Which set of numbers can represent the side lengths, in millimeters, of an obtuse triangle? a. 8, 10, 14b. 9, 12, 15c. 10, 14, 17d. 12, 15, 19

  A.  8, 10, 14

Step-by-step explanation:

As a rough cut, a triangle will be obtuse if the longest side is about 1.4 or more times the length of the second-longest side. This derives from the relationship in an isosceles right triangle, where the hypotenuse is √2 ≈ 1.414 times the length of the two equal-length sides. If one side is shorter than the other, and the hypotenuse is still 1.414 times the length of the second-longest side, then the triangle is no longer a right triangle, but is an obtuse triangle.

Here, the first selection has a middle-length side of 10 and a longest side of 14, about 1.4 times 10. It is an obtuse triangle.

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More rigorously, you can see if the sum of the squares of the short sides is less than the square of the longest side. If so, the triangle is obtuse. (The Law of Cosines will tell you the angle opposite the longest side must have a negative cosine, so must be greater than 90°.)

Our answer choices are ...

  A. 8^2 + 10^2 = 164 < 14^2 = 196 . . . . . obtuse

  B. 9^2 + 12^2 = 225 = 15^2 . . . . . . . . . . right

  C. 10^2 +14^2 = 296 > 17^2 = 289 . . . . . acute

  D. 12^2 +15^2 = 369 > 19^2 = 361 . . . . . acute

i need the options

step-by-step explanation: