Question 1056340
{{{drawing(400,400,-1,17,-1,17,
locate(.7,0,matrix(1,2,3,cm)),locate(9.7,0,matrix(1,2,3k,cm)),
locate(3.2,2.2,matrix(1,2,4,cm)),locate(-.1,2.4,matrix(1,2,5,cm)),
locate(15,5.5,matrix(1,2,4k,cm)), locate(7.6,5.5,matrix(1,2,5k,cm)),
triangle(0,0,3,0,3,4),  triangle(6,0,6+3sqrt(8),0,6+3sqrt(8),4sqrt(8)))}}}

The three sides of a right triangle 
<pre><b><font size = 4>
That's the one on the right.
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are proportional to another right triangle 
whose sides measure 3 cm, 4 cm, and 5 cm. 
<pre><b><font size = 4>
That's the one on the left.

The triangle on the right has sides whose lengths 
are some constant k times the lengths of the sides
of the triangle on the left. 
</pre></b></font size>
Find the perimeter of the right triangle if 
its area is 96 cm^2.
<pre><b><font size = 4>
The formula for the area of a triangle is

{{{Area}}}{{{""=""}}}{{{expr(1/2)base*height}}}

For the larger triangle on the right, the area
is given as 96 cm², the base is 3k and the height
is 4k, so we substitute:

{{{96}}}{{{""=""}}}{{{expr(1/2)3k*4k}}}

{{{96}}}{{{""=""}}}{{{6k^2}}}

{{{18}}}{{{""=""}}}{{{k^2}}}

{{{sqrt(18)}}}{{{""=""}}}{{{k}}}

{{{sqrt(9*2)}}}{{{""=""}}}{{{k}}}

{{{3sqrt(2)}}}{{{""=""}}}{{{k}}}

The perimeter is the sum of the three sides.

{{{P}}}{{{""=""}}}{{{3k + 4k + 5k}}}

{{{P}}}{{{""=""}}}{{{12k}}}

{{{P}}}{{{""=""}}}{{{12*3sqrt(2)}}}

{{{P}}}{{{""=""}}}{{{matrix(1,2,36sqrt(2),cm)}}}

Edwin</pre></b></font>