In a triangle ABC, measure of angle B is 90 degrees. AB is 3x-2 units and BC is x+3. If the area of the triangle is 17 sq cm, form an equation in terms of x and solve it.​

Accepted Solution

Answer:[tex]x=\frac{8}{3}\ cm[/tex] Step-by-step explanation:we know thatThe area of the right triangle ABC is equal to[tex]A=\frac{1}{2}(AB)(BC)[/tex]we have[tex]A=17\ cm^2[/tex][tex]AB=(3x-2)\ cm[/tex][tex]BC=(x+3)\ cm[/tex]substitute the values[tex]17=\frac{1}{2}(3x-2)(x+3)[/tex][tex]34=(3x-2)(x+3)[/tex][tex]34=3x^2+9x-2x-6[/tex][tex]3x^2+7x-6-34=0[/tex][tex]3x^2+7x-40=0[/tex]The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to [tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex] in this problem we have [tex]3x^2+7x-40=0[/tex]so [tex]a=3\\b=7\\c=-40[/tex] substitute in the formula [tex]x=\frac{-7(+/-)\sqrt{7^{2}-4(3)(-40)}} {2(3)}[/tex] [tex]x=\frac{-7(+/-)\sqrt{529}} {6}[/tex] [tex]x=\frac{-7(+/-)23} {6}[/tex] [tex]x=\frac{-7(+)23} {6}=\frac{16}{6}=\frac{8}{3}[/tex] [tex]x=\frac{-7(-)23} {6}=-5[/tex] thereforeThe solution is[tex]x=\frac{8}{3}\ cm[/tex]