Given points A (2,1) and B (5,1), which could be the coordinates for point C to form a triangle with area 9 square units?

Given points A (2,1) and B (5,1), which could be the coordinates for point C to form a triangle with area 9 square units?

• A. (3,4)
• B. (4,6)
• C. (5,7)
• D. (5,4)

Answer: (C)

To determine which coordinates for point C will create a triangle with points A (2,1) and B (5,1) that has an area of 9 square units, we can use the formula for the area of a triangle given its vertex coordinates. The area \(A\) of a triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be calculated using the formula: \[ A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right| \] In this case, we have points A (2,1) and B (5,1), and we are looking for a point C with coordinates \((x_C, y_C)\) such that the area equals 9 square units. Using the coordinates of points A and B and substituting into the area formula, we want: \[ 9 = \frac{1}{2} \left| 2(1 - y_C) + 5

To determine which coordinates for point C will create a triangle with points A (2,1) and B (5,1) that has an area of 9 square units, we can use the formula for the area of a triangle given its vertex coordinates.

The area (A) of a triangle formed by three points ((x_1, y_1)), ((x_2, y_2)), and ((x_3, y_3)) can be calculated using the formula:

[

A = \frac{1}{2} \left| x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2) \right|

]

In this case, we have points A (2,1) and B (5,1), and we are looking for a point C with coordinates ((x_C, y_C)) such that the area equals 9 square units.

Using the coordinates of points A and B and substituting into the area formula, we want:

[

9 = \frac{1}{2} \left| 2(1 - y_C) + 5