Right ∆ABC has vertices at A(−5, 0), B(0, 0) and C(0,12). What is the volume of the solid figure formed when ∆ABC is rotated about side BC? Express your answer in terms of ð.

To find the volume of the solid figure formed when triangle ABC is rotated about side BC, we can use the method of cylindrical shells.

Here's how you can calculate the volume step by step:

1. Find the length of side BC: Based on the given coordinates, the length of BC can be calculated using the distance formula:

Length of BC = √((x2 - x1)^2 + (y2 - y1)^2)
= √((0 - 0)^2 + (12 - 0)^2)
= √(0 + 144)
= √144
= 12 units

2. Determine the height of the cylinder: The height of the cylinder will be the same as the height of triangle ABC, which is the vertical distance between points B and C. In this case, the height is 12 units.

3. Calculate the circumference of the cylindrical shell: The circumference of the cylindrical shell will be equal to the perimeter of triangle ABC, which is the sum of the lengths of sides AB, BC, and AC:

Circumference = AB + BC + AC
= 5 + 12 + 13
= 30 units

4. Now, we have all the necessary measurements to calculate the volume of the cylindrical shell. The volume of a cylindrical shell can be calculated using the formula:

Volume = 2π * radius * height * circumference

In this case, the radius is equal to the length of BC, which is 12 units, and the height is 12 units:

Volume = 2π * 12 * 12 * 30
= 864π cubic units

Therefore, the volume of the solid figure formed when triangle ABC is rotated about side BC is 864π cubic units.