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So, we'll need to know the length of , , and
Remember, you can do this using the distance formula, or the Pythagorean Theorem. They're actually the same thing, but:
Pythagorean Theorem is easier if you can imagine the triangles, or draw on the picture
a2 + b2 = c2
Distance formula is easier if you don't have a picture at all
Your company has been hired to remediate it (clean it up). The project manager has asked you to figure out two things
The amount of fencing needed to fence off the site while your company remediates it
The area of contaminated soil
Let's start with the fencing. The fence will wrap around the edge of the site, so we'll need to know the perimeter of the site. Let's label the points.
Objective
Common Core Text:
[G.GPE.7] (*) Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula
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Explanation
We'll learn this one through an example, and summarize once we're finished
Example 1
The map below shows the location of a giant chemical spill. The numbers shown are miles.
D =
We'll go with Pythagorean Theorem here. Time to sketch in some little right triangles.
= 62 + 22
= 36 + 4
= 40
= 62 + 62
= 36 + 36
= 72
is not diagonal, so we don't need the Pythagorean Theorem
= 12 - 4
We can add the sides to find the perimeter
Perimeter = 6.32 miles + 8.49 miles + 8 miles
Perimeter = 22.81 miles
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Now we need to find the area of the spill. We'd like to use our equation for area of a triangle
A = 1/2 x b x h
but we need a right angle to do that. We clearly don't have a right angle in this picture. That means we need to draw our own in.
We could draw it through any point, but C would be easiest because then we get a nice vertical line. We'll use this line for our height. This means our base is AB, which we said was 8 miles long. We can see our height goes from y = 2 to y = 8, so the length of our height is 8 - 2 = 6 miles.
A = 1/2 x b x h
A = 1/2 x 8 miles x 6 miles
A = 24 miles2 = 24 square miles
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So you would report to the project manager that
We need 22.81 miles of fencing to fence off the site.
There will be 24 square miles of contaminated soil to remediate.
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Practice your skills
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