A triangle has sides 6, 8, and 11. What type of triangle is it?
ARight — 6² + 8² = 100 is close enough to 11² = 121
BAcute — because the two shorter sides together exceed the longest side
CObtuse — because 6² + 8² = 100 < 11² = 121
DCannot be classified using side lengths alone
Assign the longest side to c: c = 11. Compute a² + b² = 36 + 64 = 100, and c² = 121. Since 100 < 121, we have a² + b² < c², which means the triangle is obtuse. The common error is reversing the inequalities — students often think 'the sum falls short, so it must be acute,' but the logic runs the other way: an obtuse angle makes the opposite side grow, so c² grows larger than a² + b².
Question 2 Multiple Choice
Which set of side lengths forms a right triangle?
A5, 12, 14
B7, 24, 25
C3, 4, 6
D8, 15, 18
Test each by assigning the largest side to c and checking a² + b² = c². For 7, 24, 25: 7² + 24² = 49 + 576 = 625 = 25². ✓ For 5, 12, 14: 25 + 144 = 169 ≠ 196. For 3, 4, 6: 9 + 16 = 25 ≠ 36. For 8, 15, 18: 64 + 225 = 289 ≠ 324. Only 7-24-25 satisfies the equation — this is a well-known Pythagorean triple.
Question 3 True / False
If a triangle's sides satisfy a² + b² > c² (where c is the longest side), the triangle is obtuse.
TTrue
FFalse
Answer: False
This is the most common inequality mix-up. When a² + b² > c², the triangle is ACUTE. The logic: an obtuse angle causes the opposite side to grow longer, making c² exceed a² + b². Conversely, if a² + b² is MORE than c², the angle opposite c is less than 90° — the triangle is acute. Memory aid: 'acute means the sum is too big' (the sides are more than needed for a right angle).
Question 4 True / False
The converse of the Pythagorean Theorem requires its own separate proof because a theorem being true does not guarantee its converse is also true.
TTrue
FFalse
Answer: True
This is a fundamental point in logic: the converse of a true statement is not automatically true. The standard proof of the converse constructs a companion right triangle with the same leg lengths, then uses SSS congruence to conclude that the original triangle must also have a right angle. Without this proof, we couldn't legitimately reverse the theorem's direction.
Question 5 Short Answer
Why must c be assigned to the longest side before applying the Pythagorean converse? What goes wrong if you don't?
Think about your answer, then reveal below.
Model answer: c must be the longest side because the inequality test is built around the relationship between the largest angle and its opposite side. The Pythagorean theorem says the hypotenuse (longest side, opposite the largest angle) satisfies a² + b² = c². If you assign c to a shorter side, a² + b² will almost always exceed c² — giving a misleading 'acute' result — even for right or obtuse triangles. The test only works correctly when c is the candidate for the role of hypotenuse.
Example: for sides 3, 4, 5, if you mistakenly set c = 3, you get 4² + 5² = 41 > 9, suggesting 'acute' — but the triangle is actually right. The converse test classifies the angle opposite the longest side; only the longest side has any chance of being the hypotenuse.