Question

Find the area of the triangle below.

Answer 1

To find the area of a right triangle, we can use the formula:

Area = (base * height) / 2

In this case, the base of the triangle is x, and the height is the side to the left, which is x+6.

Therefore, the area of the right triangle can be calculated as:

Area = (x * (x+6)) / 2

Simplifying this expression:

Area = (x^2 + 6x) / 2

Area = (1/2)x^2 + 3x

So, the area of the right triangle with the given side lengths would be (1/2)x^2 + 3x.

Answer 2

54 square units

Explanation:

To find the area of the right triangle, we first need to determine the value of x by using Pythagoras Theorem.

`\boxed{\begin{array}{l}\underline{\sf Pythagoras \;Theorem} \\\\\large\text{$a^2+b^2=c^2$}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ and $b$ are the legs of the right triangle.}\\\phantom{ww}\bullet\;\textsf{$c$ is the hypotenuse (longest side) of the right triangle.}\\\end{array}}`

From observation of the given right triangle:

• 
  `a = x`
• 
  `b = x + 3`
• 
  `c = x + 6`

Substitute these expressions into the formula:

`x^2+(x+3)^2=(x+6)^2`

Solve for x:

`\begin{aligned}x^2+x^2+6x+9&=x^2+12x+36\\2x^2+6x+9&=x^2+12x+36\\x^2-6x-27&=0\\x^2-9x+3x-27&=0\\x(x-9)+3(x-9)&=0\\(x+3)(x-9)&=0\\\\x+3&=0\implies x=-3\\x-9&=0\implies x=9\end{aligned}`

Therefore, the two possible values of x are x = -3 and x = 9.

As length is positive, the only valid solution is x = 9.

The area of a right triangle is equal to half the product of the lengths of its two legs.

The legs of the given right triangle are:

• 
  `x=9\;\sf units`
• 
  `x + 3=9+3=12\;\sf units`

Therefore, the area of the triangle can be expressed as:

`\begin{aligned}\textsf{Area}&=(1)/(2)\cdot 9 \cdot 12\\\\&=(9)/(2) \cdot 12\\\\&=(9\cdot 12)/(2)\\\\&=(108)/(2)\\\\&=54\; \sf square\;units\end{aligned}`

Therefore, the area of the given right triangle is:

`\Large\boxed{\boxed{\sf Area=54\;square\;units}}`