probability

To find the probability that the area of triangle XYD is at most 20, we can use geometric probability.

Let's first find the area of triangle XYZ using the given leg lengths:

Area of triangle XYZ = (1/2) * XY * YZ Area of triangle XYZ = (1/2) * 12 * 6 Area of triangle XYZ = 36 square units

Now, let A be the event that the area of triangle XYD is at most 20 square units.

To calculate the probability of event A, we need to find the ratio of the area of triangle XYD to the area of triangle XYZ.

Let h be the length of the altitude from vertex Y to side XD (the height of triangle XYD).

The area of triangle XYD can be expressed as:

Area of triangle XYD = (1/2) * XY * h

Since XY = 12, the area of triangle XYD is:

Area of triangle XYD = (1/2) * 12 * h Area of triangle XYD = 6h square units

Now, to find the value of h, we can use similar triangles. Triangles XYD and XYZ are similar because they share angle Y, and their sides are in proportion. Therefore, the ratio of corresponding sides is the same:

XY/XZ = XD/XY

Substitute the given values:

12/6 = XD/12

Solve for XD:

XD = (12 * 12) / 6 XD = 24

Now, we can find the length of the altitude h using the Pythagorean theorem in triangle XYD:

h^2 + XD^2 = XY^2 h^2 + 24^2 = 12^2 h^2 + 576 = 144 h^2 = 144 - 576 h^2 = -432 (Since h is positive, we discard the negative value)

Since h^2 is negative, there is no real value for h, meaning that point D must lie outside triangle XYD. In this case, the area of triangle XYD is 0.

Therefore, the probability of the area of triangle XYD being at most 20 is 0, as it's impossible for point D to lie within the triangle for an area less than 36 (the area of triangle XYZ).

XY = 6

YZ = 12
 

Area of triangle  = 1/2 x base x height 

Area of triangle  = 1/2 x XY x YZ

              = 1/2 x 6 x 12

              = 3 x 12

              = 36

The area of triangle  XYZ is 36 square units.

The area of triangle XYD is at most 20 

P(at most 20) = 20 / 36

                         = 5 / 9

The probability that area of triangle XYD is at most 20 is 5 / 9

 

 

5 / 9