91Ó°ÊÓ

The slope of a line indicates its steepness and direction. It is calculated as:

• slope = \frac{y_2 - y_1}{x_2 - x_1}

For this exercise, we need to find the slopes of the lines formed by points A (3,1), B (6,0), and C (4,4).
First, let’s calculate the slope of line AB:

\( \text{slope of } AB = \frac{0 - 1}{6 - 3} = \frac{-1}{3} \)
Next, let’s find the slope of line BC:
\( \text{slope of } BC = \frac{4 - 0}{4 - 6} = \frac{4}{-2} = -2 \)
Finally, compute the slope of line CA:
\( \text{slope of } CA = \frac{1 - 4}{3 - 4} = \frac{-3}{-1} = 3 \)
For a right triangle, two lines must be perpendicular. Perpendicular lines have slopes that are negative reciprocals. For lines AB and CA, the slopes are \(\frac{-1}{3} \) and 3, and their product is -1, indicating they are perpendicular.

Coordinate geometry helps to determine points' positions and relationships on a plane using coordinates, like (x, y).
It can also prove geometric properties algebraically. In this problem, we utilize point coordinates to prove a triangle is right-angled.
The three points A (3,1), B (6,0), and C (4,4) form a triangle. By finding the slopes of the lines between these points, we prove perpendicularly and right angles.
To summarize:

• Analyze both x-coordinates and y-coordinates changes to find slopes.
• Confirm perpendicular lines by checking if slopes are negative reciprocals.

This geometric approach provides a precise method to confirm that angles are indeed 90 degrees, supporting the proof of a right triangle.

The area of a triangle can be determined using the formula:
\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
For our triangle, we used points A, B, and C.
We identified AB as the horizontal base and the vertical distance from C to the line through A and B as the height.
Measurements:

• Base (distance AB ) = 3 units, because from (3,1) to (6,0), the horizontal change is 3 units.
• Height (vertical distance from C to horizontal line through B at y = 0 ) = 4 units, measured from (4,4) to y=0.

Using the formula:
\( \frac{1}{2} \times 3 \times 4 = 6 \text{ square units} \)
This area calculation shows the size of the right triangle directly based on the coordinates.