Question

The lengths of the sides of a right triangle are \[5x + 2\], \[5x\] and \[3x - 1\]. If \[x > 0\] then the length of each side is?
A) 17, 15, 8
B) 17, 15, 9
C) 18, 15, 10
D) 20, 5, 12

Answer 1

Hint:
Here, we are required to find the length of each side of a right angled triangle. In a right angled triangle the largest side is the hypotenuse. So, we will compare the given sides and check which side is the largest. We will use Pythagoras theorem to find the value of \[x\]. Then, we will substitute the value of \[x\] in the expression of sides to find the required length of each side.

Formula Used:
By Pythagoras Theorem, \[{\left( {{\rm{Hypotenuse}}} \right)^2} = {\left( {{\rm{Perpendicular}}} \right)^2} + {\left( {{\rm{Base}}} \right)^2}\].

Complete step by step solution:
According to the question, the sides of a triangle are \[5x + 2\], \[5x\]and \[3x - 1\].
Now, it is given that this is a right angled triangle.
Right angle triangle is a triangle in which one angle is a right angle or in other words, its measure is \[90^\circ \].
In this question, it is given that \[x > 0\], hence this means that the value of \[x\] will always remain positive.
Now, let us assume that \[x = 1\].
Hence, in this case, the three given sides will become:
\[5x + 2 = 5\left( 1 \right) + 2 = 7\],
\[5x = 5 \times 1 = 5\]
And \[3x - 1 = 3\left( 1 \right) - 1 = 2\]
We can see that \[5x + 2\] is the longest side.
Hence, in the given right angle triangle, \[5x + 2\] is the hypotenuse.
Now, we will use Pythagoras theorem to find the exact value of \[x\].
By Pythagoras Theorem, \[{\left( {{\rm{Hypotenuse}}} \right)^2} = {\left( {{\rm{Perpendicular}}} \right)^2} + {\left( {{\rm{Base}}} \right)^2}\]
Substituting the values, we get
\[{\left( {5x + 2} \right)^2} = {\left( {5x} \right)^2} + {\left( {3x - 1} \right)^2}\]
Now, using the formulas \[{\left( {a \pm b} \right)^2} = {a^2} \pm 2ab + {b^2}\], we get
\[ \Rightarrow 25{x^2} + 20x + 4 = 25{x^2} + 9{x^2} - 6x + 1\]
Now, cancelling out the similar terms and solving further, we get
\[ \Rightarrow 9{x^2} - 26x - 3 = 0\]
Now the above equation is a quadratic equation.
Splitting the middle term, we get
\[ \Rightarrow 9{x^2} - 27x + x - 3 = 0\]
\[ \Rightarrow 9x\left( {x - 3} \right) + 1\left( {x - 3} \right) = 0\]
Factoring out common terms, we get
\[ \Rightarrow \left( {9x + 1} \right)\left( {x - 3} \right) = 0\]
Now applying zero product property, we get
Hence, \[\left( {9x + 1} \right) = 0\]
\[ \Rightarrow x = \dfrac{{ - 1}}{9}\]
But according to the question, \[x > 0\], Hence, we will reject this negative value.
Also,
\[\left( {x - 3} \right) = 0\]
\[ \Rightarrow x = 3\]
Therefore the value of \[x\] is 3.
Now we will use the value of \[x\] to find all the sides.
Now, the lengths of the sides of a right triangle are:
\[5x + 2 = 5 \times 3 + 2 = 17\]
\[5x = 5 \times 3 = 15\]
And, \[3x - 1 = 3 \times 3 - 1 = 8\]
Therefore, the required length of each side is 17, 15, 8 respectively.

Hence, option A is the correct answer.

Note:
In this question, in order to identify which side is the largest, we assumed the value of \[x = 1\]. But we could have found which side is the largest by just comparing them. Since, the given sides of a right angled triangle are \[5x + 2\], \[5x\] and \[3x - 1\]. Hence, \[5x + 2\] will definitely be the largest side as 2 is being added hence, making it larger than the other two lengths.
Also, as we have discussed, Pythagoras Theorem states that in a right angled triangle, the square of the longest side or the hypotenuse is equal to the sum of the squares of the other two sides. It is applicable only for the right angled triangles.