Question

Based on her findings, how much rope does she need? (Hint: Solve for h using the Geometric Mean Theorem).

Answer 1

In the figure, the large triangle and the two small triangles are all similar.
That means the ratioof 6:12 is equal to the ratio 12:(h-6)

Answer 2

So, this is like a geometric mean
\[12^2 = 6(h-6)\]

Answer 3

So I solve it like a algebra equation?

Answer 4

Yes. The geometric mean thingie is just saying that
\[\frac{12}{h-6} = \frac{6}{12}\]

Answer 5

12 is the geometric mean of 6 and h-6

Answer 6

Alright, the equation makes sense :) But I'm confused on how to solve the equation because of the h-6.

Answer 7

Well, \[12 \times 12 = 6(h-6) = 6h - 36 \]

Answer 8

can you solve for h

Answer 9

Do I divide both sides by 6-h ?

Answer 10

No, the next step is to add 36 to both sides

Answer 11

So I add 36 to 144 ?

Answer 12

Yes

Answer 13

\[144 + 36 = 180 = 6x - 36 + 36 = 6x\]

Answer 14

Do I divide 6x from both sides?

Answer 15

Divide both side by 6, not 6x

Answer 16

So I divide 6x by 6 ?

Answer 17

Yep, and 180 too

Answer 18

\[30=1x-36=36=6x\] So it looks like this now?

Answer 19

I mean +36

Answer 20

30=1x−36+36=6x

Answer 21

We had
\[144 + 36 = 180 = 6x - 36 + 36 = 6x\]
we divide 180 and 6x by 6 and obtain
\[\frac{180}{6} = \frac{6x}{6}\]
\[30 = x\]

Answer 22

A couple things. 1. I switched to x by mistake, it's supposed to be h. 2. We should substitute the answer into the original equation to check.

Answer 23

So x=30
So H=30?

Answer 24

We would still get 30 for H~ Am I right?

Answer 25

Original problem was
\[\frac{12}{h-6} = \frac{6}{12}\]
We solved this and found h = 30
To check that, we compute h-6 = 24 and see whether the ratio looks correct:
\[\frac{12}{24} =\frac{6}{12}\]
And it does, both sides equal 1/2

Answer 26

Thank you so much :)

Answer 27

yw