Answer:
The numbers associated with the triangle are 5.437 and 6.437 centimeters, respectively.
Step-by-step explanation:
Two numbers are consecutives, when their difference is equal to 1. The area of the triangle is determined by the following formula:
[tex]A = \frac{1}{2}\cdot n \cdot (n+1)[/tex]
[tex]A = \frac{1}{2}\cdot (n^{2}+n)[/tex]
[tex]A = \frac{n^{2}}{2}+\frac{n}{2}[/tex]
[tex]n^{2}+n = 2\cdot A[/tex]
Where [tex]n[/tex] is the shortest length of the triangle, measured in centimeters.
And we get the following second-order polynomial:
[tex]n^{2}+n -2\cdot A = 0[/tex] (1)
If we know that [tex]A = 17.5\,cm^{2}[/tex], then the shortest length of the triangle is:
[tex]n_{1} = 5.437\,cm[/tex] and [tex]n_{2} \approx -6.437\,cm[/tex]
Since length is a positive variable, the only possible solution is:
[tex]n\approx 5.437\,cm[/tex]
Then, the numbers associated with the triangle are 5.437 and 6.437 centimeters, respectively.
