Ratio of Sides and Ratio of Areas

Date: 02/11/99 at 00:45:40
From: Danny Bradley
Subject: High school Geometry

The corresponding sides of two similar triangles are in the ratio of 
1:7. What is the ratio of their areas?

Date: 02/11/99 at 18:17:50
From: Doctor Pat
Subject: Re: High school Geometry

Danny,

When two triangles are similar, if a pair of corresponding sides is in 
the ratio of 1:7 then every other pair of corresponding LENGTHS will be 
in the ratio 1:7. The bases are 1:7, the heights are 1:7, etc. Now 
let's find the areas. 

Call one triangle's area A and the other triangle's area a. Call the 
base of the larger triangle B, and the smaller triangle's base b. Call 
the height of the larger triangle H, and the smaller triangle's height 
h.   

So the area of the big triangle A = .5 B H, and the area of the 
smaller triangle is a = .5 b h. Now B = 7b and H = 7h, so 

   A =.5(7b*7h) = .5 49 bh   

That makes the ratio 

    A    .5 * 49 * b h
   --- = ------------- 
    a    .5      * b h

since the .5 bh all cancels out, we are left with A/a = 49:1. You might 
notice that this is the square of the ratio of the lengths. You can 
convince yourself that this always works by drawing two squares (all 
squares are similar) with different sides. The ratio of the areas is 
ALWAYS the square of the ratios of the sides.  

Good luck,

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