Answer:
The length of the base is [tex]b=17cm[/tex]
Step-by-step explanation:
We start by writing the equation of the base :
Given that the base of the triangle is seven less than twice its height we can write that
[tex]b=2h-7[/tex] (I)
Where ''b'' is the base and ''h'' is the height.
Now, the area of a triangle is [tex]\frac{bh}{2}[/tex]
We know that the area is [tex]102(cm^{2})[/tex] ⇒
[tex]\frac{bh}{2}=102(cm^{2})[/tex]
Now we can replace ''b'' by the expression in (I) ⇒
[tex]\frac{bh}{2}=102(cm^{2})[/tex]
[tex]\frac{(2h-7)(h)}{2}=102(cm^{2})[/tex]
Rearraging the expression :
[tex]\frac{(2h-7)(h)}{2}=102(cm^{2})[/tex]
[tex](2h-7)(h)=204(cm^{2})[/tex]
[tex]2(h^{2})-7h=204(cm^{2})[/tex]
[tex]2(h^{2})-7h-204(cm^{2})=0[/tex]
We need the values of ''h'' that satisfy the equation.
We can use the quadratic formula with
[tex]a=2\\b=-7\\c=-204[/tex]
The quadratic formula will be
[tex]h1=\frac{-b+\sqrt{b^{2}-4ac}}{2a}[/tex]
and [tex]h2=\frac{-b-\sqrt{b^{2}-4ac}}{2a}[/tex]
Using [tex]a=2\\b=-7\\c=-204[/tex] in the expression of h1 and h2 we find that
[tex]h1=12\\h2=-8.5[/tex]
Given that ''h'' is a length ⇒ [tex]h\geq 0[/tex]
We conclude that h1 = 12 cm is the correct value of ''h''
With this value of ''h'' we go to the expression of ''b'' ⇒
[tex]b=2h-7\\b=2(12)-7\\b=24-7\\b=17[/tex]
The value of the base ''b'' that satisfies the problem is [tex]b=17cm[/tex]
