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### Properties of Equilateral Triangles

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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?
```

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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/
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Associated Topics:
High School Geometry
High School Triangles and Other Polygons
Middle School Geometry
Middle School Triangles and Other Polygons

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