To solve the problem, we need to:

1. Plot points A, B, and C on the coordinate plane.

   • A = (-5, 3)
   • B = (6, 0)
   • C = (4, 6)

2. Verify if the triangle is a right triangle.
   To check if the triangle ABC is a right triangle, we will apply the Pythagorean theorem, which states: $a^2 + b^2 = c^2$ where $a$ and $b$ are the legs (two sides that meet at a right angle) and $c$ is the hypotenuse (the side opposite the right angle).

   We will calculate the lengths of the three sides of the triangle:

   • Side AB (distance between A and B): $\text{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
   • Side BC (distance between B and C): $\text{BC} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
   • Side AC (distance between A and C): $\text{AC} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

3. Find the area of the triangle.
   The area $A$ of a right triangle can be calculated using: $A = \frac{1}{2} \times \text{base} \times \text{height}$ where the base and height are the lengths of the two perpendicular sides (legs).

I will now calculate the distances and verify the right triangle property. math

Given points A, B, and C

A = (-5, 3) B = (6, 0) C = (4, 6)

Calculate the distances (lengths of sides AB, BC, and AC)

def distance(p1, p2): return math.sqrt((p2[0] - p1[0]) ** 2 + (p2[1] - p1[1]) ** 2)

AB = distance(A, B) BC = distance(B, C) AC = distance(A, C)

Verify if it's a right triangle using the Pythagorean theorem

sides = sorted([AB, BC, AC]) # Sort to ensure AB, BC are the legs and AC is the hypotenuse is_right_triangle = math.isclose(sides[0] ** 2 + sides[1] ** 2, sides[2] ** 2)

Calculate the area of the triangle (half of base * height)

Since it's a right triangle, we use AB and BC as base and height

area = 0.5 * sides[0] * sides[1]

AB, BC, AC, is_right_triangle, areaHere are the results:

1. Lengths of the sides:

   • AB = 11.40 units
   • BC = 6.32 units
   • AC = 9.49 units

2. Verification of right triangle: Using the Pythagorean theorem: $11.40^2 + 6.32^2 = 9.49^2$ This holds true, confirming that triangle ABC is indeed a right triangle.

3. Area of the triangle: The area is $30 \, \text{square units}$.

Now, regarding the plot of the points, I cannot directly view or interpret images from the graph choices in your problem, but based on the calculated distances, you can use this information to identify the correct plot.

Would you like more detailed steps on how to plot the points or any further clarification on this?