Question

mr. barth 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, 6), (12, 0), (5, -6), (5, 2), (-2, 2). what is the area of the arrow he is painting? enter your answer in the box. square units

Explanation:

Step1: Analyze the arrow's shape (assuming it's a composite figure, maybe rectangle and triangle or trapezoid, but let's check coordinates. Wait, the points are (-2,2), (5,2), (5,6), (12,0), (5,-6), (5,2), (-2,2)? Wait, maybe a typo, but let's assume the arrow is made of a rectangle and a triangle or two trapezoids. Wait, maybe the base rectangle from (-2,2) to (5,2) to (5,-6) to (-2,-6) and a triangle on top? Wait, no, the point (12,0) is there. Wait, maybe the arrow has a rectangle part and a triangular part. Wait, let's list the coordinates properly: Let's correct the points (maybe a typo, let's assume the correct points for an arrow: maybe the shaft is a rectangle, and the head is a triangle. Alternatively, use the shoelace formula.

Wait, the given points (probably a typo, let's assume the correct points are (-2,2), (5,2), (5,6), (12,0), (5,-6), (5,-2), (-2,-2)? No, original points: (-2,2), (5,2), (5,6), (12,0), (5,-6), (5,2), (-2,2)? Wait, maybe the arrow is symmetric. Let's use the shoelace formula. Let's list the coordinates in order:

Let's list the vertices in order: A(-2,2), B(5,2), C(5,6), D(12,0), E(5,-6), F(5,-2), A(-2,2)? Wait, no, maybe the correct order is A(-2,2), B(5,2), C(5,6), D(12,0), E(5,-6), B(5,2), A(-2,2)? No, that's not right. Wait, maybe the arrow has a rectangular part from (-2,2) to (5,2) to (5,-6) to (-2,-6), and a triangular part from (5,6) to (12,0) to (5,-6). Let's check:

First, area of the rectangle: length is 5 - (-2) = 7, height is 2 - (-6) = 8? Wait, no, y from 2 to -6 is 8 units? Wait, 2 - (-6) = 8? No, 2 - (-6) = 8? Wait, 2 - (-6) = 8? Wait, 2 - (-6) = 8? Yes, because 2 + 6 = 8. So length is 5 - (-2) = 7, height is 8? Wait, no, the rectangle from x=-2 to x=5 (width 7), y=-6 to y=2 (height 8). Area of rectangle: 7*8=56.

Then the triangle: vertices (5,6), (12,0), (5,-6). Let's calculate its area. The base of the triangle: from (5,6) to (5,-6) is 12 units (6 - (-6) = 12). The height is the horizontal distance from x=5 to x=12, which is 7 units (12 - 5 = 7). Area of triangle: (base * height)/2 = (12 * 7)/2 = 42.

Total area: 56 + 42 = 98? Wait, no, maybe the rectangle is from x=-2 to x=5 (width 7), y=-6 to y=2 (height 8), area 7*8=56. Then the triangle: base is from (5,6) to (5,-6) (length 12), height is 12 - 5 = 7? Wait, no, the base of the triangle is vertical, length 12 (from y=6 to y=-6), and the height is horizontal from x=5 to x=12, which is 7. So area of triangle is (12*7)/2=42. Then total area 56+42=98. Wait, but let's check with shoelace formula.

List the coordinates in order: A(-2,2), B(5,2), C(5,6), D(12,0), E(5,-6), F(5,-2), A(-2,2)? No, that's not correct. Wait, maybe the correct order is A(-2,2), B(5,2), C(5,6), D(12,0), E(5,-6), B(5,2), A(-2,2)? No, that's a polygon with a hole? No, better to split into two parts: the rectangle and the triangle.

Wait, another approach: the arrow can be divided into a rectangle and a triangle. The rectangle has length 7 (from x=-2 to x=5) and height 8 (from y=-6 to y=2). The triangle has a base of 12 (from y=6 to y=-6) and height 7 (from x=5 to x=12). Wait, no, the triangle's base is vertical, length 12 (6 - (-6) = 12), and the height is horizontal distance from x=5 to x=12, which is 7. So area of triangle is (12*7)/2=42. Area of rectangle: 7*8=56. Total area: 56+42=98.

Wait, but let's verify with shoelace formula. Let's list the correct vertices in order. Let's assume the arrow is composed of the rectangle from (-2,2) to (5,2) to (5,-6) to (-2,-6) and the triangle from (5,6) to (12,0) to (5,-6). So the combined vertices in order: (-2,2), (5,2), (5,6), (12,0), (5,-6), (-2,-6), (-2,2). Wait, no, (-2,-6) to (-2,2) is the left side. Let's apply shoelace formula:

Coordinates in order: (-2,2), (5,2), (5,6), (12,0), (5,-6), (-2,-6), (-2,2)

Shoelace formula:

Sum1 = (-2)*2 + 5*6 + 5*0 + 12*(-6) + 5*(-6) + (-2)*2

= (-4) + 30 + 0 + (-72) + (-30) + (-4)

= (-4 + 30) + (0 -72) + (-30 -4)

= 26 -72 -34 = 26 - 106 = -80

Sum2 = 2*5 + 2*5 + 6*12 + 0*5 + (-6)*(-2) + (-6)*(-2)

= 10 + 10 + 72 + 0 + 12 + 12

= 10+10=20; 20+72=92; 92+0=92; 92+12=104; 104+12=116

Area = |Sum1 - Sum2| / 2 = |-80 - 116| / 2 = |-196| / 2 = 98. Yes, that matches.

Step1: Identify the shape as composite (rectangle + triangle).

Step2: Calculate area of rectangle: width = 5 - (-2) = 7, height = 2 - (-6) = 8, area = 7*8 = 56.

Step3: Calculate area of triangle: base = 6 - (-6) = 12, height = 12 - 5 = 7, area = (12*7)/2 = 42.

Step4: Total area = 56 + 42 = 98.

Answer:

98