Question

5.) scale factor = 1.5
original lengths × scale factor = new lengths

Explanation:

First, we need to determine the original lengths of the triangle's sides. From the grid, let's assume the vertical side (height) of the original triangle is 3 units, the horizontal side (base) is 6 units, and the hypotenuse can be calculated using the Pythagorean theorem, but for scaling, we can also measure it from the grid. Let's find the original lengths:

Step 1: Determine Original Lengths

Looking at the grid, the vertical side (height) of the triangle spans 3 grid squares, so original length = 3. The horizontal side (base) spans 6 grid squares, so original length = 6. The hypotenuse: using the distance formula, if the vertical change is 3 and horizontal change is 6, hypotenuse \( = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \approx 6.708 \), but from the grid, we can count the units. Alternatively, since it's a right triangle with legs 3 and 6, the hypotenuse is \( 3\sqrt{5} \), but for scaling, we can use the grid count. Let's proceed with the legs first.

Step 2: Scale the Vertical Side (Height)

Original length (height) = 3. Scale factor = 1.5. New length = \( 3 \times 1.5 = 4.5 \).

Step 3: Scale the Horizontal Side (Base)

Original length (base) = 6. Scale factor = 1.5. New length = \( 6 \times 1.5 = 9 \).

Step 4: Scale the Hypotenuse

Original length (hypotenuse) = \( 3\sqrt{5} \) (or approximately 6.708). New length = \( 3\sqrt{5} \times 1.5 = 4.5\sqrt{5} \approx 10.062 \), or using the grid count, if the hypotenuse spans, say, from (0,0) to (6,3), the length is \( \sqrt{(6-0)^2 + (3-0)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \), so scaling by 1.5 gives \( 4.5\sqrt{5} \). But maybe the grid has integer units, so let's check the original triangle:

Looking at the grid, the vertical side (height) is 3 units (from y=0 to y=3), horizontal side (base) is 6 units (from x=0 to x=6). So:

- For the height (vertical side):

Step 1 (Height):

Original Length = 3, Scale Factor = 1.5, New Length = \( 3 \times 1.5 = 4.5 \)

- For the base (horizontal side):

Step 2 (Base):

Original Length = 6, Scale Factor = 1.5, New Length = \( 6 \times 1.5 = 9 \)

- For the hypotenuse:
Original Length: Let's calculate the length of the hypotenuse. The horizontal change is 6, vertical change is 3. So length \( = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \approx 6.708 \)

Step 3 (Hypotenuse):

New Length = \( 3\sqrt{5} \times 1.5 = 4.5\sqrt{5} \approx 10.062 \)

Now, filling the table:

First row (height):
Original Lengths: 3, \( \times \) 1.5 \( = \) 4.5

Second row (base):
Original Lengths: 6, \( \times \) 1.5 \( = \) 9

Third row (hypotenuse):
Original Lengths: \( 3\sqrt{5} \) (or 6.708), \( \times \) 1.5 \( = \) \( 4.5\sqrt{5} \) (or 10.062)

But maybe the grid is such that the vertical side is 3, horizontal is 6, and hypotenuse is, say, 6.7 (approx). Alternatively, maybe the original triangle has sides 3, 6, and hypotenuse. Let's confirm the grid:

Looking at the black triangle, the vertical side (left side) is from the bottom to the top, spanning 3 grid squares (assuming each grid square is 1 unit). The horizontal side (bottom) spans 6 grid squares. So:

- Height (vertical): 3 units
- Base (horizontal): 6 units
- Hypotenuse: \( \sqrt{3^2 + 6^2} = \sqrt{45} = 3\sqrt{5} \approx 6.708 \) units

Now, scaling each by 1.5:

- Height: \( 3 \times 1.5 = 4.5 \)
- Base: \( 6 \times 1.5 = 9 \)
- Hypotenuse: \( 3\sqrt{5} \times 1.5 = 4.5\sqrt{5} \approx 10.06 \)

So the table can be filled as:

| Original Lengths | \( \times \) | Scale Factor | \( = \) | New Lengths |
|------------------|-------------|--------------|--------|-------------|
| 3 | \( \times \) | 1.5 | \( = \) | 4.5 |
| 6 | \( \times \) | 1.5 | \( = \) | 9 |
| \( 3\sqrt{5} \) | \( \times \) | 1.5 | \( = \) | \( 4.5\sqrt{5} \) |

(Note: If we use approximate values for the hypotenuse, original length ≈ 6.71, new length ≈ 6.71 × 1.5 ≈ 10.06, which matches \( 4.5\sqrt{5} \approx 10.06 \))

Answer:

The table filled with original lengths (3, 6, \( 3\sqrt{5} \) or ≈6.71), scale factor 1.5, and new lengths (4.5, 9, \( 4.5\sqrt{5} \) or ≈10.06) as shown above.