The base of a triangular pyramid is a triangle with corners at #(6 ,8 )#, #(2 ,7 )#, and #(7 ,3 )#. If the pyramid has a height of #2 #, what is the pyramid's volume?

To calculate the volume of a triangular pyramid, you can use the formula:

Volume = (1/3) * base area * height

To find the base area of the triangular pyramid, you first need to find the area of the base triangle. You can use the formula for the area of a triangle:

Area = (1/2) * base * height

Once you have the base area, you can then calculate the volume using the formula mentioned earlier.

Here are the steps:

1. Find the lengths of the sides of the base triangle using the given coordinates.
2. Use the side lengths to calculate the area of the base triangle.
3. Once you have the area of the base triangle, plug it into the volume formula along with the height of the pyramid to find the volume.

Let's go through the calculations:

1. Calculate the lengths of the sides of the base triangle using the distance formula:

Side 1: √((6-2)^2 + (8-7)^2) = √(16 + 1) = √17 Side 2: √((2-7)^2 + (7-3)^2) = √(25 + 16) = √41 Side 3: √((6-7)^2 + (8-3)^2) = √(1 + 25) = √26

2. Use Heron's formula to find the area of the triangle:

s = (side1 + side2 + side3) / 2 s = ( √17 + √41 + √26 ) / 2 s ≈ (4.123 + 6.403 + 5.099) / 2 s ≈ 15.625 / 2 s ≈ 7.8125

Area = √(s * (s - side1) * (s - side2) * (s - side3)) Area = √(7.8125 * (7.8125 - √17) * (7.8125 - √41) * (7.8125 - √26))

3. Calculate the area:

Area ≈ √(7.8125 * (7.8125 - 4.123) * (7.8125 - 6.403) * (7.8125 - 5.099)) Area ≈ √(7.8125 * 3.6895 * 1.4095 * 2.7135) Area ≈ √(72.682) Area ≈ 8.527

4. Finally, calculate the volume:

Volume = (1/3) * base area * height Volume ≈ (1/3) * 8.527 * 2 Volume ≈ 5.685 cubic units