Trigonometry Story Problem

Date: 06/04/2002 at 22:33:13
From: Kyle
Subject: Trig. Problem of the Week 

I need help on this problem. I went to the teacher but he said that I 
should know how to do this but I don't! Could you please help me?

In Coordinateville, all roads run east-west or north-south, so a map 
of the town looks like a coordinate plane with all roads intersecting 
at right angles. The superintendent of schools wants to create two 
new roads in order to have a more direct route from the middle school 
to the high school and from the superintendent's office to the high 
school. The current locations of the high school, middle school and 
superintendent's office can be seen in the map below. 

Looking at the map, you can see that to go from the high school to 
the middle school, you need to travel 4 blocks south and 3 blocks 
east. To go from the high school to the superintendent's office, you 
travel 7 blocks east and 4 blocks south. 

If two new roads are constructed, one from the middle school to the 
high school and one from the superintendent's office to the high 
school, as shown in the diagram below, what will be the measure, in 
degrees, of the angle formed by their intersection? 

Pleeeeeeeease get back to me ASAP.
Thank you.

Date: 06/05/2002 at 08:55:55
From: Doctor Ian
Subject: Re: Trig. Problem of the Week

Hi Kyle,

If I understand your description correctly, the map looks 
like this: 

               7
    H ---------------------B
    |                      |
    |                      | 4
  4 |                      |
    |                      |
    A---------M            S
         3

where H is the high school, M is the middle school, and S is the 
superintendent's office.  Have I got it right? 

If so, then the tangent of angle MHA is 3/4, which means you can 
look up angle MHA.  And the tangent of angle SHB is 4/7, so you 
can look up angle SHB. 

Can you take it from here?

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