Find the point whether lies in the Cartesian plane.

acemi123

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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

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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

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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

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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.
 
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