Question

the vertices of △ghi are g (2,4), h (4,8), and i (8,4). the vertices of △jkl are j (1,1), k (2,3), and l (4,1). which conclusion is true about the triangles? they are congruent by the definition of congruence in terms of rigid motions. they are similar by the definition of similarity in terms of a dilation. the ratio of their corresponding sides is 1:3. the ratio of their corresponding angles is 1:3

Explanation:

Step1: Recall similarity and congruence criteria

Two triangles are congruent if all corresponding sides and angles are equal (rigid - motion based). Two triangles are similar if the ratios of corresponding sides are equal and corresponding angles are equal.

Step2: Calculate side - length ratios

Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ to find the side - lengths of the two triangles.
For $\triangle GHI$:
$GH=\sqrt{(4 - 2)^2+(8 - 4)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}$
$HI=\sqrt{(8 - 4)^2+(4 - 8)^2}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}$
$GI=\sqrt{(8 - 2)^2+(4 - 4)^2}=\sqrt{36}=6$
For $\triangle JKL$:
$JK=\sqrt{(2 - 1)^2+(3 - 1)^2}=\sqrt{1 + 4}=\sqrt{5}$
$KL=\sqrt{(4 - 2)^2+(1 - 3)^2}=\sqrt{4+4}=\sqrt{8}=2\sqrt{2}$
$JL=\sqrt{(4 - 1)^2+(1 - 1)^2}=\sqrt{9}=3$
The ratio of $GH$ to $JK$ is $\frac{GH}{JK}=\frac{2\sqrt{5}}{\sqrt{5}} = 2$, the ratio of $HI$ to $KL$ is $\frac{HI}{KL}=\frac{4\sqrt{2}}{2\sqrt{2}} = 2$, and the ratio of $GI$ to $JL$ is $\frac{GI}{JL}=\frac{6}{3}=2$.
The ratio of corresponding sides is $2:1$ (not $1:3$ as stated in some options). And since the ratio of corresponding sides is constant, the triangles are similar by the definition of similarity in terms of a dilation.

Answer:

They are similar by the definition of similarity in terms of a dilation.