We are given the coordinates of the vertices of â–ł NPO and â–ł NMO, and are asked to write a coordinate proof to show that these triangles are congruent.
If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
We have to show that three sides of â–ł NPO are congruent to three sides of â–ł NMO. We know that the triangles share the side ON. By the Reflexive Property of Congruence, this side is congruent to itself. Therefore, we can say that â–ł NPO and â–ł NMO have one pair of congruent sides. Let's find the other side lengths using the Distance Formula.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Triangle
Side
Points
Substitute
Simplify
â–ł NPO
NP
N( h, h) and P( 0, 2h)
NP=sqrt(( 0- h)^2+( 2h- h)^2)
NP=sqrt(2)h
â–ł NPO
PO
P( 0, 2h) and O( 0, 0)
PO=sqrt(( 0- 0)^2+( 0- 2h)^2)
PO=2h
â–ł NMO
NM
N( h, h) and M(2h, )
NM=sqrt((2h- h)^2+( - h)^2)
NM=sqrt(2)h
â–ł NMO
MO
M(2h, ) and O( 0, 0)
MO=sqrt(( 0-2h)^2+( 0- )^2)
MO=2h
We can see that NP=NM and PO=MO, so we have identified two more pairs of congruent sides.
Since three sides of â–ł NPO are congruent to three sides of â–ł NMO, the triangles are congruent.
△ NPO≅△ NMO
