property Lines

Plot the points that represent the trees that mark the edges of your property. Use the locations given on the previous page. To make things easier, use the origin (0, 0) as the position of your house. Remember that each grid represents 10 feet. Finally, connect the points using a straightedge.

Name: ______Date: ______

Part 2: a) & b) Break your property into smaller parts and use the Pythagorean Theorem to find the length of each side. Remember that each grid line on your graph represents 10 feet. Also remember to round to the nearest hundredth if necessary. Show your work below.

Work for the distance from Tree A to Tree B
Distance: ______ft / Work for the distance from Tree B to Tree C
Distance: ______ft
Work for the distance from Tree C to Tree D
Distance: ______ft / Work for the distance from Tree D to Tree E
Distance: ______ft
Work for the distance from Tree E to Tree A
Distance: ______ft / Work for the Perimeter of Your Property
Perimeter: ______ft

c)Company 1 charges $12 per foot of fence. Find the cost if Company 1 completes the job. Show work!

Cost: $ ______

Company 2 charges $250 for the first 100 feet of fence and $13 for each additional foot. Find the cost if Company 2 completes the job. Show work!

Cost: $ ______

d) Both companies must charge a 6% sales tax. Determine which company will be the cheapest. Then, find the cost including tax. Round to the nearest cent. Show work!

Company ______

Cost: $ ______

Part 3a) Calculate the area and volume of the swimming pool (to 1 decimal place). Don’t forget to add the correct units.

Area: ______

Volume: ______

c)Calculate the area of the grassed area by first calculating the area of the entire property and then subtracting the area of the pool and the barn. Show your working.

Calculate the cost of the grassing at $2 per square foot:

Cost: $______

Calculate the discount given:

Discount: $______

Calculate the final cost of the grassing:

Cost: $______

Part 4: If your property was dilated by a factor of ½, what effect will this have on the perimeter? How could you prove your answer is correct? Use the Graph below to help.

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