Question

use the drop - down menus to explain if the two figures below are congruent, similar, or neither. if the figures are similar, state the scale factor.

Explanation:

Step1: Check for Congruence

Congruent figures have equal corresponding side - lengths. Let's assume coordinates (by counting grid units). For example, if in the upper - right figure (say \( \triangle DFG\)) and the lower - left figure (\( \triangle KLM\)), if we assume \(DF = 2\) units (from \(D(0,0)\) to \(F(2,0)\)) and \(KL= 2\) units (from \(K(- 1,-1)\) to \(L(-1,1)\)), but the orientation and other side - length ratios (by visual inspection of slopes and side - length counts) are not the same in a way that would make all corresponding sides equal.

Step2: Check for Similarity

Similar figures have proportional corresponding side - lengths.

• Calculate side - length ratios. Let's assume we count the lengths of the sides using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For a side in the upper - right figure, say from \(F(2,0)\) to \(G(9,9)\): \(d_1=\sqrt{(9 - 2)^2+(9 - 0)^2}=\sqrt{49 + 81}=\sqrt{130}\). For a side in the lower - left figure, say from \(L(-1,1)\) to \(M(-5,8)\): \(d_2=\sqrt{(-5+1)^2+(8 - 1)^2}=\sqrt{16 + 49}=\sqrt{65}\). Another side: from \(F(2,0)\) to \(D(0,0)\) has length \(2\), and from \(K(-1,-1)\) to \(L(-1,1)\) has length \(2\). But if we consider the ratio of non - equal sides \(\frac{\sqrt{130}}{\sqrt{65}}=\sqrt{2}\), and other side ratios (e.g., if there was a third side in upper - right figure from \(G(9,9)\) to \(D(0,0)\): \(d_3=\sqrt{81+81}=\sqrt{162}\), and in lower - left from \(M(-5,8)\) to \(K(-1,-1)\): \(d_4=\sqrt{(-1 + 5)^2+(-1 - 8)^2}=\sqrt{16+81}=\sqrt{97}\), the ratios are not consistent.

Answer:

The two figures are neither congruent nor similar.