Question

1. is there a mathematical relationship between a scale factor from figure a to figure b and its reverse scale factor from figure b to figure a? describe how you figured out the relationship so that a 5th grader can understand how you did it.

Explanation:

Step1: Understand Scale Factor

Scale factor from A to B: If A has a side length (or some measurement) of \( a \) and B has the corresponding side length \( b \), then scale factor \( k_{A
ightarrow B}=\frac{b}{a} \).

Step2: Reverse Scale Factor

Scale factor from B to A: Now, using the same sides, B has length \( b \) and A has length \( a \), so scale factor \( k_{B
ightarrow A}=\frac{a}{b} \).

Step3: Relationship

Notice that \( k_{A
ightarrow B}\times k_{B
ightarrow A}=\frac{b}{a}\times\frac{a}{b} = 1 \). So they are reciprocals (multiplicative inverses) of each other. Let's take an example. Suppose A is a square with side 4 units, and B is a square with side 2 units (so A is scaled down to B). Scale factor from A to B: \( \frac{2}{4}=\frac{1}{2} \). Scale factor from B to A: \( \frac{4}{2} = 2 \). Now, \( \frac{1}{2}\times2 = 1 \), so they multiply to 1 (reciprocals).

Answer:

The scale factor from figure A to figure B and the scale factor from figure B to figure A are reciprocals (their product is 1). To find this, pick a measurement (like side length) of A and B. If A’s length is \( a \) and B’s is \( b \), scale factor A→B is \( \frac{b}{a} \), and B→A is \( \frac{a}{b} \). Multiplying them gives \( \frac{b}{a}\times\frac{a}{b}=1 \), so they are reciprocals (e.g., if A→B scale factor is \( \frac{1}{2} \), B→A is 2, and \( \frac{1}{2}\times2 = 1 \)).