Find the lengths of the sides of the triangle with the indicated vertices. $( 1 , - 4,5 ) , ( - 5,1,3 ) , ( 5,4,0 )$
A)d1= $\sqrt { 65 }$ ,d2= $\sqrt { 118 }$ ,d3= $\sqrt { 105 }$
B)d1= $\sqrt { 118 }$ ,d2= $\sqrt { 105 }$ ,d3= $\sqrt { 118 }$
C)d1= $\sqrt { 105 }$ ,d2= $\sqrt { 105 }$ ,d3= $\sqrt { 118 }$
D)d1= $\sqrt { 65 }$ ,d2= $\sqrt { 65 }$ ,d3= $\sqrt { 118 }$
E)d1= $\sqrt { 65 }$ ,d2= $\sqrt { 105 }$ ,d3= $\sqrt { 65 }$

Question

Find the lengths of the sides of the triangle with the indicated vertices.  $( 1 , - 4,5 ) , ( - 5,1,3 ) , ( 5,4,0 )$ 
A)d1= $\sqrt { 65 }$ ,d2= $\sqrt { 118 }$ ,d3= $\sqrt { 105 }$
B)d1= $\sqrt { 118 }$ ,d2= $\sqrt { 105 }$ ,d3= $\sqrt { 118 }$
C)d1= $\sqrt { 105 }$ ,d2= $\sqrt { 105 }$ ,d3= $\sqrt { 118 }$
D)d1= $\sqrt { 65 }$ ,d2= $\sqrt { 65 }$ ,d3= $\sqrt { 118 }$
E)d1= $\sqrt { 65 }$ ,d2= $\sqrt { 105 }$ ,d3= $\sqrt { 65 }$

Quizplus · Accepted Answer

To find the lengths of the sides of the triangle formed by the points $ ( 1 , - 4,5 ) $, $ ( - 5,1,3 ) $, and $ ( 5,4,0 ) $, we use the distance formula in 3D: $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$. Calculating each pair of points gives us the distances as $\sqrt{65}$, $\sqrt{118}$, and $\sqrt{105}$, matching option A.

Find the Lengths of the Sides of the Triangle with the Indicated