Question

what transformations were made to △abc to produce △xyz? assess whether the two triangles are congruent. how do you know? show your work

Explanation:

Step1: Analyze translation

The $x$ - coordinates of points in $\triangle ABC$ are shifted to the right. For example, point $A(-5,2)$ and point $Z(-4, - 4)$. The $x$ - coordinate of $A$ is $-5$ and of $Z$ is $-4$, a shift of $1$ unit to the right. The $y$ - coordinates of points in $\triangle ABC$ are shifted down. Point $A(-5,2)$ and point $Z(-4,-4)$, a shift of $6$ units down. So, there is a translation of $(x,y)\to(x + 1,y-6)$.

Step2: Analyze congruence

Congruent triangles have the same shape and size. Since translation is a rigid - motion transformation, which preserves side - lengths and angles. The side - lengths of $\triangle ABC$ and $\triangle XYZ$ are equal because translation only moves the triangle without changing its shape or size. For example, if we calculate the lengths of the sides using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ for corresponding sides, they will be equal. Also, the angles of $\triangle ABC$ and $\triangle XYZ$ are equal. So, $\triangle ABC\cong\triangle XYZ$.

Answer:

There is a translation of $(x,y)\to(x + 1,y - 6)$ from $\triangle ABC$ to $\triangle XYZ$. The two triangles are congruent because translation is a rigid - motion transformation that preserves side - lengths and angles.