Drawing Diagrams

Date: 08/02/98 at 06:51:18
From: Emma
Subject: Drawing diagrams

Hi, 

In my extension maths class we have been given some problems to work 
on. I think that I will be able to solve them; however, at the moment I 
cannot draw them out. I have spent hours trying to draw them out on 
paper and with my computer program Cabri. I would be grateful if you 
could help draw them out for me. Is there a way to put 
it on a web page where I can go and see how the correct diagrams would 
look?

The first one I am having trouble is about congruent triangles. 
In a quadrilateral ABCD, BC = AD, angle ABC = 150 degrees and angle 
BAD = 70 degrees. E is a point on the diagonal BD such that angle 
EAD = 20 degrees. It is also given that angle BDA = 30 degrees. Prove 
that CE = BD.

The second one is: In a quadrilateral ABCD, AD is parallel to BC and 
angle ABD = 45 degrees. X and Y are the feet of the perpendiculars 
dropped from D on the sides AB and BC respectively, so that X is 
between A and B and Y is between B and C. The line YX meets and 
extended line DA at point in Z. Prove that ZD = YD.

All I need really is the correct diagrams. Maybe with your more 
expansive maths knowledge you can see how to draw these diagrams 
correctly. Thanks for you time!

Date: 08/02/98 at 23:32:24
From: Doctor Jaffee
Subject: Re: Drawing diagrams

Hi Emma,

It is possible to put diagrams on a Web site, but we'll start with a 
verbal explanation that may help.

First of all, in problem 1, I would suggest that you start with a 150 
degree angle and label the vertex B. Pick a point on one ray and call 
it A. Next draw an 80 degree angle, ABX. (Draw the angle in the 
interior of the 150 degree angle and I'll explain the 80 degrees 
later).  

Next draw angle BAE measuring 50 degrees so that E is on the ray BX.
Follow that with the angle EAD measuring 20 degrees so that D is on 
the ray BX.

You can now locate the point C on the original 150 degree angle by 
measuring the length of AD and marking off that distance on BC.  
Finally, connect C and D and you have your diagram.

Now, when I originally made the drawing myself, I guessed at the 
measure of angle ABD, but after I was finished I was able to calculate 
that it had to be 80 degrees. So, I started over and redrew the 
diagram with that 80 degree angle.

My guess is that one of the reasons you are having so much trouble 
with these diagrams is that you are trying to draw them too exactly.  
Try just drawing approximations and labelling the sides and angles to 
indicate measures. Then as you learn more about the diagram, you can 
revise it and get a more accurate drawing. I know this technique has 
helped a lot of students at the school where I teach.

Let's take a look at the second one. Start off by drawing two 
horizontal lines, the lower one being the line AE.  Locate B on the 
upper line a little to the right of A, then draw the 45 degree angle 
ABD such that D is on AE. Locate C on the upper line so that is to the 
right of D. Now if you draw DY perpendicular to BC with Y on the line 
BC, Y will be between B and C.

Furthermore, if you draw DX perpendicular to AB with X on AB, X will 
be between A and B. Finally, the line YX will intersect the line AD at 
Z, and Z will not be between A and D.

I hope these verbal explanations help. Now that you have read through 
them, here are the accompanying diagrams to help you with anything 
that might not have been clear.

Question 1:

              

Question 2:

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