```Question 276490
<pre><font size = 4 color = "indigo"><b>
{{{drawing(360,400,-3,15,-3,17, locate(6,0,14), locate(9.5,7,15),
locate(1,6,13),
triangle(0,0,14,0,5,12) )}}}

Let's draw a rectangle in which that triangle is circumscribed:

{{{drawing(360,400,-3,15,-3,17, locate(6,0,14), locate(9.5,7,15),
locate(1,6,13), rectangle(0,0,14,12),
triangle(0,0,14,0,5,12) )}}}

Let's draw an altitude h of that triangle

{{{drawing(360,400,-3,15,-3,17, locate(8,0,14-x), locate(2,0,x), locate(9.5,7,15),
locate(1,6,13), rectangle(0,0,14,12), triangle(0,0,5,0,5,12), locate(5.3,6,h),
triangle(0,0,14,0,5,12) )}}}

Now you can see that the original triangle's area is half of the
large rectangle's area because each of the two right triangles w
which the altitude divides the original triangle into are congruent
to the right triangles above it with which it shares a hypotenuse.

This would be true regardless of which side of the original
triangle we chose for a side of the rectangle in which it is
inscribed.  So we know that the sheet of paper would have to
have twice the area of the triangle.

Since the base of that triangle is 14, we will call the left part of
that base x, then the right part of it is 14-x.

Applying the Pythagorean theorem to both right triangles into which
the original triangle is split we have this system:

{{{system(x^2+h^2=13^2,(14-x)^2+h^2=15^2)}}}

{{{system(x^2+h^2=169,(14-x)^2+h^2=225)}}}

Simplifying the second one:

{{{(14-x)^2+h^2=225)}}}

{{{196-28x+x^2+h^2=225}}}

{{{x^2+h^2=29+28x}}}

So our system is now:

{{{system(x^2+h^2=169,x^2+h^2=29+28x)}}}

and therefore

{{{169=29+28x}}}

since things equal to the same thing are equal to each other.

{{{140=28x}}}

{{{5=x}}}

Substituting in

{{{x^2+h^2=169}}}

{{{5^2+h^2=169}}}

{{{25+h^2=169}}}

{{{h^2=144}}}

{{{h=12}}}

{{{drawing(360,400,-3,15,-3,17, locate(8,0,9), locate(2,0,5), locate(9.5,7,15),
locate(1,6,13), rectangle(0,0,14,12), triangle(0,0,5,0,5,12), locate(5.3,6,12),
triangle(0,0,14,0,5,12) )}}}

And since the height of the rectangle equals
the altitude, h, of the triangle, the area
of the rectangle in which the original triangle
is inscribed is found by

{{{A=(base)(height)=(14)(12)=168}}}

So the correct choice is (a) 168

Edwin</pre>

```