Question

Quadrilateral ABCD in the figure below represents a scaled-down model of a walkway around a historic site. Quadrilateral EFGH represents the actual walkway. ABCD is similar to EFGH.

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What is the total length, in feet, of the actual walkway?

Answer 1

@jim_thompson5910

Answer 2

you need to find these segment lengths:

FG
GH
HE

then add up all the sides of the larger figure

Answer 3

and how do I find those?

Answer 4

by setting up ratios after using the fact that EFGH ~ ABCD

so for instance

EF/AB = FG/BC
45/5 = x/4
x = ???

Answer 5

we'd cross multiply, right?

Answer 6

yes

Answer 7

okay, so
45*4 = 180
then 180 / 5 = x
so x = 36?

Answer 8

yep

Answer 9

FG = 36

Answer 10

and how do we set up the rest? I can do the math for it, but setting them up confuses me.

Answer 11

if you're wondering as to how I set up EF/AB = FG/BC from EFGH ~ ABCD

take note of the letters highlighted in red

this similarity statement

\[\Large \color{red}{\text{EF}}\text{GH} \sim \color{red}{\text{AB}}\text{CD}\]

forms this ratio

\[\Large \frac{\color{red}{\text{EF}}}{\color{red}{\text{AB}}} = \frac{\text{FG}}{\text{BC}}\]

they are the first pair of letters in EFGH and ABCD respectively

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Now look at the blue letters highlighted

this similarity statement

\[\Large \text{E}\color{blue}{\text{FG}}\text{H} \sim \text{A}\color{blue}{\text{BC}}\text{D}\]

forms this ratio

\[\Large \frac{\text{EF}}{\text{AB}} = \frac{\color{blue}{\text{FG}}}{\color{blue}{\text{BC}}}\]

now we're looking at the middle pair of letters for each EFGH and ABCD

Answer 12

let me know if that makes sense or not

Answer 13

Okay, so to form one for GH, which we need to find...
We could use CD / GH and BC / FG?

Answer 14

Now look at the purple letters highlighted

\[\Large \text{EF}\color{purple}{\text{GH}} \sim \text{AB}\color{purple}{\text{CD}} \rightarrow \frac{\text{EF}}{\text{AB}} = \frac{\color{purple}{\text{GH}}}{\color{purple}{\text{CD}}}\]

we're looking at the last two letters for each EFGH and ABCD

Answer 15

We keep the EF / AB for all of them? Just change out the last one?

Answer 16

since EF is the only thing known about EFGH, it makes sense to keep EF

so keep EF/AB

Answer 17

okay, so the one for GH would look like this:
45 / 5 = x / 4?
which would give you the same value for GH as we had for FG?

Answer 18

sure after we discover what FG, GH, etc are, we can use other ratios on the left side. But the path of least resistance in my opinion is to keep EF/AB

Answer 19

yes now solve `45 / 5 = x / 4` to get x = 36

FG = 36
GH = 36

Answer 20

FG = GH makes sense because BC = CD = 4

Answer 21

and for HE, it would look like this:
45 / 5 = x / 3?

Answer 22

yep you got it

Answer 23

which would be 45 * 3 = 135
135 / 5 = x
x = 27

so HE = 27

Answer 24

then we add the 36 + 36 + 27 together and that's the final answer, the total for the entire 'walkway'?

Answer 25

don't forget about EF

Answer 26

Oh, I didn't remember that lol

So EF would look like this:

45 / 5 = x / 5
45*5 = 227
227 / 5 = 45

so EF = 45?

Answer 27

because EF corresponds to AB

Answer 28

EF is already known (it's shown on the figure as 45)

there's no need to solve for what is already known

Answer 29

I guess I just like doing the math way too much lol sorry

so then 36 + 36 + 27 + 45 would give us the total answer?
which would be 144?

Answer 30

hey nothing wrong with that lol

yes
EF + FG + GH + HE = 45 + 36 + 36 + 27 = 144

Answer 31

okay, great!
that was actually kind of fun lol

thanks :)

Answer 32

I'm glad it makes more sense. You're welcome