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QUESTION: If I had an equalateral triangle that is 5 units on all sides but needed to determine the height I'm told I would use the pythagtherion theorem when halfing the equalateral. Which apparently gives 4.3301270... for the height, which actually goes out to more than 50 numbers. Are there whole number values that can be used for the other sides that would make the height a whole number or at least clearly terminating within 10 decimal places?

James
ANSWER: No. If the sides are integers, then the height is an irrational number, and thus non-terminating.

An equilateral triangle divided in half gives you two congruent 30-60-90 triangles.
The sides of a 30-60-90 triangle are in the ratio 1:√3:2.

Suppose the equilateral triangle has sides of length 2n, where n is an integer. Divide the triangle in half as before.
The legs of the 30-60-90 triangle have lengths n and n√3. Since n is a positive integer but √3 is irrational, n√3 is irrational. It is a non-terminating decimal.

---------- FOLLOW-UP ----------

QUESTION: I would like to plot out on graph paper a perfect equalateral triangle. I'm probably not fully understanding your last answer. Is there any set of two non-whole numbers that could create a height that is either whole number or with decimals that does in fact terminate?

By definition, a rational number can be written as a fraction with whole numbers for both numerator and denominator, such as 3/5, 8/1, etc.
You can extend the definition to include fractions with terminating decimals for numerator and denominator. For example, 1.23/4.5 is rational, because it can be written as 123/450.

Note that for an equilateral triangle, the ratio of height to base is always √3/2,
which is a non-terminating decimal. Therefore, either the height or the base of the triangle must be non-terminating.

To construct a perfect equilateral triangle, you need a straightedge and compass.
Draw one side of the triangle, then set the compass to the length of the side.
Using an endpoint of the side as center, draw a circle with radius = length of side.

Draw a second circle using the other endpoint. The circles will intersect at two points.

Choose one of the points to be a vertex of the triangle, then draw the remaining two sides.

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