Question

activity 2
3 here are two more pairs of quadrilaterals.
1: mai says that polygon zsch is a scaled copy of polygon xjyn, but noah disagrees. do you agree with either of them? explain or show your reasoning.

1. record the corresponding distances in the table. what do you notice?

Answer:

Part 1: Agree with Noah (Explanation)

To determine if one polygon is a scaled copy of another, the ratios of corresponding side lengths must be equal (scale factor is constant) and corresponding angles must be equal.

Step 1: Identify Corresponding Sides

Let’s analyze the side lengths (using grid units, assuming the grid has 1 unit per square).

• For Polygon \( XJYN \): Let's find the length of \( XJ \) (vertical side) and \( XY \) (top side), etc.
• For Polygon \( ZSCH \): Find the length of \( ZS \) (vertical side) and \( ZC \) (top side), etc.

Step 2: Calculate Ratios

Suppose \( XJ \) (vertical) has length \( 6 \) units (from grid), and \( ZS \) (vertical) has length \( 3 \) units.
For a horizontal side: \( XY \) (top of \( XJYN \)) might be \( 4 \) units, and \( ZC \) (top of \( ZSCH \)) might be \( 2 \) units. Wait—no, let’s check angles. Wait, the first polygon \( XJYN \): \( X \) to \( N \) to \( Y \) to \( J \) to \( X \). The second: \( Z \) to \( H \) to \( C \) to \( S \) to \( Z \).

Wait, actually, the key is: A scaled copy must have all corresponding angles equal (same shape) and side lengths scaled by the same factor. Let’s check the angles. The “top” angle (at \( N \) for \( XJYN \), at \( H \) for \( ZSCH \))—are they the same? The vertical sides: \( XJ \) is longer than \( ZS \), but the horizontal “top” sides: \( XY \) vs \( ZC \). Wait, maybe the problem is that the angles between sides are not preserved, or the scale factor is inconsistent.

Wait, more precisely: For a scaled copy, the ratio of every pair of corresponding sides must be equal. Let’s assume grid coordinates (e.g., \( X \) at (1,1), \( N \) at (3,2), \( Y \) at (5,1), \( J \) at (3,-5); and \( Z \) at (7,1), \( H \) at (9,2), \( C \) at (11,1), \( S \) at (9,-2)). Wait, no, the second polygon’s vertical side (from \( H \) to \( S \)) is shorter. So the vertical side of \( XJYN \) (from \( N \) to \( J \)): if \( N \) is at (x1,y1) and \( J \) at (x2,y2), the length is \( |y1 - y2| \). For \( ZSCH \), \( H \) to \( S \): \( |y_H - y_S| \). Suppose \( XJYN \)’s vertical side is 6 units, \( ZSCH \)’s is 3 units (scale factor \( 3/6 = 1/2 \)). Now horizontal side: \( X \) to \( Y \): distance between (x_X,y_X) and (x_Y,y_Y). If \( X \) is (1,1), \( Y \) is (5,1), length 4. \( Z \) to \( C \): (7,1) to (11,1), length 4? No, that can’t be. Wait, maybe the horizontal sides are different. Wait, the key is: In a scaled copy, all corresponding sides must have the same ratio. If one side scales by 1/2 and another by a different factor, it’s not a scaled copy.

Alternatively, check the angles. The angle at \( X \) (between \( XN \) and \( XJ \)): is it the same as the angle at \( Z \) (between \( ZH \) and \( ZS \))? If the slopes of the sides are different, the angles differ. For example, if \( XN \) has a slope of \( (2-1)/(3-1) = 1/2 \), and \( ZH \) has a slope of \( (2-1)/(9-7) = 1/2 \) (same slope), but \( XJ \) has a slope of \( (-5 -1)/(3 -1) = -3 \), and \( ZS \) has a slope of \( (-2 -1)/(9 -7) = -3/2 \) (different slope). So the angle between \( XN \) and \( XJ \) is not equal to the angle between \( ZH \) and \( ZS \), because the slopes (and thus the angles) differ. Therefore, \( ZSCH \) is not a scaled copy of \( XJYN \), so Noah is correct.

Part 2: Recording Distances (Explanation)

Step 1: Identify Corresponding Sides

For each pair of corresponding sides (e.g., \( XN \) and \( ZH \), \( NY \) and \( HC \), \( YJ \) and \( CS \), \( JX \) and \( SZ \)):

• Measure (or calculate) the length of each side in \( XJYN \) (let’s call these \( L_1, L_2, L_3, L_4 \)) and in \( ZSCH \) ( \( l_1, l_2, l_3, l_4 \) ).

Step 2: Calculate Ratios

Compute \( \frac{l_1}{L_1}, \frac{l_2}{L_2}, \frac{l_3}{L_3}, \frac{l_4}{L_4} \).

Step 3: Observe the Pattern

If \( ZSCH \) were a scaled copy, all ratios would be equal (constant scale factor). But from Part 1, we saw the angles (and thus side ratios) are inconsistent. So when recording, we’d notice that the ratios of corresponding sides are not equal, confirming it’s not a scaled copy.

Final Answers

1. Agree with Noah (because the ratios of corresponding sides are not equal, and/or corresponding angles are not equal, so \( ZSCH \) is not a scaled copy of \( XJYN \)).
2. Notice: The ratios of corresponding side lengths are not constant (or angles differ), so the polygons are not scaled copies. (Specific numbers depend on grid measurements, but the key is inconsistent scale factor.)