Find the point whether lies in the Cartesian plane.

acemi123

New member
Joined
Apr 14, 2019
Messages
22
In the Cartesian plane, let us consider the triangle formed by the three lines r1 : y = 0, r2 : y = 2x, r3 : y = -x + 7. Which of the following points lies inside the triangle?

(A) P=(-3;2)
(B) P=(1;-3)
(C) P=(3;3)
(D) P=(4;4)
(E) P=(3;5)

Explain why, please.
 

MarkFL

Super Moderator
Staff member
Joined
Nov 24, 2012
Messages
1,819
In order for a point \((x,y)\) to lie within the resulting triangle, it must simultaneously satisfy the following 3 inequalities:

\(\displaystyle 0<y\)

\(\displaystyle y<2x\)

\(\displaystyle y<7-x\)

Can you find the single given point that does?
 

acemi123

New member
Joined
Apr 14, 2019
Messages
22
In order for a point \((x,y)\) to lie within the resulting triangle, it must simultaneously satisfy the following 3 inequalities:

\(\displaystyle 0<y\)

\(\displaystyle y<2x\)

\(\displaystyle y<7-x\)

Can you find the single given point that does?
the answer is (3;3) Your formula is correct, can you explain how this formula works, please. Depend on which conditions? Does the order of lines matter? Is there a general rule?
 

MarkFL

Super Moderator
Staff member
Joined
Nov 24, 2012
Messages
1,819
the answer is (3;3) Your formula is correct, can you explain how this formula works, please. Depend on which conditions? Does the order of lines matter? Is there a general rule?
Consider a plot of the 3 given lines, with the triangular region bounded by these lines shaded in green:

fmh_0043.png

Now, we see that the bottom of the triangle lies along the line \(y=0\) and so the \(y\)-coordinates of all points within the triangle must satisfy:

\(\displaystyle 0<y\)

Next, observe that the left side of the triangle lies along the line \(y=2x\), so all points must be to the right of that line. If we pick such a point like \((1,1)\), and substitute that into the line, we find:

\(\displaystyle 1<2(1)\)

And so all points within the triangle also must satisfy:

\(\displaystyle y<2x\)

Finally, we see the right side of the triangle lies along the line \(y=7-x\) and if we again use the point \((1,1)\), we find:

\(\displaystyle 1<7-1\)

And so all points within the triangle also must satisfy:

\(\displaystyle y<7-x\)

And thus we may conclude that any point \((x,y)\) that simultaneously satisfies all 3 conditions must lie within the triangular region bounded by the 3 given lines.
 
Top