How to Solve the Points of an Equilateral Triangle
As students progress in geometry classes, they will find that the problems become harder - not necessarily because the concepts are harder themselves, but rather because they involve combining a series of earlier concepts. Graphing an equilateral triangle falls into this category: when you are given only a small amount of information, it can be difficult to find other pieces of the puzzle. However, by relying on basic concepts learned earlier in the class, you can solve for the points of an equilateral triangle.
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Instructions
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Examine the information you have. To solve for all three points of a graphed equilateral triangle, you will either need the coordinates of two of the triangle's points, or one point and the length of one side (which, as the triangle is equilateral, will be equal to the other sides). If you only have one point and the triangle begins at the origin, you can use this information, with the second point being (0,0).
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Find the length of one side of the triangle. You may have been given this information; however, if you begin with only two coordinates, you will have to determine the length from that. If the triangle has its base at the same value (i.e. it remains at a value of 2 throughout the graph), simply subtract the smaller x-coordinate from the larger x-coordinate. For example, if your two points are (2,2) and (2,6), the length of the triangle's sides is 4.
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Find the second coordinate. If you are given an equilateral triangle with one point and the length of the sides, find the second coordinate by adding the leg's length to the first x-coordinate. For example, if you know that the triangle begins at the origin and has a leg length of 4, the second point will be (4,0).
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Find the third coordinate. Drop an altitude, or a straight line, from the top vertex of the triangle to halfway across the triangle's base, to create two smaller triangles. The base of the new triangle is now 1/2 of the original base, which you will add to the x-coordinate of the first point to get the x-coordinate of the third point. For example, if you have an original point of (2,2) and leg lengths of 4, add 2 to the first x-coordinate to get (4, y). To find the y-coordinate of the third point, calculate the length of the altitude and add it to the original y-coordinate. Using the Pythagorean theorem, where a^2 + b^2 = c^2, you have a (the base, in this example 2), and c (the hypotenuse, or the leg length; in this case 4). Plug them into the formula to arrive at 4 + b^2 = 16, subtracting 4 from both sides to get b^2 = 12. B is thus the square root of 12, or about 3.5. Add this to your original y-coordinate to get the final y-coordinate for your triangle: (4, 5.5).
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References
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