SOLUTION: You are on a jungle expedition and come to a raging river. You need to build a bridge across the river. You spot a tall tree directly across from you on the opposite bank (point A)

Algebra ->  Triangles -> SOLUTION: You are on a jungle expedition and come to a raging river. You need to build a bridge across the river. You spot a tall tree directly across from you on the opposite bank (point A)      Log On


   

Question 1041156: You are on a jungle expedition and come to a raging river. You need to build a bridge across the river. You spot a tall tree directly across from you on the opposite bank (point A). You place a pole in the ground to mark the position directly across from the tree. From the spot you are standing, you walk directly downstream 16 feet and place another pole (point C). You keep walking down stream and mark a third spot with a pole that is 7 feet from your last pole (point D). Turning perpendicular from the last pole, you walk away from the river 9 feet to another position and place a fourth pole (point E).
https://byuis.brainhoney.com/Resource/46824920,0/Assets/Media/Images/41.7-HOT2-River.jpg
a. Determine if the two triangles are similar. Explain which similarity theorem you used and why.
b. Use proportions to calculate the distance across the river.
A large tree is close to point C and we could chop it down at an angle to reach point A and safely cross the river.
c. How tall does the tree need to be to span the river from point C to A?
Thanks in advance!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!


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Part A)

Statement 1: Angle ABC = angle CDF

Reason 1: both are 90 degree angles. All 90 degree angles (aka right angles) are congruent to each other

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Statement 2: Angle ACB = angle ECD

Reason 2: these two angles are vertical angles

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Statement 3: triangle ABC is similar to triangle EDC
Reason 3: the angle angle (AA) similarity theorem


The statements (1 through 3), with their coupled reasons, prove that the two triangles are similar
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Part B)

Because the triangles are similar (as shown in part A), this means that the corresponding sides form ratios which form a proportion. That proportion would be

AB%2FBC+=+ED%2FDC

we know the following

  • BC = 16
  • ED = 9
  • DC = 7


based on the information given in the problem. We want to find the value of AB (length of segment AB)

So...

AB%2FBC+=+ED%2FDC


AB%2F16+=+ED%2FDC Plug in BC = 16


AB%2F16+=+9%2FDC Plug in ED = 9


AB%2F16+=+9%2F7 Plug in DC = 7


7%2AAB+=+16%2A9 Cross multiply


7%2AAB+=+144


%287%2AAB%29%2F7+=+144%2F7


AB+=+20.5714285714286 This is approximate

The approximate distance from A to B is 20.5714285714286 feet.

Side Note: Because points A and B are not right on the river edge, the width of the river itself is slightly less than 20.5714285714286 feet, but this is a good approximate. It's better to overestimate than underestimate in this case.
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Part C)

They now want us to find the distance from A to C. Notice how ABC is a right triangle with legs of AB and BC. The hypotenuse is AC

AB = 20.5714285714286 (from part b)
BC = 16 (given in problem)

Let's use the pythagorean theorem to find AC

%28AB%29%5E2+%2B+%28BC%29%5E2+=+%28AC%29%5E2


%2820.5714285714286%29%5E2+%2B+%2816%29%5E2+=+%28AC%29%5E2


423.183673469389+%2B+256+=+%28AC%29%5E2


679.183673469389+=+%28AC%29%5E2


%28AC%29%5E2+=+679.183673469389


sqrt%28%28AC%29%5E2%29+=+sqrt%28679.183673469389%29


AC+=+26.0611525736947

So the distance from A to C is 26.0611525736947 feet approximately. The tree needs to be at least 26.0611525736947 feet tall for the bridge to be safely long enough to cross.