Pre-Algebra worksheet featuring Pythagorean Theorem word problems with diagrams and numerical exercises.

Here are the step-by-step solutions for each problem on the worksheet. We will use the Pythagorean Theorem, which is $a^2 + b^2 = c^2$, where $a$ and $b$ are the legs (the sides touching the right angle) and $c$ is the hypotenuse (the longest side opposite the right angle).

1)
* Identify sides: Leg $a = 6$, Hypotenuse $c = 10$. We need to find leg $b$.
* Formula: $6^2 + b^2 = 10^2$
* Calculate squares: $36 + b^2 = 100$
* Subtract 36 from both sides: $b^2 = 64$
* Square root: $\sqrt{64} = 8$
* Answer: 8.0

2)
* Identify sides: Leg $a = 7$, Leg $b = 5$. We need to find hypotenuse $c$.
* Formula: $7^2 + 5^2 = c^2$
* Calculate squares: $49 + 25 = c^2$
* Add: $74 = c^2$
* Square root: $\sqrt{74} \approx 8.602$
* Round to nearest tenth: 8.6

3)
* Identify sides: $a = 2$, $c = 3$. We need to find $b$.
* Formula: $2^2 + b^2 = 3^2$
* Calculate squares: $4 + b^2 = 9$
* Subtract 4 from both sides: $b^2 = 5$
* Square root: $\sqrt{5} \approx 2.236$
* Round to nearest tenth: 2.2

4)
* Identify sides: $a = 10$, $b = 12$. We need to find $c$.
* Formula: $10^2 + 12^2 = c^2$
* Calculate squares: $100 + 144 = c^2$
* Add: $244 = c^2$
* Square root: $\sqrt{244} \approx 15.62$
* Round to nearest tenth: 15.6

5)
* Scenario: A ladder forms a right triangle with the wall and ground.
* Identify sides: Distance from wall (leg) = 3 ft, Ladder length (hypotenuse) = 12 ft. Find height up the wall (leg).
* Formula: $3^2 + b^2 = 12^2$
* Calculate squares: $9 + b^2 = 144$
* Subtract 9 from both sides: $b^2 = 135$
* Square root: $\sqrt{135} \approx 11.61$
* Round to nearest tenth: 11.6 feet

6)
* Scenario: Running diagonally across a rectangular field creates a right triangle.
* Identify sides: Width (leg) = 90 m, Length (leg) = 120 m. Find diagonal (hypotenuse).
* Formula: $90^2 + 120^2 = c^2$
* Calculate squares: $8100 + 14400 = c^2$
* Add: $22500 = c^2$
* Square root: $\sqrt{22500} = 150$
* Answer: 150 meters

7)
* Scenario: A ladder leaning against a house.
* Identify sides: Ladder length (hypotenuse) = 15 ft, Wall height (leg) = 12 ft. Find distance from base (leg).
* Formula: $a^2 + 12^2 = 15^2$
* Calculate squares: $a^2 + 144 = 225$
* Subtract 144 from both sides: $a^2 = 81$
* Square root: $\sqrt{81} = 9$
* Answer: 9 feet

8)
* Part 1: Side length. Area of a square is $side \times side$. Since Area = 81, we find $\sqrt{81} = 9$. So, side = 9 cm.
* Part 2: Diagonal. The diagonal cuts the square into two right triangles with legs of 9.
* Formula: $9^2 + 9^2 = c^2$
* Calculate squares: $81 + 81 = c^2$
* Add: $162 = c^2$
* Square root: $\sqrt{162} \approx 12.72$
* Round to nearest tenth: 12.7
* Answer: Side = 9 cm, Diagonal = 12.7 cm

9)
* Scenario: Riding south then east creates a right angle.
* Identify sides: South distance (leg) = 9 km, East distance (leg) = 12 km. Find distance from start (hypotenuse).
* Formula: $9^2 + 12^2 = c^2$
* Calculate squares: $81 + 144 = c^2$
* Add: $225 = c^2$
* Square root: $\sqrt{225} = 15$
* Answer: 15 km

10)
* Scenario: Finding the length of a rectangle given width and diagonal.
* Identify sides: Width (leg) = 15 ft, Diagonal (hypotenuse) = 25 ft. Find Length (leg).
* Formula: $a^2 + 15^2 = 25^2$
* Calculate squares: $a^2 + 225 = 625$
* Subtract 225 from both sides: $a^2 = 400$
* Square root: $\sqrt{400} = 20$
* Answer: 20 feet

Final Answer:
1) 8.0
2) 8.6
3) 2.2
4) 15.6
5) 11.6 feet
6) 150 meters
7) 9 feet
8) Side = 9 cm, Diagonal = 12.7 cm
9) 15 km
10) 20 feet