Question

Which of the following could be the side lengths of a right triangle?
A. 3, 13 and 14
B. 4, 5 and 6
C. 4, 9 and 10
D. 5, 10 and 15
E. 5, 12 and 13

Answer 1

Hint:
First, we will use Pythagoras theorem to all the given side lengths given to us one by one, The side lengths which satisfy the Pythagoras test must be a right-angled triangle. Hence, that would be our required answer.

Complete step by step solution:
We know that 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). Hence, we can say that
In a right triangle,
 \[Hypotenuse{e^2} = bas{e^2} + heigh{t^2}\]

For option A, we have the three sides as 3,13, and 14
Here the hypotenuse is 14,
Hence we get
 \[ \Rightarrow {14^2} = {3^2} + {13^2}\]
On simplification we get,
 \[ \Rightarrow 196 = 9 + 169\]
On adding terms on RHS we get,
 \[ \Rightarrow 196 \ne 178\]
Hence, option A is eliminated.

For option B, we have the three sides as 4,5, and 6
Here the hypotenuse is 6,
Hence we get
 \[ \Rightarrow {6^2} = {4^2} + {5^2}\]
On simplification we get,
 \[ \Rightarrow 36 = 16 + 25\]
On adding terms on RHS we get,
 \[ \Rightarrow 36 \ne 41\]
Hence, option B is eliminated.

For option C, we have the three sides as 4,9, and 10
Here the hypotenuse is 10,
Hence we get
 \[ \Rightarrow {10^2} = {4^2} + {9^2}\]
On simplification we get,
 \[ \Rightarrow 100 = 16 + 81\]
On adding terms on RHS we get,
 \[ \Rightarrow 100 \ne 97\]
Hence, option C is eliminated.

For option D, we have the three sides as 5,10, and 15
Here the hypotenuse is 15,
Hence we get
 \[ \Rightarrow {15^2} = {10^2} + {5^2}\]
On simplification we get,
 \[ \Rightarrow 225 = 100 + 25\]
On adding terms on RHS we get,
 \[ \Rightarrow 225 \ne 125\]
Hence, option D is eliminated.

For option E, we have the three sides as 5,12, and 13
Here the hypotenuse is 13, Hence we get
 \[ \Rightarrow {13^2} = {5^2} + {12^2}\]
On simplification we get,
 \[ \Rightarrow 169 = 25 + 144\]
On adding terms on RHS we get,
 \[ \Rightarrow 169 = 169\]

Hence, option E is the required answer.

Note:
For these types of questions, it is advisable to learn some basic Pythagorean triples to get the answer quickly, Some of the basic Pythagorean triplets are
(1) 3, 4, and 5
(2) 5, 12, and 13
(3) 7, 24, and 25
Also, their multiples are considered to be triplets for example
(1) 6, 8, and 10
(2) 10, 24, and 26
Are also Pythagorean triplets.