Find the coordinates of R if you know two vertices of the isosceles triangle having coordinates as P(2,3,4) and Q (4,5,6) and distance of two the vertices PR and QR is 5

Question

Find the coordinates of R if you know two vertices of the isosceles triangle having coordinates as P(2,3,4) and Q (4,5,6) and distance of two the vertices PR and QR is 5. Find out the coordiantes of R

Answer 1

The solution to this will be a circle in 3 dimensions.

Since Q - P = [2, 2, 2] and the midpoint between P and Q is M = [3, 4, 5], and 2[1, 1, 1] = [2, 2, 2],the points on the circle are in the plane x + y + z = 3 + 4 + 5 = 12.

MR = √(PR² - MP²) = √(5² - (1²+1²+1²) ) = √22

Thus, our solution is the set of points on x + y + z = 12 that are √22 from [3, 4, 5]

For example, one definition of the points, as z = 12 - x - y, is

(x - 3)² + (y - 4)² + (12 - x - y - 5)² = 22

z = 12 - x - y

or

(x - 3)² + (y - 4)² + (7 - x - y)² = 22

z = 12 - x - y