Question

6.) scale factor = \\(\frac{4}{3}\\)

original lengths	×	scale factor	=	new lengths
	×		=
	×		=

Explanation:

First, we need to determine the original lengths of the triangle's sides. Let's assume the base of the triangle (horizontal side) spans 6 grid units, and the two equal sides (the legs of the isosceles triangle) can be calculated using the grid. Wait, actually, looking at the grid, the base of the triangle: from the leftmost point to the rightmost point, let's count the grid squares. Let's say the base length is 6 (since from column 2 to column 8, that's 6 units), and the height is 3 (from row 4 to row 7, that's 3 units). Wait, no, the triangle is isosceles? Wait, maybe the original triangle has a base of 6 and a height of 3. But actually, for the sides: the base is 6, and the two equal sides (the slant sides) can be calculated using the distance formula, but maybe in the grid, each square is 1 unit. So let's find the original lengths:

1. Base: Let's count the horizontal units. From the left end to the right end, it's 6 units (since it spans 6 squares).
2. Height: From the base to the top vertex, it's 3 units (spans 3 squares).
3. The two equal sides (the legs of the isosceles triangle): Using the Pythagorean theorem, if the base is 6, then half of the base is 3, and the height is 3, so each slant side is $\sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2}$? Wait, maybe I'm overcomplicating. Wait, maybe the original triangle has a base of 6 and a height of 3, and the two equal sides (the non - base sides) have a length that we can calculate, but maybe the problem is simpler. Wait, the table is for original lengths, scale factor, and new lengths. So first, we need to find the original lengths of the triangle's sides. Let's look at the grid:

- The base of the triangle (the bottom side) is 6 units (since it goes from x - coordinate 2 to x - coordinate 8, so 8 - 2 = 6).
- The height of the triangle (vertical distance from the base to the top vertex) is 3 units (from y - coordinate 4 to y - coordinate 7, so 7 - 4 = 3).
- The two equal sides (the left and right slant sides): Let's take the left side. From (2,4) to (4,7). The horizontal change is 4 - 2 = 2, vertical change is 7 - 4 = 3? Wait, no, maybe the top vertex is at (4,7), the left end is at (1,4), and the right end is at (7,4). Wait, maybe I miscounted. Let's do it again. Let's assume each grid square has a side length of 1. Let's find the coordinates:

Suppose the leftmost point of the base is at (1,4), the rightmost at (7,4), so the base length is 7 - 1 = 6. The top vertex is at (4,7). So the height is 7 - 4 = 3. Now, the length of the left side (from (1,4) to (4,7)): using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, so $d=\sqrt{(4 - 1)^2+(7 - 4)^2}=\sqrt{9 + 9}=\sqrt{18}=3\sqrt{2}\approx4.24$, but maybe the problem is using integer lengths? Wait, maybe the original triangle has a base of 6, a height of 3, and the two equal sides (the legs) are, well, maybe the problem is simpler. Wait, the scale factor is 4/3, so we need to multiply the original lengths by 4/3.

Let's assume the original lengths are:

- Base: 6
- Height: 3
- Left side: $3\sqrt{2}$ (but maybe the problem is considering the base and the two equal sides as, say, base = 6, and the two equal sides (the non - base sides) have a length that we can calculate, but maybe the problem is using the base and the two equal sides (the legs) as, for example, if the triangle is isosceles with base 6 and height 3, then the two equal sides (the legs) are each $\sqrt{3^2 + 3^2}=3\sqrt{2}$, but maybe the problem is using the base, height, and the two equal sides as the three lengths. Wait, maybe the original triangle has a base of 6, a height of 3, and the two equal sides (the slant sides) are each, let's see, from the base to the top, the horizontal distance from the end of the base to the top is 3 (since base is 6, half is 3), and vertical distance is 3, so the slant side is $\sqrt{3^2+3^2}=3\sqrt{2}$. But maybe the problem is simpler, and the original lengths are 6 (base), 3 (height), and $3\sqrt{2}$ (slant sides), but maybe the problem is using the base and the two equal sides (the legs) as, for example, if the triangle is a different type. Wait, maybe the original triangle has a base of 6, and the two equal sides (the non - base sides) have a length of, let's say, 5? No, that doesn't fit. Wait, maybe the problem is just about the base and the height. Let's proceed with the base and height.

Step 1: Find the original base length

Looking at the grid, the base of the triangle spans 6 units (let's assume each grid square is 1 unit). So original base length = 6.

Step 2: Calculate the new base length

Multiply the original base length by the scale factor $\frac{4}{3}$. So new base length = $6\times\frac{4}{3}=8$.

Step 3: Find the original height length

The height of the triangle (vertical distance from base to top vertex) is 3 units. So original height length = 3.

Step 4: Calculate the new height length

Multiply the original height length by the scale factor $\frac{4}{3}$. So new height length = $3\times\frac{4}{3}=4$.

Step 5: Find the original length of the slant side

Using the Pythagorean theorem, for a right triangle with legs 3 (half of base) and 3 (height), the slant side length $l=\sqrt{3^2 + 3^2}=\sqrt{18}=3\sqrt{2}$.

Step 6: Calculate the new length of the slant side

Multiply the original slant side length by the scale factor $\frac{4}{3}$. So new slant side length = $3\sqrt{2}\times\frac{4}{3}=4\sqrt{2}$.

Now, filling the table:

| Original Lengths | × | Scale Factor | = | New Lengths |
| ---- | ---- | ---- | ---- | ---- |
| 6 | × | $\frac{4}{3}$ | = | 8 |
| 3 | × | $\frac{4}{3}$ | = | 4 |
| $3\sqrt{2}$ | × | $\frac{4}{3}$ | = | $4\sqrt{2}$ |

Answer:

The table is filled as follows:

Original Lengths	×	Scale Factor	=	New Lengths
3	×	$\frac{4}{3}$	=	4
$3\sqrt{2}$	×	$\frac{4}{3}$	=	$4\sqrt{2}$

(Note: The original lengths may vary depending on the exact grid - based measurement. If we consider the base as 6, height as 3, and slant side as $3\sqrt{2}$, the new lengths are 8, 4, and $4\sqrt{2}$ respectively after multiplying by the scale factor $\frac{4}{3}$)