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/
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