Question

The base of the triangle is seven less than twice its height. If the area of a triangle is 102cm², find the length of the base.

Answer 1

Answer:

The length of the base is $b=17cm$

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

$b=2h-7$ (I)

Where ''b'' is the base and ''h'' is the height.

Now, the area of a triangle is $\frac{bh}{2}$

We know that the area is $102(cm^{2})$ ⇒

$\frac{bh}{2}=102(cm^{2})$

Now we can replace ''b'' by the expression in (I) ⇒

$\frac{bh}{2}=102(cm^{2})$

$\frac{(2h-7)(h)}{2}=102(cm^{2})$

Rearraging the expression :

$\frac{(2h-7)(h)}{2}=102(cm^{2})$

$(2h-7)(h)=204(cm^{2})$

$2(h^{2})-7h=204(cm^{2})$

$2(h^{2})-7h-204(cm^{2})=0$

We need the values of ''h'' that satisfy the equation.

We can use the quadratic formula with

$a=2\\b=-7\\c=-204$

The quadratic formula will be

$h1=\frac{-b+\sqrt{b^{2}-4ac}}{2a}$

and $h2=\frac{-b-\sqrt{b^{2}-4ac}}{2a}$

Using $a=2\\b=-7\\c=-204$ in the expression of h1 and h2 we find that

$h1=12\\h2=-8.5$

Given that ''h'' is a length ⇒ $h\geq 0$

We conclude that h1 = 12 cm is the correct value of ''h''

With this value of ''h'' we go to the expression of ''b'' ⇒

$b=2h-7\\b=2(12)-7\\b=24-7\\b=17$

The value of the base ''b'' that satisfies the problem is $b=17cm$

Answer 2

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