Question

how can δabc be mapped to δxyz? first, translate ________. next, rotate δabc about b to align the sides and angles. options: vertex b to vertex z; vertex b to vertex y; vertex a to vertex z; vertex a to vertex y. (image shows δabc with ab=25 cm, ac=29 cm, ∠a=36°; δxyz with xy=25 cm, xz=29 cm, ∠x=36°)

Explanation:

Step1: Analyze Corresponding Parts

In congruent triangles (or triangles to be mapped), corresponding sides and angles should match. In $\triangle ABC$, side $AB = 25$ cm, $\angle A = 36^\circ$, side $AC = 29$ cm. In $\triangle XYZ$, side $XY = 25$ cm, $\angle X = 36^\circ$, side $XZ = 29$ cm. So vertex $A$ corresponds to vertex $X$, vertex $B$ corresponds to vertex $Y$, and vertex $C$ corresponds to vertex $Z$. But the translation step is to move a vertex of $\triangle ABC$ to a vertex of $\triangle XYZ$ so that rotation about $B$ can align sides. Wait, the rotation is about $B$, so we need to translate $B$ to a corresponding vertex? Wait no, let's check the sides. $AB = 25$ cm, $XY = 25$ cm. $\angle A = 36^\circ$, $\angle X = 36^\circ$. $AC = 29$ cm, $XZ = 29$ cm. So to map $\triangle ABC$ to $\triangle XYZ$, first, we should translate vertex $B$ to vertex $Y$ (since $AB$ corresponds to $XY$, and $B$ should map to $Y$ to align for rotation about $B$? Wait, no, let's re-examine. The rotation is about $B$, so after translation, $B$ should be at a vertex of $\triangle XYZ$ such that rotating about $B$ aligns the sides. Looking at the sides: $AB = 25$ (matches $XY = 25$), $AC = 29$ (matches $XZ = 29$), $\angle A = 36^\circ$ (matches $\angle X = 36^\circ$). Wait, maybe the correspondence is $A \to X$, $B \to Y$, $C \to Z$. But the first step is translation: we need to move a vertex of $\triangle ABC$ to a vertex of $\triangle XYZ$. The rotation is about $B$, so we should translate $B$ to $Y$ (since $B$ is a vertex of $\triangle ABC$, and $Y$ is a vertex of $\triangle XYZ$ with corresponding side length $XY = AB = 25$). So the correct translation is vertex $B$ to vertex $Y$.

Step2: Verify the Options

• Option 1: vertex $B$ to vertex $Z$: $BZ$ doesn't correspond to any equal side, so no.
• Option 2: vertex $B$ to vertex $Y$: $AB = XY = 25$, $\angle A = \angle X = 36^\circ$, $AC = XZ = 29$, so translating $B$ to $Y$ then rotating about $B$ (now at $Y$) will align the sides.
• Option 3: vertex $A$ to vertex $Z$: $A$ to $Z$ doesn't match corresponding sides.
• Option 4: vertex $A$ to vertex $Y$: $A$ to $Y$ doesn't match corresponding sides.

Answer:

B. vertex B to vertex Y