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. 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. 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: 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/ |
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