The equation of sides of a triangle are $7x-5y-11=0, 8x+3y+31=0, x+8y-19=0.$ Then the point (0,0) lies

The equations of the sides of the triangle are:

• 1. \(7x - 5y - 11 = 0\)
• 2. \(8x + 3y + 31 = 0\)
• 3. \(x + 8y - 19 = 0\)

Step-by-Step Calculation:

Step 1: Substituting (0,0) into the first equation

For the equation \(7x - 5y - 11 = 0\), substituting \(x = 0\) and \(y = 0\):

7(0) - 5(0) - 11 = -11

This result is negative, so the point (0,0) lies on the opposite side of the line compared to the side where the inequality \(7x - 5y - 11 \geq 0\) holds.

Step 2: Substituting (0,0) into the second equation

For the equation \(8x + 3y + 31 = 0\), substituting \(x = 0\) and \(y = 0\):

8(0) + 3(0) + 31 = 31

This result is positive, so the point (0,0) lies on the side of the line where the inequality \(8x + 3y + 31 \geq 0\) holds.

Step 3: Substituting (0,0) into the third equation

For the equation \(x + 8y - 19 = 0\), substituting \(x = 0\) and \(y = 0\):

0 + 8(0) - 19 = -19

This result is negative, so the point (0,0) lies on the opposite side of the line compared to the side where the inequality \(x + 8y - 19 \geq 0\) holds.

Conclusion:

The point (0,0) lies on the opposite side of the first and third lines, and on the same side of the second line. This suggests that the point lies inside the triangle.

Correct Answer:

The point (0,0) lies inside the triangle, so the correct answer is:

Option a: inside of triangle