Question

from unit 1, lesson 2
suppose quadrilaterals a and b are both squares. are a and b necessarily scaled copies of one another? explain.

Explanation:

Step1: Recall scaled copy definition

A scaled copy has proportional corresponding lengths (scale factor \( k \), \( k>0 \)), same shape (angles equal), and all linear dimensions scaled by \( k \). For squares, all angles are \( 90^\circ \), so angle condition is met.

Step2: Analyze square side lengths

Let square A have side length \( s_A \), square B have side length \( s_B \). For a scaled copy, \( \frac{s_B}{s_A}=k \) (constant scale factor). But if \( s_A
eq s_B \) and no fixed \( k \) (e.g., A: side 2, B: side 3; \( k = 1.5 \); but if A: side 2, B: side 5, \( k = 2.5 \)—still a scaled copy? Wait, no—wait, all squares are similar (same shape, proportional sides). Wait, no—wait, a scaled copy requires that one is a scale of the other, meaning there exists a scale factor \( k \) such that all sides of B are \( k \times \) sides of A. Since squares have all sides equal, if square A has side \( a \), square B has side \( b \), then the scale factor is \( \frac{b}{a} \), so B is a scaled copy of A with scale factor \( \frac{b}{a} \), and A is a scaled copy of B with scale factor \( \frac{a}{b} \). Wait, but the question is "necessarily"? Wait, no—wait, maybe I made a mistake. Wait, no—all squares are similar, so they are scaled copies. Wait, no, wait: a scaled copy must have the same shape and proportional sides. Since all squares have 4 right angles and sides in proportion (ratio of sides is constant, as all sides of a square are equal). So for any two squares, the ratio of their side lengths is constant (scale factor), so they are scaled copies. Wait, but maybe the problem is about "necessarily"—but all squares are similar, hence scaled copies. Wait, no, wait: suppose square A has side 1, square B has side 2. Then B is a scaled copy of A with \( k = 2 \). If square A has side 3, square B has side 6, \( k = 2 \). So any two squares have proportional sides (ratio of sides is constant, since all sides of a square are equal), so they are scaled copies. Wait, but maybe the confusion is: is there a case where two squares are not scaled copies? No, because for any two squares, the ratio of their side lengths is a constant (scale factor), and all angles are equal (90°), so they satisfy the scaled copy definition.

Answer:

Yes, they are necessarily scaled copies. A scaled copy requires proportional corresponding lengths (same shape, angles equal). All squares have four \( 90^\circ \) angles (same shape) and side lengths in a constant ratio (e.g., if square A has side \( s_A \) and square B has side \( s_B \), the scale factor is \( \frac{s_B}{s_A} \), so all sides of B are \( \frac{s_B}{s_A} \times \) sides of A). Thus, any two squares are scaled copies of one another.