In the following sets, which set represents the sides of a right-angled triangle?
- (a){5, 10, 12}
- (b){3, 4, 6}
- (c){5, 12, 13}
- (d){8, 15, 16}
Answer
Why
Correct — C. Apply the converse of Pythagoras to each set: the squares of the two smaller sides must add to the square of the largest.
5² + 12² = 25 + 144 = 169
13² = 169
The two agree, so the angle opposite 13 is a right angle and {5, 12, 13} is the set — option (c).
Why the others are wrong
- (a){5, 10, 12} — 5² + 10² = 125, while 12² = 144. The square on the longest side is the larger, so the angle facing 12 is obtuse rather than right.
- (b){3, 4, 6} — 3² + 4² = 25, while 6² = 36. The familiar triple is 3, 4, 5 — stretching the 5 to a 6 breaks the equality.
- (d){8, 15, 16} — 8² + 15² = 289, which is 17², not 16². The triple hiding here is 8, 15, 17, so with 16 the triangle comes out acute.
Concept
Pythagoras says a right-angled triangle satisfies a² + b² = c². Its converse runs the other way and is what this item uses: if the equality holds, the triangle must be right-angled.
The comparison also tells you the shape when it fails. If c² is bigger than a² + b² the angle opposite c is obtuse, and if it is smaller that angle is acute.
Three of the four sets here are near-misses of a triple a candidate already knows, which is why reading all three numbers matters.
The sides are given as an unordered set, so identify the largest number first — it is the only candidate for the hypotenuse.
Key facts
- Converse of Pythagoras: if a² + b² = c², the triangle is right-angled at the vertex opposite side c.
- Standard triples worth memorising: 3-4-5, 5-12-13, 8-15-17, 7-24-25 and 9-40-41.
- Any whole multiple of a triple is itself a triple, so 10-24-26 works exactly as 5-12-13 does.
- If c² is greater than a² + b² the triangle is obtuse, and if it is less it is acute.
Study next
Common traps
- Recognising two numbers of a known triple and not checking the third
- Treating {8, 15, 16} as a triple because 8-15-17 is one
- Comparing the largest side with the sum of the other two instead of comparing squares
Here the converse is asked as a spot-the-triple item, while the theorem itself turns up as a length calculation — 17 Sep 2024, 12:30, Quant Q.11 drops an altitude to the hypotenuse of a right-angled triangle and asks for its length.
In the same shift, Quant Q.17 uses Pythagoras in three dimensions for the longest stick that fits in a cube.
Related PYQs
No directly related past PYQ was found.