Question 1062658
<pre><b>{{{drawing(400,800/3,-7,23,-4,16,
line(-8,0,22,0), line(0,-5,0,17),
locate(7,-2,"A(7,-2)"),locate(19,7,"B(19,7)"),locate(-5,15.3,"C(-5,14)"),
circle(11/2,0,.2),circle(29/3,0,.2),
locate(5,0,D),locate(29/3,0,E),

triangle(7,-2,19,7,-5,14) )}}}

The part of triangle ABC that lies in the fourth quadrant
is the triangle DAE.

You find the area of triangle ABC by the determinant:

{{{matrix(1,5,Area,""="",
abs(matrix(3,3,7,-2,1,19,7,1,-5,14,1)),
""="",
300
)}}}

The area of triangle ABC is 300.

You must now find the area of triangle DAE

But to do that you must find the coordinates of D and E:

You find the equation of line CA by using the slope formula: 

{{{m}}}{{{""=""}}}{{{(y[2]-y[1])/(x[2]-x[1])}}}

and the point-slope formula 

{{{y-y[1]}}}{{{""=""}}}{{{m(x-x[1])}}}

and after doing that and simplifying, you get the equation 
of line CA, which is,
 
4x + 3y = 22 

Then point D is the x-intercept of line CA, so substitute 0 for y
and solve for x and get the coordinates of D as

{{{D(matrix(1,3,11/2,",",0))}}} 

Now exactly the same way, you'll find the equation of the line BE
as 

3x - 4y = 29

Then point E is the x-intercept of line BA, so substitute 0 for y
and solve for x and get the coordinates of E as

{{{E(matrix(1,3,29/3,",",0))}}}

Then use the matrix method again to find the area of triangle DAE:

{{{matrix(1,5,Area,""="",
abs(matrix(3,3,11/2,0,1,7,-2,1,29/3,0,1)),
""="",
25/3
)}}}

Finally you need to find what percent 25/3 is of 300.

So you divide 

{{{matrix(1,5,
matrix(1,3,25/3,"÷",300),
""="",
matrix(1,3,25/3,""*"",1/300),
""="",
1/36)}}}

So the area of triangle DAE is 1/36th of the area of triangle ABC.

To find out what percent that is, we multiply by 100%

{{{matrix(1,3,(1/36)*"100%",""="","2.7778%")}}}

Edwin</pre></b>