Question

All points lying inside the triangle formed by the points (1,3), (5,0) and (-1,2) satisfy the equation
a)$3x+2y\ge 0$
b)$2x+y-13\ge 0$
c)$2x-3y-12\le 0$
d)$-2x+y\ge 0$

Answer 1

Hint: In this question, we need to find the inequality satisfied by all the points inside the given triangle. Now, we see that all the options given in the question are first degree linear equations. Thus, they correspond to straight lines. Thus, if the three vertices of the triangle satisfy the inequality, then all the points inside the triangle will automatically satisfy the same inequality.

Complete step-by-step answer:
The given triangle has its vertices at the points (1,3), (5,0) and (-1,2). Now, we should check which inequalities given in the options are satisfied by all points inside the triangle. Now, we see that all the options correspond to first degree linear equations and thus are equations of straight lines. Thus, if all the vertices of the triangle satisfy a given inequality, then all the points inside the triangle will automatically satisfy the same inequality.

In the figure, the lines corresponding to the options are shown. We can find if all the points of the triangle lie on the same side and satisfy the given inequality as follows:
Checking whether the vertices satisfy the inequality, we find that
For option (a), the inequality is $3x+2y\ge 0$
Putting x=1, y=3, we find that $3\times 1+2\times 3=9\ge 0\text{ (true)}$
Putting x=5, y=0, we find that $3\times 5+2\times 0=15\ge 0\text{ (true)}$
Putting x=-1, y=2, we find that $3\times -1+2\times 2=1\ge 0\text{ (true)}$
Thus, the first inequality is satisfied by all the vertices and hence all the points inside the triangle…………..(1.1)
For option (b), the inequality is $2x+y-13\ge 0$
Putting x=1, y=3, we find that $2\times 1+3-13=-8\ge 0\text{ (false)}$
Putting x=5, y=0, we find that $2\times 5+0-13=-3\ge 0\text{ (false)}$
Putting x=-1, y=2, we find that $2\times -1+2-13=-13\ge 0\text{ (false)}$
Thus, the second inequality is not satisfied by all the vertices and hence all the points inside the triangle………….. (1.2)
For option (c), the inequality is $2x-3y-12\le 0$
Putting x=1, y=3, we find that $2\times 1-3\times 3-12=-19\le 0\text{ (true)}$
Putting x=5, y=0, we find that $2\times 5-3\times 0-12=-2\le 0\text{ (true)}$
Putting x=-1, y=2, we find that $2\times -1-3\times 2-12=-20\le 0\text{ (true)}$
Thus, the third inequality is satisfied by all the vertices and hence all the points inside the triangle………….. (1.3)
For option (d), the inequality is $-2x+y\ge 0$
Putting x=1, y=3, we find that $-2\times 1+3=1\ge 0\text{ (true)}$
Putting x=5, y=0, we find that $-2\times 5+0=-10\ge 0\text{ (false)}$
Putting x=-1, y=2, we find that $-2\times -1+2=4\ge 0\text{ (true)}$
Thus, the fourth inequality is not satisfied by all the vertices and hence all the points inside the triangle………….. (1.4)
Thus, from equations 1.1, 1.2, 1.3 and 1.4, we find that the inequalities stated in option (a) and (c) are satisfied by all the points inside the triangle. Therefore, the answer to this question is (a) and (c).
Note: We should note that if the inequality is not satisfied even for one vertex, the inequality will not be satisfied for all the points inside the triangle because the points close to that vertex will then not satisfy the inequality.