Question

The length of the sides of a right triangle are \[{\text{5x + 2,5x,3x - 1}}\]. If \[{\text{x > 0}}\] 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: Using the above given condition of \[{\text{x > 0}}\] in the question , we can apply Pythagoras theorem in the given question. As the length of each side is known and also the triangle is the right angle triangle. For the triangle with side length A, B, C and C being the largest side. Pythagoras theorem can be applies as \[{{\text{A}}^{\text{2}}}{\text{ + }}{{\text{B}}^{\text{2}}}{\text{ = }}{{\text{C}}^{\text{2}}}\]

Complete step-by-step answer:
As from the above given side lengths we can easily determine that \[{\text{5x + 2}}\] is the largest side of the given triangle.
So by applying Pythagoras theorem,
\[{{\text{(5x + 2)}}^{\text{2}}}{\text{ = (5x}}{{\text{)}}^{\text{2}}}{\text{ + (3x - 1}}{{\text{)}}^{\text{2}}}\]
On squaring we get,
\[ \Rightarrow {\text{25}}{{\text{x}}^{\text{2}}}{\text{ + 20x + 4 = 25}}{{\text{x}}^{\text{2}}}{\text{ + 9}}{{\text{x}}^{\text{2}}}{\text{ - 6x + 1}}\]
On simplification we get,
\[ \Rightarrow {\text{9}}{{\text{x}}^{\text{2}}}{\text{ - 26x - 3 = 0}}\]
On factorisation we get,
\[
   \Rightarrow {\text{9}}{{\text{x}}^{\text{2}}}{\text{ - 27x + x - 3 = 0}} \\
   \Rightarrow {\text{9x(x - 3) + 1(x - 3) = 0}} \\
   \Rightarrow {\text{(9x + 1)(x - 3) = 0}} \\
   \Rightarrow {\text{x = 3, - }}\dfrac{{\text{1}}}{{\text{9}}} \\
 \]
\[
  \because {\text{x > 0}} \\
   \Rightarrow {\text{x = 3}} \\
 \]
By putting the value of x in the above given sides of triangle
\[
  {\text{5x + 2 = 5(3) + 2 = 17}} \\
  {\text{5x = 5(3) = 15}} \\
  {\text{3x - 1 = 3(3) - 1 = 8}} \\
 \]
The sides of triangle are \[{\text{17,15,8}}\]
Hence option (a) is the correct answer.

Note: In a right triangle, the hypotenuse is the longest side, an "opposite" side is the one across from a given angle, and an "adjacent" side is next to a given angle. We use special words to describe the sides of right triangles.
The Pythagorean theorem states that, in a right triangle, the square of the length of the hypotenuse (the side across from the right angle) is equal to the sum of the squares of the other two sides.