In triangle T with vertices at (0,0), (6,0), and (0,6), what is the probability that a point selected within this triangle has a y-coordinate less than twice its x-coordinate?

To determine the probability that a point selected within triangle T has a y-coordinate less than twice its x-coordinate, we first need to analyze the area of triangle T and the region where the condition ( y < 2x ) holds.

The vertices of triangle T are at (0,0), (6,0), and (0,6). The triangle is right-angled and lies in the first quadrant with a base and height of 6 units each. The area of the triangle can be calculated using the formula for the area of a triangle:

[

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 6 = 18.

]

Next, we need to find the area within this triangle where the condition ( y < 2x ) is satisfied. The line ( y = 2x ) intersects the triangle at some point. To find that intersection, we can set up inequalities based on the limits of the triangle:

1. The line ( y = 2x ) intersects the side of the triangle along the x-axis (where ( y =