Question

look at the two triangles below. triangle a: vertices at (1, 1), (3, 1), (2, 3) triangle b: vertices at (4, 1), (6, 1), (5, 3) b) what do you notice about the two triangles? b) do you think they are congruent? why or why not?

Explanation:

Step1: Recall the distance - formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Calculate side - lengths of Triangle A

For the side between $(1,1)$ and $(3,1)$: $d_1=\sqrt{(3 - 1)^2+(1 - 1)^2}=\sqrt{4+0}=2$.
For the side between $(3,1)$ and $(2,3)$: $d_2=\sqrt{(2 - 3)^2+(3 - 1)^2}=\sqrt{1 + 4}=\sqrt{5}$.
For the side between $(2,3)$ and $(1,1)$: $d_3=\sqrt{(1 - 2)^2+(1 - 3)^2}=\sqrt{1+4}=\sqrt{5}$.

Step3: Calculate side - lengths of Triangle B

For the side between $(4,1)$ and $(6,1)$: $d_4=\sqrt{(6 - 4)^2+(1 - 1)^2}=\sqrt{4+0}=2$.
For the side between $(6,1)$ and $(5,3)$: $d_5=\sqrt{(5 - 6)^2+(3 - 1)^2}=\sqrt{1 + 4}=\sqrt{5}$.
For the side between $(5,3)$ and $(4,1)$: $d_6=\sqrt{(4 - 5)^2+(1 - 3)^2}=\sqrt{1+4}=\sqrt{5}$.

Step4: Analyze the relationship between the two triangles

We notice that the side - lengths of Triangle A are equal to the side - lengths of Triangle B respectively: $d_1 = d_4$, $d_2 = d_5$, $d_3 = d_6$.

Answer:

b) The two triangles have the same side - lengths.
b) They are congruent because all three pairs of corresponding sides are equal (by the SSS - Side - Side - Side congruence criterion).