Geometry/Math

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Question
The base of a triangular prism is traced onto paper, the height is 1.75 inches and the width this 2 inches. What are the areas of the sides of each prism?

Answer
Let the triangle with sides of 2" be called A.
Make a new triangle with one of the corners as a corner of A,
one of the corners as the center of A, and
one of the corners as the midpoint in one of the sides of A near the corner chosen.

Drawing this will reveal there are 6 triangles that can be drawn this way,
all of equal size.

On a triangle such as this, if the short side is 1,
the hypotenuse is 2, and the long side is root(3).
As can be seen, 1+root(3) = 4, and this is the length of the hypotenuse squared.

The outside of one of these triangles is half of 2, so it is 1.
This corresponds to the length of the short side being root(3).

We have the length of the short side as 1, so divide the lengths of the others by root(3) as well to get side lengths of 1/root(3), 2/root(3), and 1.

Taking another triangle where the base is the line from the center of the triangle out to the center of one of the sides, the other leg is the height of the triangle, so the hypotenuse of this one is the center of one of the sides.  The base of this triangle is 1/root(3), the height is given as 1.75, so the hypotenuse is the square root of the sum of the squares of these two.

When 1/root(3) is squared, the result is 1/3.
To look at the result of 1.75 squared, 1.75 is 7/4, so squaring it gives 49/16.
This is the same as 3 1/16.

If we add these, 48 is the common denominator, so were get 16/48 + 3 3/48 = 3 19/48.
Taking the square root of that gives 1.843.

Now of the triangle that is on the side, we have the base is 2 and the height is 1.843,
so the area is 2*1.843/2 = 1.843.  

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