The person and his shadow make a right triangle as well as the tree and it's shadow. they will be similar right triangles containing angles that are equal. I'm in similar triangles the angles are proportional so a ratio could be used to determine the shadow length. this ratio is:
25/5 = x/15
(Notice that both the height of the person and
the height of the tree height of the tree are on
the bottom because these would be similar
sides and the same for the shadows with both
on top. this could easily have been switched
with the shadows on bottom and heights on
top like:
5/25 = 15/x
however I noticed the 25/5 could easily be
reduced. this eliminated the need for cross
multiplication.)

The 25/5 can be reduced to 5:
5 = x/15
and then multiply both sides by 15 and you get:
x = 75
so the answer is 75 feet long.

this can be checked various ways. using trigonometry we have the opposite and adjacent sides so tangent could be used to find the angle between the shadow and the hypotenuse. this is:
tan (x) = opposite/adjacent
opposite = height
adjacent = shadow
so:
tan (x) = 5/25 for person
tan (x) = 15/75 for tree
these equations both reduce to:
tan (x) = 1/5
And of both equations are the same then the angLee are equal creating similar triangles and a correct answer

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3 years ago