1. hexagon differences

I attached a file since the last time I tried to just show the pic it didn't go through. I think I start with
P/T=4/5
R/T=2/5
How do I figure out the difference of n amd m?

In the following figure, the similarity ratio of regular hexagon P to regular hexagon T is 4/5 and the similarity ratio of regular hexagon R to T is 2/5. What is the difference of n and m?

2. Originally Posted by sanee66
I attached a file since the last time I tried to just show the pic it didn't go through. I think I start with
P/T=4/5
R/T=2/5
How do I figure out the difference of n amd m?

In the following figure, the similarity ratio of regular hexagon P to regular hexagon T is 4/5 and the similarity ratio of regular hexagon R to T is 2/5. What is the difference of n and m?
Here's one way

we have: $\displaystyle \frac {P}{T} = \frac {4}{5}$

$\displaystyle \Rightarrow T = \frac {5}{4} P$ ...................(1)

we also have: $\displaystyle \frac {R}{T} = \frac {2}{5}$

$\displaystyle \Rightarrow T = \frac {5}{2} R$ ...................(2)

since (1) = (2) = T, we can equate the expressions.

$\displaystyle \Rightarrow \frac {5}{4} P = \frac {5}{2} R$

$\displaystyle \Rightarrow P = 2R$

$\displaystyle \Rightarrow m + 12 = 2(6) \implies \boxed { m = 0 }$

But we have that: $\displaystyle T = \frac {5}{4} P$

$\displaystyle \Rightarrow n - 4 = \frac {5}{4} \cdot 12 = 15 \implies \boxed { n = 19 }$

Thus, $\displaystyle n - m = 19 - 0 = 19$

3. Hello, sanee!

A slightly different approach . . .

We have: .$\displaystyle \frac{P}{T} = \frac{4}{5}\quad\Rightarrow\quad \frac{m+12}{n-4} = \frac{4}{5}\quad\Rightarrow\quad 5m \:=\:4n-76$ . [1]

We have: .$\displaystyle \frac{R}{T} = \frac{2}{5}\quad\Rightarrow\quad\frac{6}{n-4} = \frac{2}{5}\quad\Rightarrow\quad\boxed{n = 19}$ . [2]

Substitute [2] into [1]: .$\displaystyle 5m \:=\:4(19) - 76\quad\Rightarrow\quad\boxed{m\,=\,0}$

Therefore: .$\displaystyle n - m \;=\;19 - 0 \;=\;\boxed{\boxed{19}}$