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

Hello, i am not really good at math but try multiplying...

Answer 2

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

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
`(bh)/(2)`

We know that the area is
`102(cm^(2))` ⇒

`(bh)/(2)=102(cm^(2))`

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

`(bh)/(2)=102(cm^(2))`

`((2h-7)(h))/(2)=102(cm^(2))`

Rearraging the expression :

`((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`