Which of the following could be the side lengths of a right triangle?
3, 13 and 14
4, 5 and 6
5, 12 and 13
5, 10 and 15

Solution:

We have to find the side lengths of a right triangle. We need to check if the lengths are the Pythagorean triples.

According to Pythagoras theorem,

In a right-angled triangle, the square of the longest side of a triangle(hypotenuse) is equal to the sum of the squares of the other two sides (base and height).

In a right triangle, (hypotenuse)2 = (base)2 + (height)2

From the options,

a) 3, 13 and 14

Hypotenuse = 14, Base = 3, Height = 13

(hypotenuse)2 = (base)2 + (height)2

LHS:

(14)2 = 196

RHS:

(3)2 + (13)2 = 9 + 169

= 179

LHS ≠ RHS

Therefore, option (a) is not true.

b) 4, 5 and 6

Hypotenuse = 6, Base = 4, Height = 5

(hypotenuse)2 = (base)2 + (height)2

LHS:

(6)2 = 36

RHS:

(4)2 + (5)2 = 16 + 25

= 41

LHS ≠ RHS

Therefore, option (b) is not true.

c) 5, 12 and 13

Hypotenuse = 13, Base = 5, Height = 12

(hypotenuse)2 = (base)2 + (height)2

LHS:

(13)2 = 169

RHS:

(5)2 + (12)2 = 25 + 144

= 169

LHS = RHS

Therefore, option (c) is true.

d) 5, 10 and 15

Hypotenuse = 15, Base = 5, Height = 10

(hypotenuse)2 = (base)2 + (height)2

LHS:

(15)2 = 225

RHS:

(5)2 + (10)2 = 25 + 100

= 125

LHS ≠ RHS

Therefore, option (d) is not true.

Therefore, 5, 12, and 13 could be the side lengths of the right triangle.

Which of the following could be the side lengths of a right triangle?

Summary:

5, 12, and 13 could be the side lengths of the right triangle.

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