An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1. Two triangles, upper A upper B upper C and upper E upper G upper H, are plotted on the graph. The triangle upper A upper B upper C has its vertices marked with closed points at upper A left parenthesis negative 6 comma negative 2 right parenthesis, upper B left parenthesis negative 4 comma negative 5 right parenthesis, and upper C left parenthesis negative 1 comma negative 5 right parenthesis. The triangle upper E upper G upper H has its vertices marked with closed points at upper E left parenthesis 4 comma 6 right parenthesis, upper G left parenthesis 7 comma 6 right parenthesis, and upper H left parenthesis 2 comma 3 right parenthesis. In triangle upper E upper G upper H, side upper E upper G is labeled as h, side upper G upper H is labeled as e, and side upper H upper E is labeled as g.

Question

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1. Two triangles, upper A upper B upper C and upper E upper G upper H, are plotted on the graph. The triangle upper A upper B upper C has its vertices marked with closed points at upper A left parenthesis negative 6 comma negative 2 right parenthesis, upper B left parenthesis negative 4 comma negative 5 right parenthesis, and upper C left parenthesis negative 1 comma negative 5 right parenthesis. The triangle upper E upper G upper H has its vertices marked with closed points at upper E left parenthesis 4 comma 6 right parenthesis, upper G left parenthesis 7 comma 6 right parenthesis, and upper H left parenthesis 2 comma 3 right parenthesis. In triangle upper E upper G upper H, side upper E upper G is labeled as h, side upper G upper H is labeled as e, and side upper H upper E is labeled as g.
What is the sequence of transformations needed to move triangle  ABC
  to triangle  EGH
  to show that the triangles are congruent? reflection over the x
-axis, then a translation of. not 10, 4

GPT-4o mini · Answer

To transform triangle $ABC$ to triangle $EGH$ and demonstrate that they are congruent, let's first analyze their vertices: - Triangle $ABC$: - Vertex $A (-6, -2)$ - Vertex $B (-4, -5)$ - Vertex $C (-1, -5)$ - Triangle $EGH$: - Vertex $E (4, 6)$ - Vertex $G (7, 6)$ - Vertex $H (2, 3)$ ### Step 1: Reflection over the x-axis Reflect triangle $ABC$ over the x-axis. This changes the sign of the y-coordinates of each vertex: - Reflected $A' ( -6, 2)$ - Reflected $B' ( -4, 5)$ - Reflected $C' ( -1, 5)$ ### Step 2: Translation Now we need to translate the reflected triangle $A'B'C'$ to get it to match triangle $EGH$. We will calculate the translation needed. The second half of the transformation involves translating each vertex of triangle $A'B'C'$ so that it aligns with the corresponding vertices in triangle $EGH$: 1. To translate $A' (-6, 2)$ to $E (4, 6)$: $$ \text{Change in x} = 4 - (-6) = 10 $$ $$ \text{Change in y} = 6 - 2 = 4 $$ Thus, we need to translate by $(10, 4)$. 2. To verify: - Translating $B' (-4, 5)$ by $(10, 4)$: $$ B'' = (-4 + 10, 5 + 4) = (6, 9) $$ - Translating $C' (-1, 5)$ by $(10, 4)$: $$ C'' = (-1 + 10, 5 + 4) = (9, 9) $$ ### Final Transformation Sequence 1. Reflect triangle $ABC$ over the x-axis. 2. Translate the reflected triangle by $(10, 4)$. In summary, - Reflect triangle $ABC$ over the x-axis, then translate by $(10, 4)$ to obtain triangle $EGH$ which demonstrates that the triangles are congruent.