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Calculate the Area of a Coordinate Plane Arrow Shape
Mathematics
Grade 9 (Junior High School)
Question Content
Mr. Burns is painting an arrow on the school parking lot. He draws the edges between the following points on the coordinate plane: (-2, 2), (5, 2), (5, 8), (12, 0), (5, -8), (5, -2), (-2, -2). What is the area of the arrow he is painting? Enter your answer in the box. ____ square units
Correct Answer
84
Detailed Solution Steps
1
Step 1: Plot the given points on a coordinate plane and divide the arrow shape into three simpler parts: a rectangle in the middle, and two congruent triangles on the top and bottom.
2
Step 2: Calculate the area of the middle rectangle. The rectangle spans from x=-2 to x=5 (width = 5 - (-2) = 7 units) and from y=-2 to y=2 (height = 2 - (-2) = 4 units). Area of rectangle = width × height = 7 × 4 = 28 square units.
3
Step 3: Calculate the area of the top triangle. The base of the triangle is the same as the width of the rectangle (7 units), and the height is from y=2 to y=8 (8 - 2 = 6 units). Area of a triangle = 1/2 × base × height = 1/2 × 7 × 6 = 21 square units.
4
Step 4: Calculate the area of the bottom triangle. It is congruent to the top triangle, so its area is also 21 square units.
5
Step 5: Calculate the area of the right triangle (the arrowhead). The base of this triangle spans from y=-8 to y=8 (8 - (-8) = 16 units), and the horizontal distance (height) is from x=5 to x=12 (12 - 5 = 7 units). Area = 1/2 × 16 × 7 = 56 square units.
6
Step 6: Sum all the areas: 28 + 21 + 21 + 56 = 84 square units.
Knowledge Points Involved
1
Coordinate Plane Plotting
This refers to marking ordered pairs (x, y) on a 2D grid, where x represents the horizontal axis and y represents the vertical axis. It is used to visualize shapes and calculate their dimensions for area/volume problems.
2
Area of a Rectangle
The formula for the area of a rectangle is Area = length × width. It applies to quadrilaterals with four right angles, used to calculate the space enclosed by the shape.
3
Area of a Triangle
The formula for the area of a triangle is Area = 1/2 × base × height, where height is the perpendicular distance from the base to the opposite vertex. It works for all triangle types and is often used to find areas of irregular shapes by decomposing them into triangles.
4
Decomposition of Irregular Shapes
This is a problem-solving method where complex, non-standard shapes are broken down into simpler, familiar shapes (like rectangles and triangles) whose area formulas are known. The total area is the sum of the areas of the individual parts.
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