Question

1. figure 2 is a scaled copy of figure 1.

image of figure 1 (a quadrilateral with points w, y, and others) and figure 2 (a smaller quadrilateral with points k, m, etc.) on a grid

a. identify the points in figure 2 that correspond to the points w and y in figure 1. label them j and l. what is the distance between j and l?

b. identify the points in figure 1 that correspond to the points k and m in figure 2. label them x and z. what is the distance between x and z?

c. what is the scale factor that takes figure 1 to figure 2? explain your reason

d. c and d are two points on figure 1, but they are not shown. the distance between c and d is 5. what is the distance between the corresponding points on figure 2? explain your reasoning.

Explanation:

Part (a)

Step1: Identify Corresponding Points

In a scaled copy, corresponding points have the same relative position. So, the point corresponding to \( W \) in Figure 2 is \( J \) (left - most point of Figure 2), and the point corresponding to \( Y \) in Figure 2 is \( L \) (right - most point of Figure 2).

Step2: Calculate Distance between \( J \) and \( L \)

Assume each grid square has a side length of 1 unit. By counting the number of horizontal grid units between \( J \) and \( L \), we find that the distance between \( J \) and \( L \) is 3 units (we can use the distance formula for horizontal points \( d=\vert x_2 - x_1\vert \), if \( J=(x_1,y_1) \) and \( L=(x_2,y_2) \) with \( y_1 = y_2 \), then \( d=\vert x_2 - x_1\vert \). From the grid, the horizontal difference is 3).

Step1: Identify Corresponding Points

The point corresponding to \( K \) in Figure 1 is \( X \) (top - most point of Figure 1), and the point corresponding to \( M \) in Figure 1 is \( Z \) (bottom - most point of Figure 1).

Step2: Calculate Distance between \( X \) and \( Z \)

Using the grid (assuming grid side length 1), by counting the vertical or using the distance formula for vertical points (\( d = \vert y_2-y_1\vert \) when \( x_1=x_2 \)) or horizontal/vertical combination. The distance between \( X \) and \( Z \) (by counting grid units) is 6 units.

Step1: Recall Scale Factor Formula

The scale factor \( k \) from Figure 1 to Figure 2 is given by \( k=\frac{\text{Length in Figure 2}}{\text{Length in Figure 1}} \) for corresponding segments.

Step2: Use Corresponding Distances

From part (a), length in Figure 2 ( \( JL \)) is 3, and from part (b), length in Figure 1 ( \( XZ \)) is 6. So \( k=\frac{3}{6}=\frac{1}{2} \). We can also check with other corresponding segments. For example, if we take the horizontal segment from \( W \) to \( Y \) in Figure 1, let's say its length is 6 (by counting grids), and the corresponding segment \( JL \) in Figure 2 is 3. So the scale factor is \( \frac{3}{6}=\frac{1}{2} \).

Answer:

The points corresponding to \( W \) and \( Y \) are \( J \) and \( L \) respectively. The distance between \( J \) and \( L \) is 3.

Part (b)