Question

All points lying inside the triangle formed by the points $(1,3)$, $(5,0)$ and $(-1,2)$ satisfy

• A. $3x+2y\ge 0$
• B. $2x+y–13\ge 0$
• C. $2x–3y–12\le 0$
• D. Both $\left(A\right)$ and $\left(C\right)$

Answer 1

The correct option is D

Both $\left(A\right)$ and $\left(C\right)$

Explanation for the correct option:

Step 1: Substituting the points in the equation (A)

The equation is $3x+2y\ge 0$

The points are $(1,3),(5,0)and(-1,2)$

Take the point $(1,3)$ and substitute in the equation,

$$\begin{gathered} \Rightarrow 3\left(1\right)+2\left(3\right)\ge 0 \\ 9\ge 0 \end{gathered}$$

Take the point $(5,0)$ and substitute in the equation,

$$\begin{gathered} \Rightarrow 3\left(5\right)+2\left(0\right)\ge 0 \\ 15\ge 0 \end{gathered}$$

Take the point $(-1,2)$ and substitute in the equation,

$$\begin{gathered} \Rightarrow 3(-1)+2\left(2\right)\ge 0 \\ 1\ge 0 \end{gathered}$$

Thus, option (A) satisfies the given points of triangle.

Step 2: Substituting the points in equation (B)

The equation is $2x+y–13\ge 0$

Take the point $(1,3)$ and substitute in the equation,

$$\begin{gathered} \Rightarrow 2\left(1\right)+3-13\ge 0 \\ -7\le 0 \end{gathered}$$

Take the point $(5,0)$ and substitute in the equation,

$$\begin{gathered} \Rightarrow 2\left(5\right)+0-13\ge 0 \\ -3\le 0 \end{gathered}$$

Take the point $(-1,2)$ and substitute in the equation,

$$\begin{gathered} \Rightarrow 2(-1)+2-13\ge 0 \\ -13\le 0 \end{gathered}$$

Thus, option (B) does not satisfies the given points of triangle.

Step 3: Substituting the points in equation (C)

The equation is $2x–3y–12\le 0$

Take the point $(1,3)$ and substitute in the equation,

$$\begin{gathered} \Rightarrow 2\left(1\right)-3\left(3\right)-12\le 0 \\ -2\le 0 \end{gathered}$$

Take the point $(5,0)$ and substitute in the equation,

$$\begin{gathered} \Rightarrow 2\left(5\right)-3\left(0\right)-12\le 0 \\ -2\le 0 \end{gathered}$$

Take the point $(-1,2)$ and substitute in the equation,

$$\begin{gathered} \Rightarrow 2(-1)-3\left(2\right)-12\le 0 \\ -20\le 0 \end{gathered}$$

Thus, option (C) satisfies the given points of triangle.

Hence, option (D), Both $\mathbf{\left(}\mathit{A}\mathbf{\right)}$ and $\mathbf{\left(}\mathit{C}\mathbf{\right)}$ is the correct answer.