Finding Line Segment Lengths

Date: 12/21/2001 at 18:49:50
From: Annie
Subject: Finding the length of segments

Can you help me find line segments AB and BC if points A,B,C, and D 
are collinear with B between A and C and C between B and D? 

The problems are: 

   1. AB = 2x + 32, BC = 10x + 20, AC = 112
   2. CD = 6.5x-2, BC = x+20, AB = CD, AD = 44

I would really appreciate it if you could help me with these two 
problems. 

Thank you very much.

Date: 12/25/2001 at 12:23:10
From: Doctor Ian
Subject: Re: Finding the length of segments

Hi Annie,

I would never try to solve a problem like this without drawing a 
picture first, to help me keep track of the information that the 
problem gives me. 

We know that the points are collinear, and in the order ABCD:

   A         B           C          D
   *---------*-----------*----------*


We're also given some of the segment lengths:

   A         B           C          D
   *---------*-----------*----------*
   |---------|
     2x+32
             |-----------|
                10x+20
   |---------------------|
           112

Now, what's obvious from the picture, but not from the problem 
description, is that if you add AB and BC together, you have to end up 
with AC.  So

  (2x + 32) + (10x + 20) = 112

Can you take it from here?

Let's see what the picture tells us in the second problem:


   A         B           C          D
   *---------*-----------*----------*
                         |----------|
                           6.5x-2
             |-----------|
                  x+20
   |---------|
     6.5x-2
   |--------------------------------|
                   44

So it's a pretty similar problem. 

Does this help?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/