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Which Side is Longer?


Date: 27 Jan 1995 12:00:21 -0500
From: Janice Cohrs
Subject: geometry

The following problem appeared on an assignment and I am having trouble 
justifying the answer key's solution.  You are given a quadrilateral, ABCD,
with diagonal DB, angle ADC = 90, angle DBC = 43, angle C = 91,
side AD = 9, side DC = 9.

Find which is longer AB or AD.

Please send me a solution with explanation at jcohrs@esu3.esu3.k12.ne.us

Thank you!


Date: 28 Jan 1995 11:51:21 -0500
From: Elizabeth Weber
Subject: Re: geometry

Hello Janice!

Draw a picture of what you've described.
 
			You are given a quadrilateral, ABCD, with diagonal DB, 
			angle ADC = 90, angle DBC = 43, angle C = 91, 
			side AD = 9, side DC = 9.

You know the measures of angle DBC and angle C.  Since the sum of the
angles of a triangle must add to 180, you know that angle BDC must be 46. 
This will let you use the rule of sines to find the length of side BC; 
which comes to about 9.5.

Now, parts of your drawing will look like this:

 A           |B  
  |          |
  |          |9.5
 9|          |
  |          |
  |__________|
D      9     C

Even if angle C measured 90 degrees, side AB would have to be longer than
9, because it would have to go a horizontal distance of 9, and an
additional vertical distance of 0.5.  Since angle C measures 91, degrees,
corner B is even further from corner A, making side AB all the longer. 
Side AB must be longer than 9;  it must be longer than side AD.

                            -Elizabeth, a math doctor
    
Associated Topics:
High School Trigonometry

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