Question

The sides of a triangle measure 4,6, and 7. If the shortest side of a similar triangle is 12, what is the perimeter of the larger triangle?

Answer 1

Given that both triangles are similar, then their corresponding sides satisfy a proportion.

The next relation must be satisfied:

`(AB)/(DE)=(BC)/(EF)=(AC)/(DF)`

Let's suppose that AB and DE are the shortest sides of each triangle, then:

`\begin{gathered} (AB)/(DE)=(4)/(12)=(1)/(3) \\ or \\ 3AB=DE \end{gathered}`

Also,

`\begin{gathered} 3BC=EF \\ 3AC=DF \end{gathered}`

The perimeter of the smaller triangle is,

`\begin{gathered} P=AB+BC+AC \\ P=4+6+7 \\ P=17 \end{gathered}`

The perimeter of the larger triangle is,

`\begin{gathered} P=DE+EF+DF \\ P=3AB+3BC+3AC \\ P=3(AB+BC+AC) \end{gathered}`

But, AB + BC + AC is the perimeter of the smaller triangle, which is 17, then the perimeter of the larger triangle is 3*17 = 51