Associated Topics || Dr. Math Home || Search Dr. Math

Will the Tree Hit the House?

```
Date: 05/18/99 at 22:08:28
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

Billy
```

```
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.

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:

B
/\
/    \
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
http://mathforum.org/dr.math/
```
Associated Topics:
High School Geometry
High School Practical Geometry
High School Triangles and Other Polygons
High School Trigonometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search