Question

4.) scale factor = 2.5
original length × scale factor = new length
× =
× =

Explanation:

Step1: Determine Original Length

The original diamond (a square rotated 45 degrees) has a diagonal or side (in terms of grid units). Looking at the grid, the original length (let's consider the side length of the square when unrotated, or the distance across the grid squares). From the grid, the original length (for the side, if we consider the horizontal/vertical span) – actually, the diamond spans 2 grid squares horizontally and vertically? Wait, no, the diamond (a square) has a side length (the distance between two adjacent vertices) that, in grid terms, for a square rotated 45°, the length of the side (the edge of the diamond) – wait, maybe better to see the original length as the length of the side of the square. Wait, the original figure: let's count the grid squares. The black diamond: from the top vertex to the bottom vertex, it's 2 grid squares? No, wait, the horizontal distance from left to right vertex is 2 grid squares? Wait, no, actually, the original length (the length we scale) – let's assume the original length is 2 (since from the grid, the diamond's side, when considering the grid, maybe the length is 2 units? Wait, no, maybe the original length is 2 (the number of grid squares along a side). Wait, actually, looking at the grid, the black diamond: the distance between two opposite vertices (horizontal or vertical) is 2 grid squares? Wait, no, the diamond is a square with side length equal to the diagonal of a 1x1 square? Wait, maybe I'm overcomplicating. Let's see the table: Original Length × Scale Factor = New Length. The scale factor is 2.5. Let's find the original length. The black diamond: if we consider the length of its side (the edge), but actually, in the grid, the original length (the length we are scaling) – let's count the number of grid squares. Wait, the diamond spans 2 grid squares horizontally (from column 3 to column 5, for example) and 2 vertically (from row 3 to row 5). Wait, no, the diamond is a square with side length 2 (in grid units, where each grid is 1 unit). Wait, maybe the original length is 2. So:

Step1: Identify Original Length

Looking at the grid, the original length (the length of the side of the diamond, or the distance across) is 2 units (since it spans 2 grid squares).

Step2: Apply Scale Factor

Multiply original length by scale factor: \( 2 \times 2.5 = 5 \)

Wait, but maybe the original length is the length of the diagonal? No, the formula is Original Length × Scale Factor = New Length. Let's check the grid again. The black diamond: if we consider the length of its side (the edge), but actually, in the grid, the original length (the length we are scaling) – let's see, the diamond is a square with side length equal to \( \sqrt{2} \) if each grid is 1x1, but that's more complicated. Wait, maybe the problem is simpler: the original length is 2 (the number of grid squares along a side, like the horizontal or vertical span). So:

Original Length: 2 (let's confirm: the diamond has a horizontal span of 2 grid squares (from x=2 to x=4, for example) and vertical span of 2 grid squares (from y=3 to y=5). So the length (the side length of the square, when unrotated) is 2. Then:

Step1: Original Length = 2

Step2: Scale Factor = 2.5

Step3: New Length = 2 × 2.5 = 5

So filling the table:

Original Length: 2, Scale Factor: 2.5, New Length: 5.

Wait, but maybe the original length is the length of the edge of the diamond (the side of the square). Let's see, the diamond is a square with side length 2 (in grid units, where each grid is 1 unit). So when we scale by 2.5, the new leng…

Answer:

In the table:

Original Length: \( 2 \), Scale Factor: \( 2.5 \), New Length: \( 5 \)

So the table rows:

First row (if original length is 2):

\( 2 \times 2.5 = 5 \)

Second row (maybe same, since it's a square, all sides are same):

\( 2 \times 2.5 = 5 \)