Question

Your classmate is unsure about how to use side lengths to determine the type of triangle. How would you explain this to your classmate?

Answer 1

Here’s a clear, step-by-step way to decide what kind of triangle you have from its side lengths.

1. Check that the sides can form a triangle
- All side lengths must be positive.
- Triangle inequality: for sides a, b, c, each pair must sum to more than the third:
a + b > c, a + c > b, and b + c > a.
- If any one of these is equality (e.g. a + b = c) the figure is degenerate (the points are collinear), not a proper triangle.

2. Classify by equality of side lengths
- Equilateral: all three sides equal (a = b = c).
- Isosceles: exactly two sides equal (a = b ≠ c, etc.).
- Scalene: all three sides different (a ≠ b ≠ c).

3. (Optional) Classify by angles using side lengths
- First sort sides so a ≤ b ≤ c (c is the longest).
- Compare a^2 + b^2 with c^2:
- If a^2 + b^2 = c^2 → right triangle.
- If a^2 + b^2 > c^2 → acute triangle (all angles < 90°).
- If a^2 + b^2 < c^2 → obtuse triangle (one angle > 90°).
- Note: do the angle test only after you’ve confirmed it’s a proper triangle.

4. Practical notes
- For approximate or decimal lengths, use a small tolerance when checking equality (e.g. if |a^2 + b^2 − c^2| < 10^−6 treat as equal).
- Always sort the sides before the Pythagorean comparison so you compare the two smallest squared-sum with the largest squared.

Examples
- (3, 4, 5): 3^2 + 4^2 = 25 = 5^2 → right triangle (also scalene).
- (4, 5, 6): 4^2 + 5^2 = 41 > 6^2 = 36 → acute scalene.
- (2, 3, 4): 2^2 + 3^2 = 13 < 4^2 = 16 → obtuse scalene.
- (5, 5, 5): all equal → equilateral (also acute).
- (5, 5, 8): satisfies triangle inequality, two equal sides → isosceles (here obtuse).

That’s the full procedure: verify the triangle inequality, check side-equality for equilateral/isosceles/scalene, and (if you want angle-type) sort and use the squared-side comparison for right/acute/obtuse.