Question

The hypotenuse of a right triangle is $3\sqrt 5 cm$. If the smaller side is tripled and the larger side is doubled, the new hypotenuse will be 15cm. Find the length of each side.

Answer 1

Hint: We will first let the smaller side be $x$cm and the larger side be $y$ cm. Then, use the given conditions to form equations in $x$ and $y$. Solve the equations to find the length of smaller side and length of larger side by using substitution method.

Complete step-by-step answer:
Let the smaller side be $x$cm and the larger side be $y$cm
As we know, Pythagoras theorem holds for every right triangle.
Pythagoras theorem states that ${\left( {{\text{Hypotenuse}}} \right)^2} = {\left( {{\text{Perpendicular}}} \right)^2} + {\left( {{\text{Base}}} \right)^2}$
Then, according to the given condition, ${\left( {3\sqrt 5 } \right)^2} = {x^2} + {y^2}$
Which is equal to $45 = {x^2} + {y^2}$ (1)
Also, we are given that if the smaller side is tripled and the larger side is doubled, the new hypotenuse will be 15cm
Then the smaller side will be $3x$ and the larger side will be $2y$, also the hypotenuse is 15cm.
On applying Pythagoras theorem, we will get,
$
  {\left( {15} \right)^2} = {\left( {3x} \right)^2} + {\left( {2y} \right)^2} \\
   \Rightarrow 225 = 9{x^2} + 4{y^2} \\
$
Substitute the value of ${y^2} = 45 - {x^2}$ from the equation (1)
Then,
 $
  225 = 9{x^2} + 4\left( {45 - {x^2}} \right) \\
   \Rightarrow 225 = 9{x^2} + 180 - 4{x^2} \\
   \Rightarrow 45 = 5{x^2} \\
   \Rightarrow {x^2} = 9 \\
   \Rightarrow x = \pm 3 \\
$
Also, the length of the side can never be negative, hence $x = 3$
On substituting the value of $x = 3$ in equation (1) to find the value of $y$
Then,
 $
  45 = {\left( 3 \right)^2} + {y^2} \\
   \Rightarrow 45 = 9 + {y^2} \\
   \Rightarrow {y^2} = 36 \\
   \Rightarrow y = \pm 6 \\
$
Also, side cannot be negative $y = 6cm$
Then, the length of the smaller side be 3cm and the length of larger side be 6cm.

Note: In a right triangle, the longest side is the hypotenuse and other two sides form an angle of ${90^ \circ }$ Also, Pythagoras theorem holds for every right triangle. The length of the side can never be negative.