Answer:
Length of base of triangle = 12 meters
Step-by-step explanation:
Let the base of the triangle be = [tex]x[/tex] meters
The height of triangle is 5 less than its base.
So, height of the triangle is given by = [tex]x-5[/tex]
Area of triangle = [tex]\frac{1}{2} \times base\times height[/tex]
⇒ [tex](\frac{1}{2} \times x\times (x-5))\ m^2[/tex]
⇒ [tex]\frac{x^2-5x}{2}\ m^2[/tex] [Using distribution.]
Area of triangle given = [tex]42\ m^2[/tex]
Thus we have:
[tex]\frac{x^2-5x}{2}=42[/tex]
Multiplying both sides by 2 to remove fraction.
[tex]2\times\frac{x^2-5x}{2}=42\times 2[/tex]
[tex]x^2-5x=84[/tex]
Subtracting both sides by 84.
[tex]x^2-5x-84=84-84[/tex]
[tex]x^2-5x-84=0[/tex]
We can now solve quadratic using formula.
[tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Plugging in values from the quadratic equation to solve for [tex]x[/tex]
[tex]x=\frac{-(-5)\pm\sqrt{(-5)^2-4(1)(-84)}}{2(1)}[/tex]
[tex]x=\frac{5\pm\sqrt{25+336}}{2}[/tex]
[tex]x=\frac{5\pm\sqrt{361}}{2}[/tex]
[tex]x=\frac{5\pm19}{2}[/tex]
So,
[tex]x=\frac{5+19}{2}[/tex] and [tex]x=\frac{5-19}{2}[/tex]
[tex]x=\frac{24}{2}[/tex] and [tex]x=\frac{-14}{2}[/tex]
[tex]x=12[/tex] and [tex]x=-7[/tex]
Since length cannot be negative, we take [tex]x=12[/tex] meters as length of base of triangle. (Answer)
