Question

consider the transformation. which statement about the transformation is true? it is isometric because the side lengths remained the same. it is isometric because all angle measures remained the same. it is not isometric because the side lengths did not remain the same. it is not isometric because the angle measures did not remain the same.

Explanation:

Step1: Recall Isometric Transformation

An isometric transformation (rigid transformation) preserves side lengths and angle measures. So we check side lengths of the two trapezoids.

Step2: Compare Side Lengths

- Original trapezoid \(PQRS\): \(PQ = 8\), \(PS = 4\), \(SR = 12\), \(RQ = 4\) (wait, no, looking at the diagram: top base \(QR = 8\), bottom base \(PS = 12\), legs \(PQ = 4\), \(RS = 4\)? Wait no, the lower trapezoid: \(Q\) to \(R\) is 8, \(P\) to \(S\) is 12, \(P\) to \(Q\) is 4, \(R\) to \(S\) is 4. The upper trapezoid: \(Q'\) to \(R'\) is 4, \(P'\) to \(S'\) is 6, \(P'\) to \(Q'\) is 2, \(R'\) to \(S'\) is 2. So side lengths changed: 4 became 2, 8 became 4, 12 became 6. So it's a scaling (dilation), not isometric. Wait, but the options: let's re - examine. Wait, the first trapezoid (upper) has \(Q'R' = 4\), \(P'S' = 6\), legs 2. The lower trapezoid: \(QR = 8\), \(PS = 12\), legs 4. So the side lengths of the upper are half of the lower? Wait, no, maybe I misread. Wait, the key is: isometric transformations (translation, rotation, reflection) preserve side lengths and angle measures. Dilation (scaling) changes side lengths (so not isometric). Now let's check the options:

Option 1: "It is isometric because the side lengths remained the same." But from the diagram, side lengths changed (e.g., \(PQ\) was 4, \(P'Q'\) is 2; \(QR\) was 8, \(Q'R'\) is 4; \(PS\) was 12, \(P'S'\) is 6). So this is wrong.

Option 2: "It is isometric because all angle measures remained the same." Isometric transformations preserve angles, but if side lengths change (like in dilation), angles can stay the same? Wait no, dilation preserves angle measures but changes side lengths. But isometric transformations must preserve side lengths. So even if angles are same, if side lengths change, it's not isometric. Wait, but let's check the side lengths. Wait, maybe I made a mistake. Wait the upper trapezoid: \(Q'R' = 4\), \(P'S' = 6\), legs 2. Lower trapezoid: \(QR = 8\), \(PS = 12\), legs 4. So the ratio of sides is 1:2. So it's a dilation with scale factor 1/2. Dilation preserves angles (so angle measures are same) but changes side lengths. But isometric transformations require side lengths to be same. So the transformation here is a dilation (not isometric) because side lengths changed. Wait, but let's check the options again.

Option 3: "It is not isometric because the side lengths did not remain the same." This matches, because isometric transformations (rigid motions) preserve side lengths. Since the side lengths (e.g., leg length 4 vs 2, base lengths 8 vs 4, 12 vs 6) changed, it's not isometric.

Option 4: "It is not isometric because the angle measures did not remain the same." But in dilation, angle measures are preserved. So this is wrong.

Wait, but maybe I misread the diagram. Wait the lower trapezoid: \(Q\) to \(R\) is 8, \(P\) to \(S\) is 12, \(P\) to \(Q\) is 4, \(R\) to \(S\) is 4. The upper trapezoid: \(Q'\) to \(R'\) is 4, \(P'\) to \(S'\) is 6, \(P'\) to \(Q'\) is 2, \(R'\) to \(S'\) is 2. So the side lengths are scaled by 1/2. So side lengths changed, so it's not isometric. So the correct option is Option 3? Wait no, wait the options:

Wait the third option: "It is not isometric because the side lengths did not remain the same." Yes, because isometric transformations must preserve side lengths. Since the side lengths (like the legs, the bases) are different (scaled), it's not isometric. So this is correct.

Wait, but let's confirm: Isometric transformation definition: a transformation that preserves the distance between every pair of points (so side lengths are preserved). So if side lengths change, it's not isometric. So the transformation here (looks like a dilation) changes side lengths, so it's not isometric, and the reason is side lengths didn't remain the same. So the correct option is the third one: "It is not isometric because the side lengths did not remain the same."

Answer:

It is not isometric because the side lengths did not remain the same. (The option corresponding to this statement, assuming the options are labeled as: the third option, e.g., if the options are A, B, C, D: C. It is not isometric because the side lengths did not remain the same.)