9514 1404 393
Answer:
Use the Pythagorean theorem: (5/8)^2 +(3/2)^2 = (13/8)^2 ⇒ right triangle
Explanation:
Check to see if the numbers satisfy the Pythagorean theorem.
(5/8)^2 + (3/2)^2 = (13/8)^2
25/64 + 9/4 = 169/64
25/65 +144/64 = 169/64 . . . . . . . true. The triangle is a right triangle.
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If a, b, c represent the lengths of the sides, least to greatest, then ...
a^2 + b^2 > c^2 . . . . acute triangle
a^2 + b^2 = c^2 . . . . right triangle
a^2 + b^2 < c^2 . . . . obtuse triangle
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Additional comment
A triple of integers that satisfy the Pythagorean theorem (form a right triangle) is called a "Pythagorean triple." One you see often, and that has the special property that it is the only triple that is an arithmetic sequence is (3, 4, 5). A couple of other common ones are (5, 12, 13) and (7, 24, 25).
Any of these can be scaled by a common factor to make sides of a right triangle. Here, we have (5, 12, 13) scaled by a factor of 1/8 to make (5/8, 12/8, 13/8) = (5/8, 3/2, 13/8) — the numbers in the problem statement.
If you're aware of these common triples, you can check to see if they are being used in any right-triangle problem you may run across. Recognizing that one of these applies can save you the effort of working out the squares and/or root that would otherwise be involved.