Question

MEDAL!!! (anyone who tries, can also deserve a medal)

Question is attached... please help...

Answer 1

@phi

Answer 2

@agent0smith

Answer 3

Know the rules for similar triangles?

Answer 4

Well I know that similar triangles have corresponding angles and their sides are in proportion.

Answer 5

Yes. Another thing is that there is a ratio between the lengths of the sides. So if one side is half the size of the matching side, all other sides will be half the size. This means that finding the ratio lets you set up the conversion for any side.

Answer 6

That is where the problem comes in because I can't find the sides.

Answer 7

Which # are you looking at? 6 or 9?

Answer 8

Question 6

Answer 9

Um, they are given right on the image.

Answer 10

But I can't figure it out for some reason.

Answer 11

Well, what is the measure of the base of the larger triangle?

Answer 12

120ft?

Answer 13

yes

Answer 14

Yep. Now, what is the measure of the base of the smaller?

Answer 15

What about the other part where it had gray area of the lighthouse? Do you need to count that?

Answer 16

Ignore the objects. Look at the triangles. They are drawn to the highest points and that is all you care about.

Answer 17

The smaller figure, would it be 9ft?

Answer 18

@e.mccormick

Answer 19

Yes. Now, a proportion is an equality of ratios, so you need the rations first. What is the ratio of the bases? Well, there are two, but chooose one, small to large or large to small.

Answer 20

\[\frac{ 9 }{ 120 } = \frac{ 15 }{ x }\] ?

Answer 21

Just a question out of curiosity. Why would the base of a lighthouse be that long?

Answer 22

Yah, you saw where I was going. That is one complete proportion for that.

Answer 23

So for the second proportion, would I just use, like, the reverse?

Answer 24

It is not the base of the lighthouse. It is the distance from directly below the highest point to the observation point. This is a survey method.

And for the other, use the inverse.

Answer 25

Okay. Thank you for that. Now question number 9.

Answer 26

If point O is the observation point, then it would be the shared corener on the right end of both bases. From O you look past the peek of the "house" (shed, it is small) to the tip of the lighthouse. Becase the shed is small enought to be measured and you can measure the distance to directly below the high point of the lighthouse, this becomes a reasonable method for calculating the taller height without havng a 200+ foot crane and tape measure.

Answer 27

For 9, look at what it says abut the distance between poles. What else can you put on the drawing with that information alone?

Answer 28

So the distance between each pole is 100ft, yet the tallest pole is 50ft in height.

Answer 29

Yes. Now, without any ratios or triangles, just as marks on the line, what can you tell me about the base line with the drawing and that info?

Answer 30

The base line is 400ft. Since there are four poles, the distance between each of the poles are 100ft.

Answer 31

You are not seeing what I intend... Let me put i a different way.

Do you see that you can turn #9 into 4 versions of #6. Also, if you understand the definition of a ratio, you can do #9 very quickly.

Answer 32

I know that this can be a pretty stupid idea, but would the ratio be of 50ft to 100ft?

Answer 33

Ummm... no. That is a height to a width, BUT, one of them actually does have that ratio in there even though you are finding it the wrong way.

Answer 34

I have no idea what the ratio will be, honestly.

Answer 35

Like I said, there are different ones. I am editing the pic to make it more clear?

Answer 36

You can edit the picture as much as you want because I need to see this explicitly, and more visually.

Answer 37

Can you give me another hint?

Answer 38

The edited pic is the hint. See, there are 4 triangles.

Answer 39

waitt.... i might have got it.

would it be:
50 to 400 and x to 400?

am i close or still far away?

Answer 40

sorry i meant

50 to 400 and x to 300?

Answer 41

50 height to 400 base works iff (if and only if) the other side is also a height to base.

Answer 42

yes...

\[\frac{ 50ft(height) }{ 400ft(base)} = \frac{ xfeet(height) }{ 300ft(base) }\]

Answer 43

Your ratios just need to match up. Now, strictly speaking you don't have a ratio if you choose things from one triangle, BUT it DOES work. That is because a Proportion is a ratio to ratio and with algebra you can move them around.

So yes, that wors for finding the second height.

Answer 44

Now, there is an easier way once you understand the concept.

Answer 45

do you think i understood the subject because i may have.

Answer 46

Lets look at JUST the base ratios.

\(\dfrac{300 \text{ft base 2}}{400 \text{ft base 1}}\)

What does that simplify as?

Answer 47

If you say, "Ratio of bases in feet from smaller to larger, we do not even need most of that in the fraction. We can then write it as:

\(\dfrac{300}{400}\)

What does that simplify to?

Answer 48

a little bit :|

Answer 49

3 to 4

Answer 50

@e.mccormick

Answer 51

Exactly! So the second, smaller triangle is 3 to 4, or \(\dfrac{3}{4}\)ths the size of the large one. That means every side is \(\dfrac{3}{4}\)ths as long.

Answer 52

So you can multiply the large height by that fraction to get the red question mark height (from my edited picture).

Answer 53

See, my point is this: If you look at the numbers, they cut the large triangle into 4, similar triangles. Nut they are not just any 4 similar triangles. They used 100 foot marks on a 400 foot line. So 100, 200, 300, and 400.

\(\dfrac{100}{400}\), \(\dfrac{200}{400}\), \(\dfrac{300}{400}\), and \(\dfrac{400}{400}\)

Those look like very simple fractios to me. I use those simple fractions in simplified form to do the same thing to the side.

Take the blue one, the middle one. \(\dfrac{200}{400}\) AKA: \(\dfrac{1}{2}\). It is half as long. So it MUST be half as tall. I can do the same thing for each of these.

So yes, you can change #9 into 4 versions of #6 and solve them, OR, by understanding the ratios, you can turn them into simple fractions.

Answer 54

Thank you so much...

Answer 55

So, can you find the 3 apparent heights now?