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Area of an Irregular Shape, a la Shmoop.
Tom is a professional Shetland pony jockey…The problem is, he’s packed on quite a few pounds,
and his pony Daisy is starting to crack under the pressure.
In an effort to lose weight before next week’s pony race, Tom goes on a diet of strictly
dark chocolate…
Here is Tom’s chocolate lunch. GHIJ is a square piece of chocolate.
He’s only allowed to have a serving of chocolate with an area of less than 55 square millimeters.
So… let’s find the area of the Figure FHIJ.
We know angle FGH is 90 degrees… because the figure on the right is a square…
…if FG and GH are the same length, we know that angle FHG and FGH are both 45 degrees…
because all three angles need to add up to 180.
Since all the sides of our square GHIJ are the same length… we can assign each side
the value of x… Making the area of our square GHIJ equal to
“x” squared. This means that the area of the triangle FGH
is one-half “x” squared.
To solve for the side lengths FG and GH, x, we can use our friend the Pythagorean theorem.
The Pythagorean theorem tells us that x squared plus x squared… in this case… equals 6
square root of 2 squared. Now we need to do a little math. Sorry.
First we add up the left side of the equation to give us 2x squared…
On the right side of the equation, we can square 6 to give us 36, and square the square
root of two to give us two. But we still need to multiply…36 times 2 gives us 72…
Our equation is now 2x squared equals 72…Divide both sides by two to give us x squared equals
36.. Taking the square root of both sides gives
us our value of x, which is 6 millimeters.
Now we know the length of each side of the square, as well as the two other sides of
our triangle! To get the area of our square, we take our
x value, the length of one of the sides and square it…making it 6 squared, giving us
36. Now we can use the area formula for a triangle…
area equals one-half base times height. Plugging in our numbers, we get ½ 6 times
6 equals 18. So now we need to add up the area of the triangle
and the area of the square, which gives us 36 plus 18… which equals 54 square milliliters.
Unfortunately, the chocolate diet didn’t pan out to well. Poor Daisy, her back will
never be the same.