Since the triangle is a right triangle we can draw the triangle like so, calling x the distance corresponding to the base of the triangle and x+5 the height oh the triangle.
now use the formula of the area
[tex]A=\frac{bh}{2}[/tex]we now the area and the corresponding variables for the base and height, replace them on the formula
[tex]52=\frac{x\cdot(x+5)}{2}[/tex]clear for the x
[tex]104=x^2+5x[/tex]equal the equation to 0 and solve the quadratic equation or by factorization
[tex]x^2+5x-104=0[/tex][tex](x-8)\cdot(x+13)=0[/tex]by factorization we know that roots are x=-13 and x=8, since we are talking about a distance the base of the triangle must be 8 ft.
check the answer with the formula of the area
[tex]\begin{gathered} A=\frac{bh}{2} \\ A=\frac{8\cdot(8+5)}{2} \\ A=\frac{8\cdot13}{2} \\ A=52 \\ 52=52 \end{gathered}[/tex]The procedure is correct, the base of the triangle is 8 ft and the height is 13 ft.
