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 DEC are congruent to three sides of â–ł BOC. To do so, we need to find the length of each side. We will start by finding the length of the horizontal sides DE and BO, and the vertical sides DC and CB.
Side
Length
Simplify
DE
DE=|2h-h|
DE=h
BO
BO=|h-0|
BO=h
DC
DC=|2k-k|
DC=k
CB
CB=|k-0|
CB=k
We found that DE and BO have the same length. Therefore, by the definition of congruent segments, these two sides are congruent. Similarly, since DC and CB have the same length, we can also say that they are congruent segments. This means that two sides of â–ł DEC are congruent to two sides of â–ł BOC.
To prove congruence, we also need to see whether the lengths of EC and OC are equal. For this purpose, we will use the Distance Formula.
d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)
Side
Points
Substitute
Simplify
EC
E( 2h, 2k) and C( h, k)
EC=sqrt(( h- 2h)^2+( k- 2k)^2)
EC=sqrt(h^2+k^2)
OC
O( 0, 0) and C( h, k)
OC=sqrt(( h- 0)^2+( k- 0)^2)
OC=sqrt(h^2+k^2)
We found that EC and OC have the same length. Therefore, by the definition of congruent segments, EC is congruent to OC.
EC=OC ⇔ EC≅OC
By the Side-Side-Side Congruence Theorem, since three sides of â–ł DEC are congruent to three sides of â–ł BOC, the triangles are congruent.
