Triangle Altitude and Area

Date: 03/07/99 at 03:07:02
From: Michelle Cheung
Subject: Altitude of a triangle

In triangle PQR, angle Q is obtuse, PQ = 11, QR = 25, and PR = 30. 
Find the altitude to line PQ; find the area of PQR.

Date: 03/07/99 at 03:29:50
From: Doctor Pat
Subject: Re: Altitude of a triangle

You are given three sides of a triangle. You can use Heron's formula 
to find the area. Once you know the area you know that for any base, 
1/2 the base times the height is also the area. 

Since the altitude to PQ is requested, let PQ be the base. The height 
you find goes perpendicular to PQ up to vertex R. If you do not know 
Heron's formula here it is:

Find the perimeter of the triangle and divide by two. This is called 
the semi-perimeter, S. In your problem it is (11+25+30)/2 = 33.
Now you need to subtract each side one at a time from S (to be brief I 
will use a, b, and c for the lengths of the three sides)

 S - a = 33 - 30 = 3
 S - b = 33 - 25 = 8
 S - c = 33 - 11 = 22.

Now use these three numbers and S, multiply them all together, and 
find the square root. This is the area.
   
 Area = sqrt(S (S - a)(S - b)(S - c))  

Now since area = 1/2 B * h  plug in this value for the area, and 11 
for the base, and solve for h, and you are done.  

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