Question

answer attempt 2 out of 2 the two figures ~ congruent because used to map figure 1 onto figure 2. point corresponding to point h:

Explanation:

Step1: Analyze the figures' transformation

First, we need to determine the transformation (translation, rotation, reflection) that maps Figure 1 to Figure 2. By observing the coordinates, Figure 1 is on the left side of the y - axis and Figure 2 is on the right side. Also, we can check the corresponding vertices. Point H in Figure 1 is at ( - 1, 0) (assuming the grid, since H is on the x - axis, left of the origin). Now, looking at Figure 2, the point U is at (5, 0), T is at (9, 0), etc. Wait, actually, when we look at the base of the two quadrilaterals, Figure 1 has a base from H (let's confirm coordinates: H is at (-1, 0), K is at (-8, 2), I is at (-5, 7), and the other point (let's say J) at (-1, 4)? Wait, maybe better to look at the corresponding vertices. The figure 1 has a vertex H at (-1, 0), and in figure 2, the vertex U is at (5, 0). Wait, maybe the transformation is a translation. But also, we can see that the two figures are congruent (same shape and size). Now, to find the point corresponding to H: in Figure 1, H is the bottom - right vertex (on the x - axis, left of origin). In Figure 2, the bottom - left vertex on the x - axis is U? Wait, no. Wait, let's list the coordinates:

For Figure 1:
- H: (-1, 0)
- K: Let's say (-8, 2)
- I: (-5, 7)
- J: (-1, 4) (assuming the fourth vertex)

For Figure 2:
- U: (5, 0)
- T: (9, 0)
- S: (11, 7)
- V: (7, 10) (Wait, maybe my coordinate estimation is off, but the key is that the base of Figure 1 is from H (-1,0) to... and Figure 2's base is from U (5,0) to T (9,0). The horizontal distance from H (-1,0) to U (5,0) is 5 - (-1)=6 units to the right. Also, the shape is congruent. So the point corresponding to H in Figure 2 is U? Wait, no, let's check the structure. Figure 1: H is the bottom vertex (on x - axis), K is bottom - left, I is top - left, and the other is top - right. Figure 2: U is bottom - left, T is bottom - right, S is top - right, V is top - left. So the correspondence is H (bottom - right of Figure 1) corresponds to T (bottom - right of Figure 2)? Wait, no, let's see the order. Let's assume the quadrilaterals are labeled in order: Figure 1: H, K, I, J (H to K to I to J to H). Figure 2: U, T, S, V (U to T to S to V to U). So the transformation: let's check the vector from H to U: U - H=(5 - (-1), 0 - 0)=(6, 0). From K to T: T - K=(9 - (-8), 0 - 2)=(17, - 2)? No, that's not a translation. Wait, maybe rotation and translation. Alternatively, since the figures are congruent (same side lengths, angles), the corresponding point to H: in Figure 1, H is at (-1, 0). In Figure 2, the point U is at (5, 0), T is at (9, 0). Wait, maybe the correct correspondence is that H (x=-1, y = 0) in Figure 1 corresponds to U (x = 5, y=0) in Figure 2? No, wait, let's look at the base length. In Figure 1, the distance from H (-1,0) to the other bottom vertex (let's say K is at (-8,2), no, maybe H is at (-1,0), and the adjacent vertex is at (-1,4) (vertical line), and K is at (-8,2), I at (-5,7). In Figure 2, U is at (5,0), T at (9,0), S at (11,7), V at (7,10). So the vertical side from U (5,0) to V (7,10) and from H (-1,0) to I (-5,7)? Wait, maybe the correct corresponding point for H is U? No, wait, the problem is to find the point corresponding to H when mapping Figure 1 to Figure 2. Since the two figures are congruent (we can see they have the same shape and size, so congruent), and by looking at the position, H is on the x - axis (y = 0) in Figure 1, and U is on the x - axis (y = 0) in Figure 2. Also, the horizontal shift: from x=-1 (H) to x = 5 (U) is a shift of 6 units to the right. So the point corresponding to H is U? Wait, no, maybe T? Wait, let's count the grid. H is at x=-1, y = 0. U is at x = 5, y=0. T is at x = 9, y=0. So the distance from H to U is 6, U to T is 4. In Figure 1, the distance from H to the other bottom vertex (let's say if H is at (-1,0) and the other bottom vertex is at (-8,2), no, maybe I made a mistake. Wait, the key is that in congruent figures, corresponding vertices have the same relative position. So if Figure 1 is transformed (translated and maybe rotated) to Figure 2, the point H (which is a vertex on the x - axis, bottom - right of Figure 1) should correspond to the vertex U (bottom - left of Figure 2) or T (bottom - right of Figure 2)? Wait, looking at the figures, Figure 1: H is the right - most bottom vertex. Figure 2: T is the right - most bottom vertex. Wait, H is at (-1,0), T is at (9,0). The distance from H to T is 9 - (-1)=10. The length of the base of Figure 1: from H (-1,0) to K (let's say K is at (-8,2), the distance is $\sqrt{(-8 + 1)^2+(2 - 0)^2}=\sqrt{49 + 4}=\sqrt{53}$. The base of Figure 2: from U (5,0) to T (9,0) is 4 units. No, that's not matching. Wait, maybe my coordinate estimation is wrong. Let's look at the x - axis: Figure 1 is on the left, with H at x=-1 (since the x - axis has - 2, - 1, 0, 1, 2,...). Figure 2 is on the right, with U at x = 5, T at x = 9. So the horizontal distance from H (-1) to U (5) is 6, from U (5) to T (9) is 4. In Figure 1, the horizontal distance from H (-1) to the other vertex (let's say the left - most bottom vertex K) is, say, from x=-1 to x=-8, which is 7 units. No, this is confusing. Alternatively, since the two figures are congruent (same shape and size), the corresponding point to H is U. Wait, the problem says "Point corresponding to point H". Let's think about the transformation: if we translate Figure 1 6 units to the right (since - 1+6 = 5, which is the x - coordinate of U) and maybe some vertical shift? No, H is at y = 0, U is at y = 0. So the translation vector is (6, 0). So H (-1,0) translated by (6,0) is ( - 1+6, 0+0)=(5,0), which is point U. So the point corresponding to H is U.

Step2: Confirm the correspondence

Since the two figures are congruent (they have the same shape and size, so we can use translation to map Figure 1 to Figure 2), and the translation vector from H to U is (6, 0) (as H is at (-1,0) and U is at (5,0)), the point corresponding to H in Figure 2 is U.

Answer:

U