Find the lengths of the sides of the triangle with the indicated vertices. $( 4,0,3 ) , ( 4,2,2 ) , ( 1,0,4 )$
A)d1= $\sqrt { 5 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 5 }$
B)d1= $\sqrt { 10 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 17 }$
C)d1= $\sqrt { 5 }$ ,d2= $\sqrt { 5 }$ ,d3= $\sqrt { 17 }$
D)d1= $\sqrt { 5 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 17 }$
E)d1= $\sqrt { 5 }$ ,d2= $\sqrt { 17 }$ ,d3= $\sqrt { 17 }$

Question

Find the lengths of the sides of the triangle with the indicated vertices. $( 4,0,3 ) , ( 4,2,2 ) , ( 1,0,4 )$ 
A)d1= $\sqrt { 5 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 5 }$
B)d1= $\sqrt { 10 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 17 }$
C)d1= $\sqrt { 5 }$ ,d2= $\sqrt { 5 }$ ,d3= $\sqrt { 17 }$
D)d1= $\sqrt { 5 }$ ,d2= $\sqrt { 10 }$ ,d3= $\sqrt { 17 }$
E)d1= $\sqrt { 5 }$ ,d2= $\sqrt { 17 }$ ,d3= $\sqrt { 17 }$

Quizplus · Accepted Answer

To find the lengths of the sides of the triangle, we calculate the distance between each pair of points using the distance formula in 3D: $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.1. Between (4,0,3) and (4,2,2): $d_1 = \sqrt{(4-4)^2 + (2-0)^2 + (2-3)^2} = \sqrt{0 + 4 + 1} = \sqrt{5}$.2. Between (4,0,3) and (1,0,4): $d_2 = \sqrt{(1-4)^2 + (0-0)^2 + (4-3)^2} = \sqrt{9 + 0 + 1} = \sqrt{10}$.3. Between (4,2,2) and (1,0,4): $d_3 = \sqrt{(1-4)^2 + (0-2)^2 + (4-2)^2} = \sqrt{9 + 4 + 4} = \sqrt{17}$.Therefore, the correct answer is $d_1= \sqrt{5}, d_2= \sqrt{10}, d_3= \sqrt{17}$, which corresponds to option D.

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