nathanl6656

Let's visualize the problem:

Imagine a right-angled corner of a room. Place a circular table in this corner so that one edge of the table touches both walls.

Point P: A point on the edge of the table that is 8 units away from one wall and 9 units away from the other.

Radius: The distance from the center of the table to any point on its edge (like point P).

Solution:

Form a right triangle:

The distance from point P to the first wall (8 units) is one leg of a right triangle.

The distance from point P to the second wall (9 units) is the other leg of the right triangle.

The hypotenuse of this right triangle is the diameter of the table, which is twice the radius.

Use the Pythagorean theorem:

a² + b² = c²

8² + 9² = c²

64 + 81 = c²

145 = c²

c = √145

Find the radius:

Since the diameter is twice the radius, the radius is half of √145.

Radius = (√145) / 2

Therefore, the radius of the table is (√145) / 2 units.

The answer is 4n + 3.

Finding the Area of Triangle XYZ

The area of a right triangle is (base * height)/2.

In this case, base = XY = 12 and height = YZ = 10.

So, area of triangle XYZ = (12 * 10)/2 = 60.

Relating the Area of XYD to the Height from D to XY

Let h be the height of triangle XYD from point D to side XY.

The area of triangle XYD = (base * height)/2 = (12 * h)/2 = 6h.

Finding the Maximum Height for an Area of 24

We want the area of XYD to be at most 24.

So, 6h <= 24.

Solving for h, we get h <= 4.

Calculating the Probability

The probability of choosing a point D with a height less than or equal to 4 is equal to the ratio of the area where the height is less than or equal to 4 to the total area of triangle XYZ.

If we draw a line parallel to XY at a distance of 4 units from XY, it will divide the triangle into two similar triangles. The smaller triangle will have a height of 4 and its area will be (1/4) * (area of triangle XYZ) = (1/4) * 60 = 15.

So, the probability of choosing a point D with a height less than or equal to 4 (and thus an area of XYD at most 24) is 15/60 = 1/4.

Therefore, the probability that the area of triangle XYD is at most 24 is 1/4.