Question

A person wishes to fit three rods together in the shape of a right-angled triangle so that the hypotenuse is to be $4cm$ longer than the base and $8cm$ longer than the altitude. The length of the rods is:
A) $3cm,4cm,5cm$
B) $1.5cm,2cm,2.5cm$
C) $6cm,8cm,10cm$
D) $12cm,16cm,20cm$

Answer 1

Hint: First assume the lengths of the base, altitude and hypotenuse. Use the Pythagoras theorem which states that the square of the length of the hypotenuse is equal to the sum of the square of the length of the base and the altitude.

Complete Step-by-Step solution:
Draw a figure of three rods which are in the shape of a right-angled triangle.

Assume the length of the hypotenuse AC is $z$, length of the base BC is $x$ and the length of the altitude AB is $y$.
We are given that the hypotenuse is to be $4cm$ longer than the base and $8cm$ longer than the altitude.
Therefore,
$
  z = x + 4.....(1) \\
  z = y + 8.....(2) \\
 $
Compare both the equations to get the relation between $x$ and $y$.
$
  x + 4 = y + 8 \\
  y = x - 4........(3) \\
 $
Now we use the Pythagoras theorem.
Pythagoras theorem states that the square of the length of the hypotenuse is equal to the sum of the square of the length of the base and the altitude.
Therefore,
${z^2} = {x^2} + {y^2}$
Substitute the value of $y$ from equation $(3)$ and the value of $z$ from equation $(1)$
Therefore,
${(x + 4)^2} = {x^2} + {(x - 4)^2}$
Use the algebraic identities
$
  {(a + b)^2} = {a^2} + 2ab + {b^2} \\
  {(a - b)^2} = {a^2} - 2ab + {b^2} \\
 $
Rewrite the above equation after using the identities.
$
  {x^2} + 16x + 16 = {x^2} + {x^2} - 16x + 16 \\
  {x^2} - 16x = 0 \\
  x(x - 16) = 0 \\
  x \ne 0 \\
  \therefore x - 16 = 0 \\
  x = 16 \\
 $
Substitute the value of $x$ in equation $(1)$and $(3)$
$
  y = x - 4 \\
  y = 16 - 4 \\
  y = 12 \\
 $
$
  z = x + 4 \\
  z = 16 + 4 \\
  z = 20 \\
 $
Hence, the length of the rods is $12cm,16cm,20cm$

Therefore, option (D) is correct.

Note:
There is an alternate approach to solve this question which is given below:
If we look at our condition that the hypotenuse is to be $4cm$ longer than the base and $8cm$ longer than the altitude and we know that hypotenuse is the largest length of the right-angled triangle.