Pythagoras and the Pythagorean Theorem: A math worksheet featuring problems to solve using the theorem.

It looks like you've uploaded an image with a math puzzle involving the Pythagorean Theorem and a sketch of Pythagoras, suggesting that this is a Pythagorean theorem problem set. The task is to fill in the missing values for $ a $, $ b $, or $ c $ in right triangles using:

$$
a^2 + b^2 = c^2
$$

where $ c $ is the hypotenuse (longest side), and $ a $, $ b $ are the legs.

Let’s go through each lettered problem one by one and solve it step by step.

---

S: $ a = 7 $, $ b = \_\_\_\_\_\_ $, $ c = 12 $

We use:
$$
a^2 + b^2 = c^2 \\
7^2 + b^2 = 12^2 \\
49 + b^2 = 144 \\
b^2 = 144 - 49 = 95 \\
b = \sqrt{95} \approx 9.75
$$

✔ $ b = \sqrt{95} $

---

O: $ a = 5 $, $ b = \_\_\_\_\_\_ $, $ c = 14 $

$$
5^2 + b^2 = 14^2 \\
25 + b^2 = 196 \\
b^2 = 171 \\
b = \sqrt{171} = \sqrt{9 \cdot 19} = 3\sqrt{19}
$$

✔ $ b = \sqrt{171} $

---

H: $ a = 8 $, $ b = \_\_\_\_\_\_ $, $ c = \sqrt{164} $

$$
8^2 + b^2 = (\sqrt{164})^2 \\
64 + b^2 = 164 \\
b^2 = 100 \\
b = 10
$$

✔ $ b = 10 $

---

I: $ a = 4 $, $ b = 11 $, $ c = \_\_\_\_\_\_ $

$$
c^2 = 4^2 + 11^2 = 16 + 121 = 137 \\
c = \sqrt{137}
$$

✔ $ c = \sqrt{137} $

---

E: $ a = 12 $, $ b = 5 $, $ c = \_\_\_\_\_\_ $

$$
c^2 = 12^2 + 5^2 = 144 + 25 = 169 \\
c = \sqrt{169} = 13
$$

✔ $ c = 13 $

---

L: $ a = \_\_\_\_\_\_ $, $ b = 7 $, $ c = 10 $

$$
a^2 + 7^2 = 10^2 \\
a^2 + 49 = 100 \\
a^2 = 51 \\
a = \sqrt{51}
$$

✔ $ a = \sqrt{51} $

---

F: $ a = \_\_\_\_\_\_ $, $ b = \sqrt{48} $, $ c = 13 $

Note: $ \sqrt{48} = \sqrt{16 \cdot 3} = 4\sqrt{3} $

$$
a^2 + (\sqrt{48})^2 = 13^2 \\
a^2 + 48 = 169 \\
a^2 = 121 \\
a = 11
$$

✔ $ a = 11 $

---

K: $ a = \_\_\_\_\_\_ $, $ b = 12 $, $ c = 15 $

$$
a^2 + 12^2 = 15^2 \\
a^2 + 144 = 225 \\
a^2 = 81 \\
a = 9
$$

✔ $ a = 9 $

---

U: $ a = 10 $, $ b = \_\_\_\_\_\_ $, $ c = 16 $

$$
10^2 + b^2 = 16^2 \\
100 + b^2 = 256 \\
b^2 = 156 \\
b = \sqrt{156} = \sqrt{4 \cdot 39} = 2\sqrt{39}
$$

✔ $ b = \sqrt{156} $

---

W: $ a = 1 $, $ b = \_\_\_\_\_\_ $, $ c = 2 $

$$
1^2 + b^2 = 2^2 \\
1 + b^2 = 4 \\
b^2 = 3 \\
b = \sqrt{3}
$$

✔ $ b = \sqrt{3} $

---

M: $ a = 1 $, $ b = 1 $, $ c = \_\_\_\_\_\_ $

$$
c^2 = 1^2 + 1^2 = 1 + 1 = 2 \\
c = \sqrt{2}
$$

✔ $ c = \sqrt{2} $

---

T: $ a = 0.8 $, $ b = 0.6 $, $ c = \_\_\_\_\_\_ $

$$
c^2 = (0.8)^2 + (0.6)^2 = 0.64 + 0.36 = 1.00 \\
c = \sqrt{1} = 1
$$

✔ $ c = 1 $

---

N: $ a = \_\_\_\_\_\_ $, $ b = 1.5 $, $ c = 2.5 $

$$
a^2 + (1.5)^2 = (2.5)^2 \\
a^2 + 2.25 = 6.25 \\
a^2 = 4 \\
a = 2
$$

✔ $ a = 2 $

---

Y: $ a = \_\_\_\_\_\_ $, $ b = 11 $, $ c = 17 $

$$
a^2 + 11^2 = 17^2 \\
a^2 + 121 = 289 \\
a^2 = 168 \\
a = \sqrt{168} = \sqrt{4 \cdot 42} = 2\sqrt{42}
$$

✔ $ a = \sqrt{168} $

---

A: $ a = \_\_\_\_\_\_ $, $ b = 24 $, $ c = 25 $

This is a classic Pythagorean triple!

$$
a^2 + 24^2 = 25^2 \\
a^2 + 576 = 625 \\
a^2 = 49 \\
a = 7
$$

✔ $ a = 7 $

---

P: $ a = \sqrt{75} $, $ b = \_\_\_\_\_\_ $, $ c = 15 $

Note: $ \sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3} $

$$
(\sqrt{75})^2 + b^2 = 15^2 \\
75 + b^2 = 225 \\
b^2 = 150 \\
b = \sqrt{150} = \sqrt{25 \cdot 6} = 5\sqrt{6}
$$

✔ $ b = \sqrt{150} $

---

R: $ a = \sqrt{87} $, $ b = \sqrt{57} $, $ c = \_\_\_\_\_\_ $

$$
c^2 = (\sqrt{87})^2 + (\sqrt{57})^2 = 87 + 57 = 144 \\
c = \sqrt{144} = 12
$$

✔ $ c = 12 $

---

G: $ a = \_\_\_\_\_\_ $, $ b = 3 $, $ c = 5 $

$$
a^2 + 3^2 = 5^2 \\
a^2 + 9 = 25 \\
a^2 = 16 \\
a = 4
$$

✔ $ a = 4 $

---

✔ Final Answers:

| Letter | Missing Value |
|--------|----------------|
| S | $ b = \sqrt{95} $ |
| O | $ b = \sqrt{171} $ |
| H | $ b = 10 $ |
| I | $ c = \sqrt{137} $ |
| E | $ c = 13 $ |
| L | $ a = \sqrt{51} $ |
| F | $ a = 11 $ |
| K | $ a = 9 $ |
| U | $ b = \sqrt{156} $ |
| W | $ b = \sqrt{3} $ |
| M | $ c = \sqrt{2} $ |
| T | $ c = 1 $ |
| N | $ a = 2 $ |
| Y | $ a = \sqrt{168} $ |
| A | $ a = 7 $ |
| P | $ b = \sqrt{150} $ |
| R | $ c = 12 $ |
| G | $ a = 4 $ |

---

Now, if this is a code puzzle, perhaps the letters correspond to something when filled in — maybe the answers form a word or name?

Let’s look at the letters and their solutions:

- S: $ \sqrt{95} $
- O: $ \sqrt{171} $
- H: 10
- I: $ \sqrt{137} $
- E: 13
- L: $ \sqrt{51} $
- F: 11
- K: 9
- U: $ \sqrt{156} $
- W: $ \sqrt{3} $
- M: $ \sqrt{2} $
- T: 1
- N: 2
- Y: $ \sqrt{168} $
- A: 7
- P: $ \sqrt{150} $
- R: 12
- G: 4

But notice that several of these are integers:

- H: 10
- E: 13
- F: 11
- K: 9
- T: 1
- N: 2
- A: 7
- R: 12
- G: 4

Wait — look at the letters that have integer values:

- H: 10
- E: 13
- F: 11
- K: 9
- T: 1
- N: 2
- A: 7
- R: 12
- G: 4

And the non-integers might be ignored.

But let's try ordering them alphabetically or see if they spell something.

Alternatively, perhaps the values correspond to positions in the alphabet?

Let’s check:

- H = 8 → value = 10 → not matching
- E = 5 → value = 13 → no
- F = 6 → value = 11 → no

Wait — what if we look at the letters themselves and see if they spell something?

List of all letters:
S, O, H, I, E, L, F, K, U, W, M, T, N, Y, A, P, R, G

That’s almost the whole alphabet — but perhaps it's a word?

Wait — "SOHILFKUWMTNYAPRG" — doesn’t make sense.

But wait — perhaps it's Pythagoras? But we have many more letters.

Alternatively, perhaps the answers are clues to a word.

But here's a better idea: look at the letters where the answer is an integer:

- H: 10 → H = 8th letter, value = 10 → no
- E: 13 → E = 5th, value = 13 → no
- F: 11 → F = 6th → no
- K: 9 → K = 11th → no
- T: 1 → T = 20th → no
- N: 2 → N = 14th → no
- A: 7 → A = 1st → yes! A=1, value=7 → no
- R: 12 → R = 18th → no
- G: 4 → G = 7th → no

No clear pattern.

Wait — perhaps the problem is just to fill in the blanks, and the picture of Pythagoras is just decorative.

But since the user said “solve the problem and explain”, likely the main goal is to compute the missing values.

So the solution is filling in the blanks as above.

If you're supposed to decode a message, maybe from the letters corresponding to values, or from the missing values being integers, then let’s extract only the ones with integer solutions:

- H: b = 10 → H → 10
- E: c = 13 → E → 13
- F: a = 11 → F → 11
- K: a = 9 → K → 9
- T: c = 1 → T → 1
- N: a = 2 → N → 2
- A: a = 7 → A → 7
- R: c = 12 → R → 12
- G: a = 4 → G → 4

Now list the letters: H, E, F, K, T, N, A, R, G

Sort them: A, E, F, G, H, K, N, R, T

Values: 10, 13, 11, 9, 1, 2, 7, 12, 4

Still no obvious pattern.

Wait — what if we take the values and map them to letters?

For example:

- 1 → A
- 2 → B
- ...
- 4 → D
- 7 → G
- 9 → I
- 10 → J
- 11 → K
- 12 → L
- 13 → M

So:

- H: 10 → J
- E: 13 → M
- F: 11 → K
- K: 9 → I
- T: 1 → A
- N: 2 → B
- A: 7 → G
- R: 12 → L
- G: 4 → D

So sequence: J, M, K, I, A, B, G, L, D → not meaningful.

Alternatively, perhaps the letters are meant to be arranged based on the value of the missing side.

But without more context, it seems most likely that the task is simply to apply the Pythagorean Theorem to fill in the blanks.

---

✔ Final Answer (filled):

$$
\begin{align*}
\text{S} & : a = 7, b = \sqrt{95}, c = 12 \\
\text{O} & : a = 5, b = \sqrt{171}, c = 14 \\
\text{H} & : a = 8, b = 10, c = \sqrt{164} \\
\text{I} & : a = 4, b = 11, c = \sqrt{137} \\
\text{E} & : a = 12, b = 5, c = 13 \\
\text{L} & : a = \sqrt{51}, b = 7, c = 10 \\
\text{F} & : a = 11, b = \sqrt{48}, c = 13 \\
\text{K} & : a = 9, b = 12, c = 15 \\
\text{U} & : a = 10, b = \sqrt{156}, c = 16 \\
\text{W} & : a = 1, b = \sqrt{3}, c = 2 \\
\text{M} & : a = 1, b = 1, c = \sqrt{2} \\
\text{T} & : a = 0.8, b = 0.6, c = 1 \\
\text{N} & : a = 2, b = 1.5, c = 2.5 \\
\text{Y} & : a = \sqrt{168}, b = 11, c = 17 \\
\text{A} & : a = 7, b = 24, c = 25 \\
\text{P} & : a = \sqrt{75}, b = \sqrt{150}, c = 15 \\
\text{R} & : a = \sqrt{87}, b = \sqrt{57}, c = 12 \\
\text{G} & : a = 4, b = 3, c = 5 \\
\end{align*}
$$

These are the complete solutions using the Pythagorean Theorem.

Let me know if there's a hidden word or code — but based on the image and content, solving each triangle is the main task.