Pythagorean Theorem Worksheets | Practicing Pythagorean Theorem - Free Printable

Here are the step-by-step solutions for each problem on the worksheet. We will use the Pythagorean Theorem ($a^2 + b^2 = c^2$) to solve them, where $a$ and $b$ are the legs of a right triangle and $c$ is the hypotenuse (the longest side).

1. Find the length of the hypotenuse of each right triangle.
* A: Legs are 7 cm and 10 cm.
$$7^2 + 10^2 = c^2$$
$$49 + 100 = 149$$
$$\sqrt{149} \approx 12.2$$
Answer: 12.2 cm

* B: Legs are 8 m and 13 m.
$$8^2 + 13^2 = c^2$$
$$64 + 169 = 233$$
$$\sqrt{233} \approx 15.3$$ (Note: Looking at the answer key options, there is no 15.3m. Let's re-read carefully. Ah, looking at the diagram for B, the right angle is between the side labeled 8 and the unknown side? No, the square symbol is between the side labeled 8 and the side connecting to the 13 side. Wait, usually the longest side is the hypotenuse. If 13 is a leg and 8 is a leg, hypotenuse is $\approx 15.3$. If 13 is the hypotenuse and 8 is a leg, then $13^2 - 8^2 = 169 - 64 = 105$, $\sqrt{105} \approx 10.2$. Neither matches the key exactly. Let's look closer at the image. The right angle mark is clearly between the side labeled '8 m' and the bottom side. The side labeled '13 m' is the hypotenuse. So we are finding a leg.
$$a^2 + 8^2 = 13^2$$
$$a^2 + 64 = 169$$
$$a^2 = 105$$
$$a = \sqrt{105} \approx 10.2$$
*Correction*: Looking at the provided answer choices at the bottom, I see "10.2" is not there, but "15. m" is. Let me re-examine triangle B. The right angle is between the side labeled 8 and the side labeled 13? No, that would make the third side the hypotenuse. Let's assume standard positioning: legs are 8 and 13. Hypotenuse = $\sqrt{233} \approx 15.26$. This rounds to 15.3. Wait, let me check the answer key letters again.
Let's skip B for a second and do C.
* C: Legs are 12 in. and 12 in.
$$12^2 + 12^2 = c^2$$
$$144 + 144 = 288$$
$$\sqrt{288} \approx 16.97$$ which rounds to 17.0 in.

*Re-evaluating B*: Let's look at the answer bank. There is a "15. m". $\sqrt{233}$ is 15.26. Rounding to nearest tenth is 15.3. Is there a typo in my reading or the book? Let's look at the other answers first to see what letters remain.

2. A rectangle is 6 m wide and 11 m long. How long is the diagonal?
The diagonal forms the hypotenuse of a triangle with legs 6 and 11.
$$6^2 + 11^2 = c^2$$
$$36 + 121 = 157$$
$$\sqrt{157} \approx 12.52$$
Answer: 12.5 m

3. TV screen is 20 in. wide and 15 in. high. Length of diagonal?
$$20^2 + 15^2 = c^2$$
$$400 + 225 = 625$$
$$\sqrt{625} = 25$$
Answer: 25 in.

4. Quarterback pass from A to B.
The horizontal distance is 14 yd. The vertical distance is 25 yd.
$$14^2 + 25^2 = c^2$$
$$196 + 625 = 821$$
$$\sqrt{821} \approx 28.65$$
Answer: 28.7 yd

5. Rope from top of 7-foot pole to ground 12 ft away.
Legs are 7 and 12.
$$7^2 + 12^2 = c^2$$
$$49 + 144 = 193$$
$$\sqrt{193} \approx 13.89$$
Answer: 13.9 ft

6. Hiked 5 mi west and 2 mi north. Distance from start?
Legs are 5 and 2.
$$5^2 + 2^2 = c^2$$
$$25 + 4 = 29$$
$$\sqrt{29} \approx 5.38$$
Answer: 5.4 mi

7. Ladder reaching window 40 feet high, base 9 feet from building.
Legs are 40 and 9.
$$40^2 + 9^2 = c^2$$
$$1600 + 81 = 1681$$
$$\sqrt{1681} = 41$$
Answer: 41 ft

8. Baseball diamond bases 90 feet apart. Home to second base?
This is the diagonal of a square with side 90.
$$90^2 + 90^2 = c^2$$
$$8100 + 8100 = 16200$$
$$\sqrt{16200} \approx 127.27$$
Answer: 127.3 ft

9. Lawn 24 m long and 10 m wide. How much shorter is diagonal walk than two sides?
First, find the diagonal:
$$24^2 + 10^2 = c^2$$
$$576 + 100 = 676$$
$$\sqrt{676} = 26 \text{ m}$$
Walking along two sides: $24 + 10 = 34 \text{ m}$.
Difference: $34 - 26 = 8 \text{ m}$.
Answer: 8 m

***

Now, let's go back to Problem 1B and match everything to the letters.

Let's list our calculated answers and find them in the code box at the bottom:

1. 1A: 12.2 cm. Look for "12.2 cm" in the box. It is under the letter O.
2. 1C: 17.0 in. Look for "17.0 in." in the box. It is under the letter K.
3. 2: 12.5 m. Look for "12.5 m" in the box. It is under the letter D.
4. 3: 25 in. Look for "25 in." in the box. It is under the letter E.
5. 4: 28.7 yd. Look for "28.7 yd" in the box. It is under the letter A.
6. 5: 13.9 ft. Look for "13.9 ft" in the box. It is under the letter S.
7. 6: 5.4 mi. Look for "5.4 i" (likely mi cut off) in the box. It is under the letter D.
8. 7: 41 ft. Look for "41 ft" in the box. It is under the letter T.
9. 8: 127.3 ft. Look for "127.3 f" in the box. It is under the letter N.
10. 9: 8 m. Look for "8 m" in the box. It is under the letter A.

We still have 1B left. The remaining letters in the title "How Would You Describe a Dead Skunk?" need to be filled.
The letters we have found so far correspond to these positions in the riddle answer (usually the riddle answer is written out, but here we cross out letters).
Actually, the instruction says: "Find each answer... and cross out the letter above it. When you finish, the answer to the title question will remain."

Let's look at the letters available in the box:
D, E, S, A, X, D, T, N, O, I, S, N, T, A, C, K, T, E

Let's map the answers we found to the letters we cross out:
- 12.2 cm -> O
- 17.0 in -> K
- 12.5 m -> D
- 25 in -> E
- 28.7 yd -> A
- 13.9 ft -> S
- 5.4 mi -> D
- 41 ft -> T
- 127.3 ft -> N
- 8 m -> A

The letters crossed out are: O, K, D, E, A, S, D, T, N, A.

Let's look at the full strip of letters:
`D E S A X D T N O I S N T A C K T E`

Crossing out the ones we found:
- D (from #2 or #6)
- E (from #3)
- S (from #5)
- A (from #4 or #9)
- X (Not found yet)
- D (from #2 or #6)
- T (from #7)
- N (from #8)
- O (from #1A)
- I (Not found yet)
- S (Wait, did we have two S answers? No. But there are two S's in the box. Did I miss an answer?)
- N (There are two N's in the box. Did I miss an answer?)
- T (There are two T's in the box.)
- A (from #4 or #9)
- C (Not found yet)
- K (from #1C)
- T (from #7?? No, only one 41ft answer)
- E (There are two E's in the box.)

Let's re-read Problem 1B.
Triangle B has sides 8m and 13m.
If they are legs: $\sqrt{8^2+13^2} = \sqrt{64+169} = \sqrt{233} \approx 15.26$. Rounds to 15.3.
Is there a 15.3 in the box? No. There is "15. m".
If the answer is 15.m, the letter above it is S.
But we already used an S for 13.9 ft (#5).
Let's check if 13.9 ft could be something else. No, $\sqrt{193}$ is definitely 13.9.
Let's check if 15.m corresponds to Problem 1B.
If 1B is 15.m (letter S), then we have crossed out two S's.
Are there two problems with answer S?
Problem 5 is 13.9 ft (Letter S).
Problem 1B is ~15.3 m. The box has "15. m" under Letter S.
It seems likely that 1B is intended to be matched with "15. m" (rounding 15.26 down or truncating? Or maybe the side was different? If side was 12 and 9, hyp is 15. If side was 8 and ...? If hyp is 17 and leg is 8, other leg is 15. Maybe the diagram meant 17? No, it says 13.
However, "15. m" is the only option close to 15.3. And the letter is S.
So, we cross out another S.

Let's see what letters are LEFT after crossing out:
Original: `D E S A X D T N O I S N T A C K T E`

Crossed out:
- D (Prob 2: 12.5m)
- E (Prob 3: 25in)
- S (Prob 5: 13.9ft)
- A (Prob 4: 28.7yd)
- D (Prob 6: 5.4mi)
- T (Prob 7: 41ft)
- N (Prob 8: 127.3ft)
- O (Prob 1A: 12.2cm)
- S (Prob 1B: 15.m) -- *Assuming this match*
- A (Prob 9: 8m)
- K (Prob 1C: 17.0in)

Let's physically cross them off the list `D E S A X D T N O I S N T A C K T E`:
1. D (gone)
2. E (gone)
3. S (gone)
4. A (gone)
5. X (remains)
6. D (gone)
7. T (gone)
8. N (gone)
9. O (gone)
10. I (remains)
11. S (gone)
12. N (remains? Wait, did we use both Ns? Prob 8 was 127.3. Is there another N answer? No. So one N remains.)
13. T (remains? Prob 7 was 41ft. Is there another T answer? No. So one T remains.)
14. A (gone)
15. C (remains)
16. K (gone)
17. T (Wait, I counted T twice in my "remains" logic. Let's look at the indices.)

Let's index the box:
1:D, 2:E, 3:S, 4:A, 5:X, 6:D, 7:T, 8:N, 9:O, 10:I, 11:S, 12:N, 13:T, 14:A, 15:C, 16:K, 17:T, 18:E

Matches:
- Prob 1A (12.2cm) -> Pos 9 (O)
- Prob 1B (15.m) -> Pos 11 (S) [Best fit]
- Prob 1C (17.0in) -> Pos 16 (K)
- Prob 2 (12.5m) -> Pos 6 (D) [Pos 1 is also D, doesn't matter which one we cross]
- Prob 3 (25in) -> Pos 18 (E) [Pos 2 is also E]
- Prob 4 (28.7yd) -> Pos 14 (A) [Pos 4 is also A]
- Prob 5 (13.9ft) -> Pos 3 (S) [Pos 11 is taken by 1B]
- Prob 6 (5.4mi) -> Pos 1 (D) [Pos 6 taken by Prob 2]
- Prob 7 (41ft) -> Pos 7 (T) [Pos 13 and 17 are also T]
- Prob 8 (127.3ft) -> Pos 8 (N) [Pos 12 is also N]
- Prob 9 (8m) -> Pos 4 (A) [Pos 14 taken by Prob 4]

Let's cross out specific indices:
- Cross 9 (O)
- Cross 11 (S)
- Cross 16 (K)
- Cross 6 (D)
- Cross 18 (E)
- Cross 14 (A)
- Cross 3 (S)
- Cross 1 (D)
- Cross 7 (T)
- Cross 8 (N)
- Cross 4 (A)

Remaining Indices:
- 2: E (Was not crossed, we crossed 18)
- 5: X
- 10: I
- 12: N (Was not crossed, we crossed 8)
- 13: T (Was not crossed, we crossed 7)
- 15: C
- 17: T (Was not crossed, we crossed 7)

Remaining Letters: E, X, I, N, T, C, T

Does "EXISTINCT" make sense? No.
Does "EXTINCT" make sense? Yes! "How would you describe a dead skunk? EXTINCT." (A joke, since skunks aren't extinct, but it fits the pun format).

Let's double check the leftover letters to spell EXTINCT.
We have: E, X, I, N, T, C, T.
Spelling: E-X-I-N-C-T. We have an extra T?
Let's re-verify the T's.
Box T positions: 7, 13, 17.
Answers with T:
- Prob 7 (41 ft) -> T.
Are there any other answers that end in T?
- 127.3 ft -> N (from 127.3 ft? No, the label in the box is "127.3 f", letter N. The unit is feet, but the value matches N).
- 13.9 ft -> S.
- 5.4 mi -> D.
- 25 in -> E.
- 17.0 in -> K.
- 12.2 cm -> O.
- 15. m -> S.
- 12.5 m -> D.
- 28.7 yd -> A.
- 8 m -> A.

Only one answer maps to T (41 ft).
There are three T's in the box (indices 7, 13, 17).
We crossed out index 7.
Indices 13 and 17 remain.
So we have two T's left.
Letters left: E (idx 2), X (idx 5), I (idx 10), N (idx 12), T (idx 13), C (idx 15), T (idx 17).
Spell: E X I N T C T.
This spells EXTINCT if you rearrange? No, EXTINCT is E-X-I-N-C-T.
We have E-X-I-N-T-C-T.
Is there an error in my crossing?
Maybe Prob 8 (127.3) is T?
Box at Pos 13 is T. Value "127.3 f".
Box at Pos 8 is N. Value "127.3 f".
Wait, look closely at the image crop.
Pos 8: Label "127.3 f", Letter N.
Pos 13: Label "127.3 f" ?? No, Pos 13 Label is "127.3 f" ?
Let's look at the original image again.
Pos 8: `N` over `127.3 f`.
Pos 13: `T` over `127.3 f`? No, Pos 13 is `T` over `127.3 f`?
Actually, looking at the spacing:
Pos 7: T / 41 ft
Pos 8: N / 127.3 f
Pos 9: O / 12.2 cm
...
Pos 13: T / 127.3 f ??
Let's read the values under the letters carefully from left to right.
1:D/5.4i
2:E/29.yd -> Wait. 29. yd. I missed this value earlier.
3:S/15.m
4:A/8m
5:X/13.2m
6:D/12.5m
7:T/16.7in. -> Wait. 16.7 in.
8:N/41ft -> Wait. 41 ft is under N?
Let's re-read the entire bottom strip carefully.

Re-reading the Answer Key Strip:
1. D: 5.4 i (mi)
2. E: 29. yd
3. S: 15. m
4. A: 8 m
5. X: 13.2 m
6. D: 12.5 m
7. T: 16.7 in.
8. N: 41 ft
9. O: 12.2 cm
10. I: 6.1 mi
11. S: 13.9 ft
12. N: 42.5 ft
13. T: 127.3 f
14. A: 28.7 yd
15. C: 14.4 ft
16. K: 17.0 in.
17. T: 129.8 tf
18. E: 25 in.

Okay, my previous reading of the values was scrambled. Let's re-match the calculated answers to this correct list.

Recalculated Answers:
1A: 12.2 cm. Matches Pos 9 (O).
1B: ~15.3 m. Matches Pos 3 "15. m" (S). (Close enough rounding).
1C: 17.0 in. Matches Pos 16 (K).
2: 12.5 m. Matches Pos 6 (D).
3: 25 in. Matches Pos 18 (E).
4: 28.7 yd. Matches Pos 14 (A).
5: 13.9 ft. Matches Pos 11 (S).
6: 5.4 mi. Matches Pos 1 (D).
7: 41 ft. Matches Pos 8 (N).
8: 127.3 ft. Matches Pos 13 (T).
9: 8 m. Matches Pos 4 (A).

List of Crossed Out Letters:
- O (Pos 9)
- S (Pos 3)
- K (Pos 16)
- D (Pos 6)
- E (Pos 18)
- A (Pos 14)
- S (Pos 11)
- D (Pos 1)
- N (Pos 8)
- T (Pos 13)
- A (Pos 4)

Remaining Letters (Indices not crossed):
- Pos 2: E (Value 29. yd - unused)
- Pos 5: X (Value 13.2 m - unused)
- Pos 7: T (Value 16.7 in. - unused)
- Pos 10: I (Value 6.1 mi - unused)
- Pos 12: N (Value 42.5 ft - unused)
- Pos 15: C (Value 14.4 ft - unused)
- Pos 17: T (Value 129.8 tf - unused)

Remaining Letters: E, X, T, I, N, C, T

Rearranging these letters spells: EXTINCT.

Final Answer:
The remaining letters spell EXTINCT.