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Will the Tree Hit the House?

Date: 05/18/99 at 22:08:28
From: Billy Madison
Subject: How to find the sides of the triangle?

Dr. Math,

There is a tree out in front of our yard. It is tilted slightly at 70 
degress. Our house is 66 1/2 feet away from the tree. The angle from 
our house to the top of the tree is 40 degrees.  

My family is worried that if we have a big storm the tree will fall 
and hit the house.  I read somewhere that if you have 2 angles and a 
side you can figure out the dimensions of the triangle. I haven't 
taken trig yet, so could you please help me out?  Thanks.


Date: 05/19/99 at 09:00:24
From: Doctor Rick
Subject: Re: How to find the sides of the triangle?

Hi, Billy. Nice question.

You don't need trigonometry to answer your basic question, which is, 
could the tree hit the house? There is a theorem in geometry (it's 
Euclid's Proposition 19) that says: "In any triangle the side 
opposite the greater angle is greater." Let's see how we can use 
this. Here is a figure:

       /    \
 TREE /        \
     /            \
    /                \
   / 70              40 \
  /________________________\ HOUSE
 A           66.5'          C

I am assuming that the tree is tilted toward the house. Now, what is 
the angle at the top of the triangle, angle B? Since the sum of the 
angles in a triangle is 180 degrees, that angle is 70 degrees.

Now we can use the theorem. The distance to the house, AC, is opposite 
a 70-degree angle. The height (or rather length) of the tree, AB, is 
opposite a 40-degree angle. The side opposite the greater angle is 
greater, so AC is greater than the length of the tree AB. The tree 
cannot hit the house.

You are correct that trigonometry can be used to find the actual 
length of the tree. We use the Law of Sines, which puts numbers into 
Euclid's theorem.

  AB/sin(C) = AC/sin(B)

  AB = AC * sin(C)/sin(B)
     = 66.5' * sin(40)/sin(70)

The sine of 40 degrees, abbreviated sin(40), is 0.642787610 from my
calculator. The sine of 70 degrees is 0.939692621. Therefore

  AB = 66.5' * 0.642787610 / 0.939692621
     = 45.5'

So you have 21 feet to spare.

- Doctor Rick, The Math Forum   
Associated Topics:
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons
High School Trigonometry

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