Properties of Equilateral Triangles

Date: 07/20/98 at 01:03:48
From: Khaliah
Subject: ABC is equilateral

ABC is equilateral, and AD is one of its heights.

a) Copy the figure on your paper and write the measures of all the 
   acute angles on the figure.

b) Is ADB equal to ADC? Prove it.

c) If AB = 2, find BD and AD. Leave your results in simple 
   radical form.

d) If AB = 10, find BD and AD. Leave your results in simple 
   radical form.

e) Write a short paragraph that describes the relationship between the 
   three sides of a 30 degree-60 degree-90 degree triangle.

Where do I start with this problem? Can you explain what simple 
radical form is, and how to answer questions like this one in the 
future?

Date: 07/20/98 at 12:45:52
From: Doctor Becky
Subject: Re: ABC is equilateral

Hi Khaliah,

You should start by drawing a picture of the triangle (as the question 
says to do.) The most important thing to notice here is that triangle 
ABC is equilateral. An equilateral triangle is a triangle that has 
three equal angles and three equal sides. As you know, there are 180 
degrees in a triangle. So if you have 180 degrees divided by three 
equal angles, each angle should be 60 degrees.

For Part B, ADB and ADC are the two angles that are formed when the 
height, AD, touches the base, BC. So what is a height, anyway? 
It's a line drawn from one of the angles of a triangle so that it is 
perpendicular to the side opposite it. In other words, it forms a right 
angle with the opposite side. 

For part C, take a look at triangle ABD. What do you know about this 
triangle? You know that AB = 2, and that angle ADB is a right angle.  
You also know something about BD. Think about it. You have a triangle 
where all the sides are the same, and you are dividing it exactly in 
half with the height AD. So since all the sides are equal, meaning the 
whole side BC = 2, half of the side, BD, should just be 1. 

Now you know two sides of a right triangle. How about using the 
Pythagorean Theorem to find the other? You can probably think about 
part D in the same way.

For part E, this question is asking you to make some conclusions from 
what you learned about your triangle ABD in parts C and D.  ABD is a 
30-60-90 triangle. (60 degrees for angle B, 90 for angle D, and you 
are left with 30 for angle A.) As it turns out, when you have a 
30-60-90 triangle, there are certain relationships that always exist 
between the three sides. Can you take a guess at these? Take a look 
at your answers from parts C and D. Notice, for example, that the 
hypotenuse (side AB) is always twice the smallest side (BD) in both 
cases. Another way to try to answer this question is to use variables 
in your sides. Set side DB equal to x. What do you know about the 
other sides? By doing it with variables, you make sure the relations 
are the same for all triangles, not just the ones you are looking at. 
Don't forget that you can use the Pythagorean Theorem with variables as 
well as with numbers.

Note that if a radical is in simple radical form, you don't want 
anything under the square root sign that doesn't have to be there. 
So if you can take out a square and leave it outside of the sign, then 
do. For more information about how this is done, take a look in the 
archives. Here's the address where you'll find the answer the a 
question about simplifying radicals:

  http://mathforum.org/dr.math/problems/seguin3.html   

If you need any more help, make sure to write  back.

- Doctor Becky, The Math Forum
Check out our web site! http://mathforum.org/dr.math/