Question

Could someone please explain?

Answer 1

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Answer:

• obtuse
• acute

Step-by-step explanation:

The Law of Cosines tells you the largest angle C will be found from ...

  C = arccos((a² +b² -c²)/(2ab))

where c is the longest side of the triangle.

For the purpose of classifying the triangle as acute, right, or obtuse, you need only look at the sign of the argument of the arccos function. Since all side lengths are positive, this means you only need to look at the sign of the "form factor" a²+b²-c².

When f = a²+b²-c² is negative, the cosine is of an angle larger than 90°, so the triangle is obtuse. When it is 0, the angle is 90°, so a right triangle. (That condition is recognizable as related to the Pythagorean theorem.) When f > 0, the triangle is acute.

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In the attached spreadsheet, we have done these calculations by summing the squares of all three sides, then subtracting twice the square of the longest side. (This makes the formula fairly simple.) It shows ...

  Triangle 1: f < 0 — obtuse triangle

  Triangle 2: f > 0 — acute triangle

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In summary, you can compute a form factor ...

  f = a² +b² -c² . . . . . . . triangle with side lengths a, b, c with c longest

• f < 0 — obtuse
• f = 0 — right
• f > 0 — acute

Answer 2

Square the two smallest values and then add them together. If they are greater than the square of the largest number in the set, than it is obtuse. If smaller it is acute. If it is the same it is a right triangle.