TenTors Math Teacher Resources: Pythagorean Theorem Activity - Free Printable

Problem Overview:

The task involves solving a series of Pythagorean theorem problems to find the value of \( x \) for each triangle. Once the values of \( x \) are determined, they will be linked to a table at the top of the image to decode a message in the four boxes at the bottom.

Pythagorean Theorem:

The Pythagorean theorem states that for a right triangle with legs \( a \) and \( b \), and hypotenuse \( c \):
\[
a^2 + b^2 = c^2
\]
We will use this formula to solve for \( x \) in each triangle.

Step-by-Step Solution:

#### 1. Triangle 1:
Given:
- One leg = 9
- Other leg = 12
- Hypotenuse = \( x \)

Using the Pythagorean theorem:
\[
9^2 + 12^2 = x^2
\]
\[
81 + 144 = x^2
\]
\[
225 = x^2
\]
\[
x = \sqrt{225} = 15
\]

#### 2. Triangle 2:
Given:
- One leg = 3
- Other leg = 4
- Hypotenuse = \( x \)

Using the Pythagorean theorem:
\[
3^2 + 4^2 = x^2
\]
\[
9 + 16 = x^2
\]
\[
25 = x^2
\]
\[
x = \sqrt{25} = 5
\]

#### 3. Triangle 3:
Given:
- One leg = 8
- Other leg = \( x \)
- Hypotenuse = 16.1

Using the Pythagorean theorem:
\[
8^2 + x^2 = 16.1^2
\]
\[
64 + x^2 = 259.21
\]
\[
x^2 = 259.21 - 64
\]
\[
x^2 = 195.21
\]
\[
x = \sqrt{195.21} \approx 13.97
\]

#### 4. Triangle 4:
Given:
- One leg = 8
- Other leg = 15
- Hypotenuse = \( x \)

Using the Pythagorean theorem:
\[
8^2 + 15^2 = x^2
\]
\[
64 + 225 = x^2
\]
\[
289 = x^2
\]
\[
x = \sqrt{289} = 17
\]

#### 5. Triangle 5:
Given:
- One leg = 12
- Other leg = \( x \)
- Hypotenuse = 14

Using the Pythagorean theorem:
\[
12^2 + x^2 = 14^2
\]
\[
144 + x^2 = 196
\]
\[
x^2 = 196 - 144
\]
\[
x^2 = 52
\]
\[
x = \sqrt{52} \approx 7.21
\]

#### 6. Triangle 6:
Given:
- One leg = 4.4
- Other leg = \( x \)
- Hypotenuse = 6.5

Using the Pythagorean theorem:
\[
4.4^2 + x^2 = 6.5^2
\]
\[
19.36 + x^2 = 42.25
\]
\[
x^2 = 42.25 - 19.36
\]
\[
x^2 = 22.89
\]
\[
x = \sqrt{22.89} \approx 4.79
\]

#### 7. Triangle 7:
Given:
- One leg = 8
- Other leg = \( x \)
- Hypotenuse = 12

Using the Pythagorean theorem:
\[
8^2 + x^2 = 12^2
\]
\[
64 + x^2 = 144
\]
\[
x^2 = 144 - 64
\]
\[
x^2 = 80
\]
\[
x = \sqrt{80} \approx 8.94
\]

#### 8. Triangle 8:
Given:
- One leg = 7
- Other leg = \( x \)
- Hypotenuse = 25

Using the Pythagorean theorem:
\[
7^2 + x^2 = 25^2
\]
\[
49 + x^2 = 625
\]
\[
x^2 = 625 - 49
\]
\[
x^2 = 576
\]
\[
x = \sqrt{576} = 24
\]

#### 9. Triangle 9:
Given:
- One leg = 6
- Other leg = 15
- Hypotenuse = \( x \)

Using the Pythagorean theorem:
\[
6^2 + 15^2 = x^2
\]
\[
36 + 225 = x^2
\]
\[
261 = x^2
\]
\[
x = \sqrt{261} \approx 16.16
\]

#### 10. Triangle 10:
Given:
- One leg = 3.6
- Other leg = \( x \)
- Hypotenuse = 8.2

Using the Pythagorean theorem:
\[
3.6^2 + x^2 = 8.2^2
\]
\[
12.96 + x^2 = 67.24
\]
\[
x^2 = 67.24 - 12.96
\]
\[
x^2 = 54.28
\]
\[
x = \sqrt{54.28} \approx 7.37
\]

#### 11. Triangle 11:
Given:
- One leg = 20
- Other leg = \( x \)
- Hypotenuse = 29

Using the Pythagorean theorem:
\[
20^2 + x^2 = 29^2
\]
\[
400 + x^2 = 841
\]
\[
x^2 = 841 - 400
\]
\[
x^2 = 441
\]
\[
x = \sqrt{441} = 21
\]

#### 12. Triangle 12:
Given:
- One leg = 85
- Other leg = \( x \)
- Hypotenuse = 36

Using the Pythagorean theorem:
\[
85^2 + x^2 = 36^2
\]
\[
7225 + x^2 = 1296
\]
This equation is not possible because \( 7225 > 1296 \). There might be an error in the problem setup or interpretation.

Final Values of \( x \):

1. 15
2. 5
3. 13.97
4. 17
5. 7.21
6. 4.79
7. 8.94
8. 24
9. 16.16
10. 7.37
11. 21
12. (Not solvable as given)

Linking to the Table:

Using the values of \( x \) and the table provided, we can decode the message in the four boxes at the bottom. However, since the exact decoding mechanism is not specified in the image, we assume the values correspond directly to letters in the table.

Final Answer:

\[
\boxed{15, 5, 13.97, 17, 7.21, 4.79, 8.94, 24, 16.16, 7.37, 21}
\]