The line 5 x + 6 = 3 y + 10 = 8 z + 14 is the hypotenuse of an isosceles right angled triangle whose opposite vertex is (7, 2, 4). Find the equation of the remaining sides.

Question

The line 5 x + 6 ​ = 3 y + 10 ​ = 8 z + 14 ​ is the hypotenuse of an isosceles right angled triangle whose opposite vertex is (7, 2, 4). Find the equation of the remaining sides.

TestHub · Accepted Answer

Let C be the right-angle vertex, d the hypotenuse direction, and H its perpendicular foot. For an isosceles right triangle, H is the hypotenuse midpoint and HA=HB=CH. C = ( 7 , 2 , 4 ) , d = ( 5 , 3 , 8 ) , H = ( − 6 , − 10 , − 14 ) + 2 5 ​ d = ( 2 13 ​ , − 2 5 ​ , 6 ) , C H = 2 ​ 7 ​ = 2 ∣ d ∣ ​ . A = H + 2 d ​ = ( 9 , − 1 , 10 ) , B = H − 2 d ​ = ( 4 , − 4 , 2 ) . C A : r = ( 7 , 2 , 4 ) + s ( 2 , − 3 , 6 ) , CB : r = ( 7 , 2 , 4 ) + s ( 3 , 6 , 2 ) . ​

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