Isosceles Triangle whose Ratio Area/Perimeter is a prime number

 
For example,

 

Area to Perim prime 1

 

Can you find another isosceles triangle whose sides are integers and

Area / Perimeter = p

where p is a prime number?

 
Note that

Area \; = \; b/4 \, \sqrt{4 \, a^2 - b^2}
Perimeter \; = \; (2 \, a + b)

and

b/4 \sqrt{4 \, a^2 - b^2} \; = \; p \, (2 \, a + b)

b \, \sqrt{4 \, a^2 - b^2} \; = \; 4 \, p \, (2 \, a + b)

squaring both sides,

b^2 \, (4 \, a^2 - b^2) \; = \; 16 \, p^2 \, (2 \, a + b)^2

b^2 \, (2 \, a - b) \,(2 \, a + b) \; = \; 16 \, p^2 \, (2 \, a + b)^2

b^2 \, (2a - b) \; = \; 16 \, p^2 \, (2 \, a + b)

b^2 \, 2 \, a \; - \; b^3 \; = \; 32 \, p^2 \, a \; + \; 16 \, p^2 \, b

b^2 \, 2 \, a \; - \; 32 \, p^2 \, a \; = \; 16 \, p^2 \, b \; + \; b^3

a \, (2 \, b^2 - 32 \, p^2) \; = \; (16 \, p^2 \, b + b^3)

a \; = \; (16 \, p^2 \, b + b^3) / (2 \, b^2 - 32 \, p^2)

 
 
 
 
 
 
 
 
 
 
 
 
 

Advertisements

About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s