Answer:
- perimeter: 27.48 units
- area: 26 square units
Explanation:
You want the perimeter and area of the trapezoid shown in the graph.
Perimeter
The perimeter of the figure is the sum of the lengths of its sides. When those sides are not horizontal or vertical, their length can be found using the distance formula:
d = √((x2 -x1)² +(y2 -y1)²)
The sides not aligned with the grid are ...
AB = √((0 -(-5))² +(3 -4)²) = √(5² +(-1)²) = √26 ≈ 5.0990
BC = √((4 -0)² +(-1 -3)²) = √(4² +(-4)²) = √32 ≈ 5.6569
DA = √((-5 -4)² +(4 -(-5))²) = √((-9)² +9²) = √162 ≈ 12.7279
The side aligned with the grid is ...
CD = vertical line, length -1 -(-5) = 4
So, the perimeter is ...
P = AB +BC +CD +DA = 5.0990 +5.6569 +4 +12.7279 = 27.4838
The perimeter is about 27.48 units.
Area
There are numerous ways to find the area. We can treat the total figure as a trapezoid and find its area using the area formula for a trapezoid.
In addition to side lengths BC and DA, we need to know its height, CE.
CE = √((2 -4)² +(-3 -(-1))²) = √((-2)² +(-2)²) = √8 ≈ 2.8284
The area of the trapezoid is about ...
A = 1/2(b1 +b2)h
A = 1/2(5.6569 +12.7279)(2.8284) = 26.00
The area of the trapezoid is 26 square units.
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Additional comment
For purposes here, we have rounded the numbers to 4 decimal places. This provides reasonable assurance that an answer rounded to 2 decimal places will be correct to that precision. Doing the actual calculations, we used the full precision of the calculator (32 significant figures).
For the area, we can use the exact measures of the lengths:
A = 1/2(4√2 +9√2)(2√2) = 1/2(13√2)(2√2) = 26 . . . . . exactly
When the figure is defined by points with integer coordinates the area can be found using those coordinates. It can be found using a formula of the form ...
A = 1/2·|x1(y2-y4) +x2(y3-y1) +x3(y4-y2) +x4(x1-x3)|
A = 1/2|(-5)(3 -(-5)) +0(-1 -4) +4(-5 -3) +4(4 -(-1))|
A = 1/2|-40 +0 -32 +20| = 1/2(52) = 26
Note that this will always be an integer multiple of 1/2.
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