Pythagorean Theorem practice worksheet with coordinate grids for calculating line segment lengths.

To find the length of each line segment, we will use the distance formula. This formula is based on the Pythagorean theorem.

The Distance Formula:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the endpoints of the segment. We will count the grid units to find these coordinates, assuming each grid square represents 1 unit. Then we will round the final answer to the nearest tenth.

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Problem 1)
* Endpoints: The segment goes from $(-2, 1)$ to $(3, 4)$.
* Calculation:
* Change in $x$: $3 - (-2) = 5$
* Change in $y$: $4 - 1 = 3$
* $d = \sqrt{5^2 + 3^2}$
* $d = \sqrt{25 + 9}$
* $d = \sqrt{34}$
* $\sqrt{34} \approx 5.83$
* Rounding: To the nearest tenth, this is 5.8.

Problem 2)
* Endpoints: The segment goes from $(-1, -3)$ to $(2, 0)$.
* Calculation:
* Change in $x$: $2 - (-1) = 3$
* Change in $y$: $0 - (-3) = 3$
* $d = \sqrt{3^2 + 3^2}$
* $d = \sqrt{9 + 9}$
* $d = \sqrt{18}$
* $\sqrt{18} \approx 4.24$
* Rounding: To the nearest tenth, this is 4.2.

Problem 3)
* Endpoints: The segment goes from $(-3, -1)$ to $(4, 3)$.
* Calculation:
* Change in $x$: $4 - (-3) = 7$
* Change in $y$: $3 - (-1) = 4$
* $d = \sqrt{7^2 + 4^2}$
* $d = \sqrt{49 + 16}$
* $d = \sqrt{65}$
* $\sqrt{65} \approx 8.06$
* Rounding: To the nearest tenth, this is 8.1.

Problem 4)
* Endpoints: The segment goes from $(-1, 3)$ to $(2, -3)$.
* Calculation:
* Change in $x$: $2 - (-1) = 3$
* Change in $y$: $-3 - 3 = -6$
* $d = \sqrt{3^2 + (-6)^2}$
* $d = \sqrt{9 + 36}$
* $d = \sqrt{45}$
* $\sqrt{45} \approx 6.708$
* Rounding: To the nearest tenth, this is 6.7.

Problem 5)
* Endpoints: The segment goes from $(-2, 1)$ to $(3, 3)$.
* Calculation:
* Change in $x$: $3 - (-2) = 5$
* Change in $y$: $3 - 1 = 2$
* $d = \sqrt{5^2 + 2^2}$
* $d = \sqrt{25 + 4}$
* $d = \sqrt{29}$
* $\sqrt{29} \approx 5.385$
* Rounding: To the nearest tenth, this is 5.4.

Problem 6)
* Endpoints: The segment goes from $(-1, 4)$ to $(4, -2)$.
* Calculation:
* Change in $x$: $4 - (-1) = 5$
* Change in $y$: $-2 - 4 = -6$
* $d = \sqrt{5^2 + (-6)^2}$
* $d = \sqrt{25 + 36}$
* $d = \sqrt{61}$
* $\sqrt{61} \approx 7.81$
* Rounding: To the nearest tenth, this is 7.8.

Problem 7)
* Endpoints: The segment goes from $(-3, 2)$ to $(2, -2)$.
* Calculation:
* Change in $x$: $2 - (-3) = 5$
* Change in $y$: $-2 - 2 = -4$
* $d = \sqrt{5^2 + (-4)^2}$
* $d = \sqrt{25 + 16}$
* $d = \sqrt{41}$
* $\sqrt{41} \approx 6.403$
* Rounding: To the nearest tenth, this is 6.4.

Problem 8)
* Endpoints: The segment goes from $(-2, -3)$ to $(3, 2)$.
* Calculation:
* Change in $x$: $3 - (-2) = 5$
* Change in $y$: $2 - (-3) = 5$
* $d = \sqrt{5^2 + 5^2}$
* $d = \sqrt{25 + 25}$
* $d = \sqrt{50}$
* $\sqrt{50} \approx 7.071$
* Rounding: To the nearest tenth, this is 7.1.

Problem 9)
* Endpoints: The segment goes from $(-1, -4)$ to $(4, 1)$.
* Calculation:
* Change in $x$: $4 - (-1) = 5$
* Change in $y$: $1 - (-4) = 5$
* $d = \sqrt{5^2 + 5^2}$
* $d = \sqrt{25 + 25}$
* $d = \sqrt{50}$
* $\sqrt{50} \approx 7.071$
* Rounding: To the nearest tenth, this is 7.1.

Final Answer:
1) 5.8
2) 4.2
3) 8.1
4) 6.7
5) 5.4
6) 7.8
7) 6.4
8) 7.1
9) 7.1