Question

the graph shows triangles bcd and wxy. is bcd congruent to wxy? justify your answer. yes, because a reflection across the x - axis maps bcd onto wxy. yes, because a translation right 1 unit and up 12 units maps bcd onto wxy. no, because $overline{cd}$ and $overline{xy}$ do not have the same length. no, because $angle b$ and $angle w$ do not have the same measure.

Explanation:

Step1: Recall congruence criteria

Two triangles are congruent if their corresponding sides and angles are equal.

Step2: Analyze side - length comparison

We can use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the lengths of sides. For example, for side \(CD\) with \(C(x_1,y_1)\) and \(D(x_2,y_2)\) and side \(XY\) with \(X(x_3,y_3)\) and \(Y(x_4,y_4)\). If the lengths of corresponding sides are not equal, the triangles are not congruent.

Step3: Analyze transformation options

A reflection across the \(x -\)axis or a translation right 1 unit and up 12 units would preserve the shape and size of a triangle only if the original triangles have the same side - lengths and angles. But if a corresponding side like \(\overline{CD}\) and \(\overline{XY}\) have different lengths, the triangles are not congruent.

Step4: Check angle - measure comparison

While angle - measure comparison is also a part of congruence, in this case, side - length comparison is more straightforward as we can visually and geometrically analyze the lengths of the sides on the coordinate - plane.

Answer:

No, because \(\overline{CD}\) and \(\overline{XY}\) do not have the same length.