In a tetrahedron ABCD, the lengths of

AB, AC, and BD are 6, 10, and
14 respectively. The distance between the midpoints M of AB and N of CD is 4. The line AB is perpendicular to AC, BD, and MN. The volume of ABCD can be written as a√b, where a and b are positive integers, and b is not divisible by the square of a prime number. What is the value of a+b?

To find the volume of tetrahedron ABCD, we can use the formula:

Volume = (1/6) * base area * height

First, let's find the base area of triangle ABC. Since AB is perpendicular to AC, we can use the formula for the area of a right triangle:

Area of triangle ABC = (1/2) * AB * AC

Area of triangle ABC = (1/2) * 6 * 10 = 30

Next, let's find the height of tetrahedron ABCD. The height is the distance between the line AB and the opposite face CD, which is the distance MN.

Given that MN = 4, we need to find the height of the tetrahedron. Let's call the height h.

Using Pythagoras' theorem, we can find h:

h^2 = BD^2 - MN^2
h^2 = 14^2 - 4^2
h^2 = 196 - 16
h^2 = 180
h = √180

Now, let's calculate the volume:

Volume = (1/6) * base area * height
Volume = (1/6) * 30 * √180
Volume = 5 * √180

Finally, we can write the volume as a√b, where a = 5 and b = 180. The value of a + b is 5 + 180 = 185.

Therefore, the value of a + b is 185.

To find the volume of a tetrahedron, you can use the formula:

V = (1/6) * base area * height

Here, the base of the tetrahedron is the triangle formed by the points A, B, and C. The height is the perpendicular distance from the vertex D to the plane of the base triangle ABC.

Let's start by finding the base area:

The length of AB is 6, and we know that AB is perpendicular to AC. Therefore, AC is the height of the base triangle ABC. Since we have a right-angled triangle ABC, we can use the Pythagorean theorem to find the length of AC.

AC^2 = AB^2 - BC^2
AC^2 = 6^2 - 10^2
AC^2 = 36 - 100
AC^2 = -64

Since AC^2 is negative, there is no real value for AC. This means that the given information is not consistent, and we cannot find the volume of the tetrahedron.