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?
is the answer 53?
no its not 53
Points A(-2, 4), B(1, 3), C(4, -1) and D form a parallelogram. What are the coordinates of D?
To find the volume of tetrahedron ABCD, we can use the formula:
V = (1/6) * base_area * height
First, let's find the base area. We know that AB is perpendicular to both AC and BD, so triangle ABC is a right triangle. We can use the Pythagorean theorem to find the length of BC:
AC^2 = AB^2 + BC^2
10^2 = 6^2 + BC^2
100 = 36 + BC^2
BC^2 = 64
BC = 8
Now that we know the lengths of all three sides of triangle ABC, we can use Heron's formula to find the area of triangle ABC:
s = (AB + AC + BC) / 2
s = (6 + 10 + 8) / 2
s = 24 / 2
s = 12
base_area = √(s * (s - AB) * (s - AC) * (s - BC))
base_area = √(12 * (12 - 6) * (12 - 10) * (12 - 8))
base_area = √(12 * 6 * 2 * 4)
base_area = √(576)
base_area = 24
Next, let's find the height of tetrahedron ABCD. Given that MN is perpendicular to AB, we can consider the right triangle AMN. The length of line segment MN is given as 4, and AM and AN are both half the length of AB, which is 6.
Applying the Pythagorean theorem:
AM^2 = AB^2 - MN^2
AM^2 = 6^2 - 4^2
AM^2 = 36 - 16
AM^2 = 20
AM = √20 = 2√5
The height of the tetrahedron is the length of line segment AM.
Now that we have the base area as 24 and the height as 2√5, we can calculate the volume:
V = (1/6) * base_area * height
V = (1/6) * 24 * 2√5
V = 4√5
Therefore, the volume of tetrahedron ABCD is 4√5.
Then, a = 4 and b = 5. The sum a + b = 4 + 5 = 9.
So, the value of a+b is 9.