Question

The area of a triangle is 243 in. If both its length and width are reduced to one- third their original length, what would its new area be?

Answer 1

The original area of the triangle is:

`A_{\text{original}}=243in^2_{}`

Let's call the original length "l" and the original width "w".

And now we remember the formula to calculate the area of a triangle using the length (or base) and the width (or height):

`A=(l* w)/(2)`

So for the original triangle:

`(l* w)/(2)=243in^2`

Now, since we are told that the length and width are reduced to 1/3 their orifinal length, the new length is:

`(l)/(3)`

And the new width is:

`(w)/(3)`

And using this length and width, the area of the new triangle will be calculated as follows:

`A_{\text{new}}=((l)/(3)*(w)/(3))/(2)`

Solving the operations in the numerator:

`\begin{gathered} A_{\text{new}}=((l* w)/(3*3))/(2) \\ \\ A_{\text{new}}=((l* w)/(9))/(2) \end{gathered}`

We can re-write this expression as follows:

`A_{\text{new}}=(1)/(9)((l* w)/(2))`

And we know that for this triangle the expression in parentheses is equal to:

`(l* w)/(2)=243in^2`

Substituting this into the expression to find the new area:

`\begin{gathered} A_{\text{new}}=(1)/(9)(243in^2) \\ \\ A_{\text{new}}=27in^2 \end{gathered}`

Answer:

the new area is 27 in^2.