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Hint: In a right angled triangle, one of the perpendicular sides is 6 centimeters longer than the other side. If the area of the triangle is 36 square centimeters, what is the length of its perpendicular sides.

Complete step-by-step answer:

Let one of the perpendicular sides of the right angled triangle be $x$.

It is given that the other side is six centimeters longer than this side.

Hence, the other side is $(6 + x)$ centimeters long.

The area of a triangle is half times the product of its base and its height. If the base of the triangle is and its height is h, then the area is given by,

\[A = \dfrac{1}{2}bh...........(1)\]

For a right angled triangle, if one of its perpendicular sides is considered as a base, then the other perpendicular side will be the height of the triangle, hence, the area of the right angled triangle is half times the product of the two perpendicular sides.

For this problem, we have the perpendicular sides as x and (6 + x) and the value of the area is 36 square centimeters.

Using equation (1), we have:

$\Rightarrow$ \[36 = \dfrac{1}{2}x(6 + x)\]

Taking 2 in the denominator to the other side and multiplying with 36 we get 72.

$\Rightarrow$ \[72 = x(6 + x)\]

Multiplying x inside the bracket and taking all terms to one side, we have:

$\Rightarrow$ \[{x^2} + 6x - 72 = 0...........(2)\]

Equation (2) is a quadratic equation and we can solve it as follows:

$\Rightarrow$ \[{x^2} + 12x - 6x - 72 = 0\]

$\Rightarrow$ \[x(x + 12) - 6(x + 12) = 0\]

$\Rightarrow$ \[(x - 6)(x + 12) = 0\]

$\Rightarrow$ \[x = 6;x = - 12\]

We have two solutions for $x$ and we must only consider the positive solution.

Hence, the value of one perpendicular side is 6 centimetres.

The value of the other perpendicular side is $(6 + x)$ which is 12 centimetres.

Hence, the answer is 6 cm and 12 cm.

Note: You might get confused with two answers for $x$, but the length of the triangle is always positive, hence, choose only the positive solution.

Complete step-by-step answer:

Let one of the perpendicular sides of the right angled triangle be $x$.

It is given that the other side is six centimeters longer than this side.

Hence, the other side is $(6 + x)$ centimeters long.

The area of a triangle is half times the product of its base and its height. If the base of the triangle is and its height is h, then the area is given by,

\[A = \dfrac{1}{2}bh...........(1)\]

For a right angled triangle, if one of its perpendicular sides is considered as a base, then the other perpendicular side will be the height of the triangle, hence, the area of the right angled triangle is half times the product of the two perpendicular sides.

For this problem, we have the perpendicular sides as x and (6 + x) and the value of the area is 36 square centimeters.

Using equation (1), we have:

$\Rightarrow$ \[36 = \dfrac{1}{2}x(6 + x)\]

Taking 2 in the denominator to the other side and multiplying with 36 we get 72.

$\Rightarrow$ \[72 = x(6 + x)\]

Multiplying x inside the bracket and taking all terms to one side, we have:

$\Rightarrow$ \[{x^2} + 6x - 72 = 0...........(2)\]

Equation (2) is a quadratic equation and we can solve it as follows:

$\Rightarrow$ \[{x^2} + 12x - 6x - 72 = 0\]

$\Rightarrow$ \[x(x + 12) - 6(x + 12) = 0\]

$\Rightarrow$ \[(x - 6)(x + 12) = 0\]

$\Rightarrow$ \[x = 6;x = - 12\]

We have two solutions for $x$ and we must only consider the positive solution.

Hence, the value of one perpendicular side is 6 centimetres.

The value of the other perpendicular side is $(6 + x)$ which is 12 centimetres.

Hence, the answer is 6 cm and 12 cm.

Note: You might get confused with two answers for $x$, but the length of the triangle is always positive, hence, choose only the positive solution.

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