Worksheet on special right triangles, including 45-45-90 and 30-60-90 triangles, with problems to find missing sides and a word puzzle.

Let’s solve each triangle one by one using the special right triangle rules.

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Isosceles Right Triangle (45-45-90):
- Legs are equal.
- Hypotenuse = leg × √2
- If hypotenuse is given, leg = hypotenuse ÷ √2

30-60-90 Triangle:
- Side opposite 30° = shortest side = let’s call it “a”
- Side opposite 60° = a√3
- Hypotenuse = 2a

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Now let’s go problem by problem:

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1. Isosceles right triangle, legs = 6 and b, angle 45° → so both legs equal → b = 6

Hypotenuse = 6√2 → but we’re only asked for missing sides — here, b is the other leg → b = 6

Wait — looking at diagram: one leg is 6, angle 45°, so yes, other leg b = 6.

But wait — in diagram, side labeled “b” is the hypotenuse? Let me check.

Actually, in problem 1: triangle with right angle, one acute angle 45°, one leg = 6, other leg = b? Or is b the hypotenuse?

Looking at standard labeling: usually, side opposite angle is labeled with same letter. But here, since it's isosceles right triangle, and one leg is 6, and angle is 45°, then the other leg must also be 6. So if b is the other leg → b=6. But if b is hypotenuse → b=6√2.

In the diagram for #1: it shows a right triangle, bottom leg = 6, left side = ?, angle at bottom right = 45°, so the two legs are adjacent to the right angle. Since one angle is 45°, the other acute angle is also 45° → so it’s isosceles → both legs equal → so the vertical leg (which might be unlabeled) equals 6. But the side labeled “b” — where is it?

Actually, looking back at original image description: in problem 1, it says “find missing sides”, and labels one leg as 6, and another side as “b”. In most such diagrams, “b” is likely the hypotenuse.

Wait — let’s assume based on common practice:

In problem 1: right triangle, one leg = 6, angles 45-45-90 → so other leg = 6, hypotenuse = 6√2.

If “b” is the hypotenuse → b = 6√2

But in the answer grid later, we’ll match letters. Let’s proceed carefully.

Actually, let’s look at the reference box:

In Isosceles Right Triangle example: they show legs = a, hypotenuse = b, and say a=4, b=4√2.

So in their notation, “b” is hypotenuse.

Similarly, in 30-60-90, they label side opposite 30° as “a”, opposite 60° as “a√3”, hypotenuse as “b”.

So following that:

In problem 1: isosceles right triangle, one leg = 6 → so other leg = 6, hypotenuse = 6√2.

The side labeled “b” — if it’s the hypotenuse, then b = 6√2.

But in the diagram for #1, it might be labeled differently. Since I can’t see the exact labeling, I’ll assume based on standard problems.

To avoid confusion, let’s solve all and match to the answer grid at the end.

Let me list all 12 problems with solutions:

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Problem 1: 45-45-90, leg = 6 → hypotenuse = 6√2

Assume “b” is hypotenuse → b = 6√2

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Problem 2: 30-60-90, hypotenuse = 10 → so side opposite 30° = 5, side opposite 60° = 5√3

Diagram: angle 60° shown, hypotenuse = 10 → so side opposite 30° (shorter leg) = 5, side opposite 60° = 5√3

Which side is missing? Probably the two legs. But the problem says “find missing sides” — likely both, but in answer grid, we pick one letter per problem? Wait no — the instruction says: “Cross out the correct answers. The remaining letters (one per space) complete the statement.”

And there are 12 problems, and 12 blanks in the final statement.

Also, the answer grid has many options, and we need to select one correct answer per problem, cross it out, and the leftover letters spell the answer.

So for each problem, there is one missing side to find, and we choose the correct value from the grid.

Looking at the problems:

In problem 1: likely asking for hypotenuse, since one leg is given.

Similarly, problem 2: hypotenuse given, probably asking for shorter leg or longer leg.

Let’s go one by one with clear assumptions.

I’ll define for each problem what is given and what is missing, based on typical presentation.

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Problem 1: Isosceles right triangle, one leg = 6, find hypotenuse.

→ hypotenuse = 6√2

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Problem 2: 30-60-90, hypotenuse = 10, find side opposite 30° (shorter leg).

→ shorter leg = 10 / 2 = 5

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Problem 3: Isosceles right triangle, hypotenuse = 2√2, find leg.

→ leg = hypotenuse / √2 = 2√2 / √2 = 2

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Problem 4: 30-60-90, side opposite 60° = 6, find hypotenuse.

Side opposite 60° = a√3 = 6 → so a = 6/√3 = 2√3

Then hypotenuse = 2a = 4√3

But let’s compute: a√3 = 6 → a = 6/√3 = (6√3)/3 = 2√3

Hypotenuse = 2 * 2√3 = 4√3

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Problem 5: Isosceles right triangle, hypotenuse = 2√2, find leg.

Same as problem 3 → leg = 2

Wait, problem 5: diagram shows hypotenuse = 2√2, angle 45°, so leg = 2

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Problem 6: 30-60-90, side opposite 30° = 4√3? Wait no.

Diagram: angle 30° shown, side adjacent to it (which is side opposite 60°) = 4√3? Let's see.

Standard: in 30-60-90, side opposite 30° is shortest.

In problem 6: it shows a right triangle, angle 30° at bottom right, side next to it (adjacent) is labeled 4√3 — that would be the side opposite 60°.

Because: from 30° angle, adjacent side is the longer leg (opposite 60°).

So: side opposite 60° = 4√3 = a√3 → so a = 4

Then side opposite 30° = a = 4

Hypotenuse = 2a = 8

But which side is missing? The diagram might be asking for the shorter leg or hypotenuse.

Looking at the label: in problem 6, it says "4√3" on the base, and angle 30° at end, so likely the side opposite 60° is 4√3, so shorter leg (opposite 30°) is 4.

Probably asking for the shorter leg.

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Problem 7: Isosceles right triangle, hypotenuse = 6, find leg.

Leg = 6 / √2 = (6√2)/2 = 3√2

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Problem 8: 30-60-90, side opposite 30° = 9? Wait.

Diagram: angle 30° shown, side opposite to it is labeled 9? Or adjacent?

In problem 8: it shows right angle, angle 30°, and side labeled 9 — likely the side opposite 30°, because it's across from the 30° angle.

In standard position, if 30° is at top, side opposite is bottom leg.

Assume side opposite 30° = 9 → then hypotenuse = 18, side opposite 60° = 9√3

But which is missing? Probably the other sides, but we need to see what is asked.

Perhaps in this case, since it's labeled "9" and angle 30°, and it's a leg, likely it's the shorter leg.

But let's confirm with answer choices later.

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Problem 9: Isosceles right triangle, leg = 4, find hypotenuse.

Hypotenuse = 4√2

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Problem 10: 30-60-90, side opposite 60° = 6, find shorter leg.

Side opposite 60° = a√3 = 6 → a = 6/√3 = 2√3

Shorter leg = a = 2√3

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Problem 11: Isosceles right triangle, leg = 3, find hypotenuse.

Hypotenuse = 3√2

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Problem 12: 30-60-90, side opposite 30° = 13, find side opposite 60°.

Side opposite 60° = 13√3

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Now, let's list the answers we have:

1. 6√2
2. 5
3. 2
4. 4√3
5. 2
6. 4 (since side opposite 30° = 4)
7. 3√2
8. ? Let's reexamine.

For problem 8: diagram shows a right triangle, angle 30°, and side labeled 9. If 9 is the side opposite 30°, then hypotenuse is 18, but perhaps they want the other leg.

But in the context, likely they want the side opposite 60°, which is 9√3.

Let me check the answer grid to see what matches.

Perhaps I should list all expected answers and match to the grid.

From above:

P1: 6√2
P2: 5
P3: 2
P4: 4√3
P5: 2
P6: 4
P7: 3√2
P8: let's say if 9 is opposite 30°, then side opposite 60° is 9√3
P9: 4√2
P10: 2√3
P11: 3√2
P12: 13√3

Now, look at the answer grid provided in the image description:

The grid has:

Row 1: S, 9, 6√2, 3, 10, 3√2, 3, 4√3, 3, 12, 2√2
EQ, HA, UA, LT, LF, OT, HE, SQ, UA, RE, RO

Row 2: 6√3, 5√3, 25, 3√3, 6√3, 5, 20, 3, 3√3, 36, 2
OT, OF, TH, ER, AD, IU, EH, SO, FT, PY

Row 3: 11, 4, 16, 6, 8, 32, 5√2, 2, 7, 8√3, 2√2
OT, TH, EN, AG, OR, US, AS, T, E, T, S

We need to select one correct answer per problem, cross it out, and the remaining letters spell the answer to the statement.

The statement is: "In a 30-60-90 degrees right triangle, the side opposite the 30-degree angle is:"

And there are 12 blanks, so 12 letters left after crossing out 12 correct answers.

Total cells in grid: 3 rows x 11 columns = 33 cells.

We cross out 12 correct answers, so 21 left, but we need only 12 letters for the statement? That doesn't match.

Perhaps each "answer" corresponds to a cell, and we cross out the cell that contains the correct numerical answer for each problem, and the letter in that cell is crossed out, and the remaining letters (not crossed out) are used to fill the blanks.

But there are 12 problems, so we cross out 12 cells, each containing a number and a letter. After crossing out 12, there are 21 cells left, but the statement has 12 blanks, so probably we take the first 12 remaining or something? That seems messy.

Perhaps the "remaining letters" means the letters that are not crossed out, and we use them in order to fill the blanks, but there are more than 12.

Another possibility: for each problem, the correct answer is listed in the grid, and we cross out that entire entry (number and letter), and the letters that are left uncrossed are to be read in order to form the answer.

But with 33 entries, crossing out 12 leaves 21, but the statement has 12 blanks, so perhaps only specific positions.

This is confusing. Perhaps I misinterpreted.

Let me read the instruction again: "Cross out the correct answers. The remaining letters (one per space) complete the statement."

And "the remaining letters" — probably means after crossing out the correct numerical answers, the letters associated with those crossed-out answers are removed, and the leftover letters are used.

But there are 33 letters initially, cross out 12, left with 21, but the statement has 12 blanks, so maybe we take the first 12 or last 12? Unlikely.

Perhaps "the remaining letters" refers to the letters in the cells that were not selected as correct answers, and we arrange them in row-major order or something.

But that seems complicated for a student worksheet.

Another idea: perhaps for each problem, there is only one correct choice in the grid, and when you cross it out, the letter in that cell is eliminated, and the letters that are still there are to be read in order to spell the answer.

But again, 21 letters for 12 blanks.

Unless the grid is to be read as a whole, and after removing 12 letters, the remaining 21 are not all used; perhaps only the letters in certain positions.

I think I need to focus on solving the triangles correctly first, then match to the grid.

Let me list the correct answers for each problem based on standard interpretation:

1. 45-45-90, leg=6, find hypotenuse: 6√2
2. 30-60-90, hyp=10, find short leg: 5
3. 45-45-90, hyp=2√2, find leg: 2
4. 30-60-90, long leg=6, find hyp: as calculated, 4√3
5. 45-45-90, hyp=2√2, find leg: 2 (same as 3)
6. 30-60-90, long leg=4√3, find short leg: 4
7. 45-45-90, hyp=6, find leg: 3√2
8. 30-60-90, short leg=9, find long leg: 9√3
9. 45-45-90, leg=4, find hyp: 4√2
10. 30-60-90, long leg=6, find short leg: 2√3
11. 45-45-90, leg=3, find hyp: 3√2
12. 30-60-90, short leg=13, find long leg: 13√3

Now, let's map these to the grid.

Grid cells (row, col):

Row 1:
Col1: S / EQ
Col2: 9 / HA
Col3: 6√2 / UA
Col4: 3 / LT
Col5: 10 / LF
Col6: 3√2 / OT
Col7: 3 / HE
Col8: 4√3 / SQ
Col9: 3 / UA
Col10: 12 / RE
Col11: 2√2 / RO

Row 2:
Col1: 6√3 / OT
Col2: 5√3 / OF
Col3: 25 / TH
Col4: 3√3 / ER
Col5: 6√3 / AD
Col6: 5 / IU
Col7: 20 / EH
Col8: 3 / SO
Col9: 3√3 / FT
Col10: 36 / PY
Col11: 2 / ? wait, in the text it's "2" and "PY" for col10, but col11 is "2" and what? In the user input, for row2 col11: "2" and "PY"? No:

User input for row2: "6√3, 5√3, 25, 3√3, 6√3, 5, 20, 3, 3√3, 36, 2" and below "OT, OF, TH, ER, AD, IU, EH, SO, FT, PY" — that's 10 items, but should be 11. Mistake.

In the user's message:

"6√3, 5√3, 25, 3√3, 6√3, 5, 20, 3, 3√3, 36, 2" — that's 11 numbers.

Then "OT, OF, TH, ER, AD, IU, EH, SO, FT, PY" — that's 10 letters. Missing one.

Perhaps "2" corresponds to a letter, but it's not listed. In the text: "36, 2" and "FT, PY" — so probably "2" is paired with a letter, but it's cut off. Looking back:

In user input: "36, 2" and then "FT, PY" — but PY is for 36? Then 2 might be without letter, but that can't be.

Perhaps the last "2" is for col11, and the letter is missing in the text. In the initial description, it might be implied.

To resolve, let's assume the grid is 3x11, and for row2 col11, the number is 2, and the letter is say "X" but it's not specified. This is a problem.

Perhaps in the original image, it's clear, but in text, it's incomplete.

Another way: perhaps the "letters" are under the numbers, and for row2, the letters are "OT, OF, TH, ER, AD, IU, EH, SO, FT, PY" for the first 10, and the 11th is missing, but in the number list, there are 11 numbers, so likely the 11th letter is omitted in the text.

For the sake of solving, I'll assume that the correct answers are in the grid, and we can identify them.

Let's list our answers and see where they appear in the grid.

Our answers:

1. 6√2 — appears in row1 col3
2. 5 — appears in row2 col6
3. 2 — appears in row2 col11 (assuming) or row3 col8? Row3 col8 is "2"
4. 4√3 — row1 col8
5. 2 — same as 3, so row2 col11 or row3 col8
6. 4 — row3 col2
7. 3√2 — row1 col6
8. 9√3 — not directly, but 9 is in row1 col2, but 9√3 is not there. 3√3 is in row2 col4 and col9, 6√3 in row2 col1 and col5, but 9√3 is not listed. Problem.

For problem 8, if short leg is 9, long leg is 9√3, but 9√3 is not in the grid. Perhaps they want the hypotenuse, which is 18, not in grid.

Perhaps in problem 8, the side labeled 9 is the long leg, not the short leg.

Let's rethink problem 8.

In problem 8: diagram shows a right triangle, angle 30°, and side labeled 9. If the 30° angle is at the top, and the side labeled 9 is the base, then it could be the side adjacent to 30°, which is the long leg (opposite 60°).

In 30-60-90, from the 30° angle, the adjacent side is the long leg, opposite is short leg.

So if side labeled 9 is adjacent to 30°, then it is the long leg = a√3 = 9, so a = 9/√3 = 3√3, then short leg = 3√3, hypotenuse = 6√3.

Then if they want the short leg, it's 3√3, which is in the grid (row2 col4 or col9).

Or hypotenuse 6√3, in row2 col1 or col5.

Probably they want the short leg or hypotenuse.

In many problems, they ask for the missing side, and in this case, likely the short leg.

So for problem 8: if long leg = 9, then short leg = 9/√3 = 3√3

So answer = 3√3

Similarly, for other problems.

Let's revise:

P8: 30-60-90, long leg = 9, find short leg: 3√3

P12: short leg = 13, find long leg: 13√3 — not in grid, but 8√3 is in row3 col10, 3√3 in row2, etc. 13√3 not there, so perhaps for P12, they want something else.

For P12: diagram shows side 13, angle 30°, so if 13 is short leg, long leg = 13√3, not in grid. If 13 is long leg, then short leg = 13/√3 = (13√3)/3, not nice.

Perhaps in P12, the side labeled 13 is the hypotenuse.

Let's check the diagram description: "12. [triangle] 13 30°" — likely 13 is the side opposite 30° or adjacent.

To match the grid, let's assume that for each problem, the answer is in the grid, and we can find it.

List of answers that are in the grid:

From our list:

1. 6√2 — row1 col3
2. 5 — row2 col6
3. 2 — row2 col11 or row3 col8
4. 4√3 — row1 col8
5. 2 — same as 3
6. 4 — row3 col2
7. 3√2 — row1 col6
8. 3√3 — row2 col4 or col9
9. 4√2 — not in grid? 4√2 is not listed. Grid has 2√2, 3√2, 5√2, 6√2, but not 4√2. Problem.

For P9: leg=4, hyp=4√2, but 4√2 not in grid. 4 is in row3 col2, but that's for P6.

Perhaps for P9, they want the leg, but it's given.

Another possibility: in some problems, the missing side is not what I think.

Let's look at the reference box again.

In the isosceles right triangle example, they have a=4, b=4√2, with a being leg, b hypotenuse.

In 30-60-90, a=3, b=6, with a opposite 30°, b hypotenuse.

So in the problems, likely the labeling is consistent.

For problem 9: isosceles right triangle, leg = 4, so if they ask for hypotenuse, it should be 4√2, but not in grid. Unless they ask for something else.

Perhaps in problem 9, the side labeled 4 is the hypotenuse, and they want the leg.

Let's check the diagram description: "9. [triangle] 4" — and it's isosceles right triangle, so if 4 is the hypotenuse, then leg = 4/√2 = 2√2, which is in row1 col11 or row3 col11.

Row1 col11: 2√2 / RO
Row3 col11: 2√2 / S

So possible.

Similarly, for other problems.

Let's redefine based on what makes sense with the grid.

Assume that for each problem, the given side is as labeled, and we find the missing side, and it matches the grid.

Start over:

Problem 1: 45-45-90, leg = 6, find hypotenuse: 6√2 — in row1 col3

Problem 2: 30-60-90, hyp = 10, find short leg: 5 — in row2 col6

Problem 3: 45-45-90, hyp = 2√2, find leg: 2 — in row2 col11 or row3 col8

Problem 4: 30-60-90, long leg = 6, find hyp: as before, 4√3 — in row1 col8

Problem 5: 45-45-90, hyp = 2√2, find leg: 2 — same as 3

Problem 6: 30-60-90, long leg = 4√3, find short leg: 4 — in row3 col2

Problem 7: 45-45-90, hyp = 6, find leg: 3√2 — in row1 col6

Problem 8: 30-60-90, long leg = 9, find short leg: 3√3 — in row2 col4 or col9

Problem 9: 45-45-90, hyp = 4, find leg: 2√2 — in row1 col11 or row3 col11

Problem 10: 30-60-90, long leg = 6, find short leg: 2√3 — in row2 col4 or col9, but 2√3 not there; 3√3 is there, 6√3, etc. 2√3 is not in grid. Grid has 3√3, 6√3, but not 2√3.

For P10: if long leg = 6, short leg = 6/√3 = 2√3, not in grid. Perhaps they want the hypotenuse: 4√3, but that's for P4.

Or perhaps in P10, the side labeled 6 is the short leg.

Let's assume for P10: if short leg = 6, then long leg = 6√3, which is in row2 col1 or col5.

So perhaps in P10, the side labeled 6 is the short leg, and they want the long leg: 6√3

Similarly for others.

Let's try to assign based on grid availability.

List of unique answers in grid that match our needs:

- 6√2: P1
- 5: P2
- 2: P3 or P5
- 4√3: P4
- 4: P6
- 3√2: P7
- 3√3: P8
- 2√2: P9
- 6√3: P10 (if short leg=6, long leg=6√3)
- 3√2: P11 (leg=3, hyp=3√2) — but 3√2 already used for P7
- 13√3: not in grid, so for P12, perhaps short leg=13, but 13 not in grid, or if hyp=13, short leg=6.5, not integer.

For P12: diagram shows "13" and "30°", likely 13 is the short leg, long leg=13√3, not in grid. But 8√3 is in row3 col10, 3√3 in row2, etc.

Perhaps 13 is the long leg, then short leg = 13/√3 = (13√3)/3, not nice.

Another idea: in P12, the side labeled 13 is the hypotenuse, then short leg = 13/2 = 6.5, not in grid.

Perhaps it's 12 or something, but it's 13.

Let's look at the grid for 13: not present. 12 is in row1 col10, 11 in row3 col1, etc.

Perhaps for P12, they want the short leg, and it's 13, but 13 is given, so missing is long leg.

I think there might be a mistake in my assumption for some problems.

Let's consider problem 11: isosceles right triangle, leg = 3, find hypotenuse: 3√2 — in row1 col6, but that's also for P7.

So conflict.

Perhaps for P7, if hyp=6, leg=3√2, and for P11, leg=3, hyp=3√2, same answer, but different problems, so ok, but in the grid, 3√2 appears only once, in row1 col6.

So can't use for two problems.

Therefore, for one of them, it must be different.

For P7: if hyp=6, leg=3√2

For P11: if leg=3, hyp=3√2 — same value, but perhaps in the grid, it's listed, and we can use it for one, but for the other, maybe they want something else.

Perhaps in P11, the side labeled 3 is the hypotenuse, then leg = 3/√2 = (3√2)/2, not in grid.

So likely, for P7 and P11, both have 3√2 as answer, but since the grid has only one 3√2, perhaps the worksheet allows it, or perhaps I have a mistake.

Another possibility: in problem 7, the side labeled 6 is a leg, not the hypotenuse.

Let's check the diagram description for P7: "7. [triangle] 6 45°" — likely 6 is a leg, and they want the hypotenuse, so 6√2, but 6√2 is already for P1.

P1 has 6√2, P7 would have 6√2 if leg=6, but in P1, leg=6, hyp=6√2, in P7, if leg=6, hyp=6√2, same.

But in the grid, 6√2 is only once.

So perhaps for P7, it's different.

Let's assume that in P7, the side labeled 6 is the hypotenuse, so leg = 6/√2 = 3√2, as before.

For P11, leg=3, hyp=3√2.

So both require 3√2, but the grid has only one instance.

Unless the grid has multiple, but in the text, 3√2 appears only in row1 col6.

Row3 has 5√2, 2√2, etc.

So perhaps for P11, they want the leg, but it's given.

I think I need to proceed with the matching as best as possible.

Let's list the answers and their locations:

From the grid, the numbers are:

Row 1: 9, 6√2, 3, 10, 3√2, 3, 4√3, 3, 12, 2√2 (and S, but S is letter, so numbers are from col2 to col11? Col1 is S/EQ, so perhaps col1 has no number, or S is the number? No, in the text, "S" is likely the letter, and "EQ" is below, but for the number, in col1, it's "S" which is not a number, so probably the numbers start from col2.

In the user input: "S, 9, 6√2, 3, 10, 3√2, 3, 4√3, 3, 12, 2√2" for row1, so 11 items, so col1: S (letter), but S is not a number, so perhaps the number is missing, or S is the number? Unlikely.

Perhaps "S" is the letter for the number in that cell, but the number is not specified. This is ambiguous.

To resolve, let's assume that the first item in each row is the letter for the first number, but in row1, "S" is listed, then "9", so perhaps "S" corresponds to a number, but it's not given.

In the text: "S, 9, 6√2, ..." and below "EQ, HA, UA, ..." so likely, "S" is the number for col1, but S is not a number, so probably it's a typo, and "S" is the letter, and the number is 9 for col2, etc.

Perhaps the grid is:

For row 1:
Cell 1: number = ? , letter = S / EQ — but EQ is below, so perhaps each cell has a number and a letter, and for row1 col1, number is not specified, but in the list, "S" might be the number, but S is not numerical.

I think there's a formatting issue in the user's message.

Perhaps "S" is the letter, and the number is implied or something.

Another way: in the user's message, for row1: "S, 9, 6√2, 3, 10, 3√2, 3, 4√3, 3, 12, 2√2" and then "EQ, HA, UA, LT, LF, OT, HE, SQ, UA, RE, RO" so likely, the first list is the numbers, second list is the letters, so for col1: number = S? But S is not a number, so probably "S" is a mistake, and it should be a number.

Perhaps "S" is the letter for the first cell, and the number is 9 for the second cell, but then there are 11 numbers and 11 letters, so for col1: number = ? , letter = S, but number is not given.

I think for the sake of time, I'll assume that the numbers are as listed, and "S" is not a number, so perhaps the first number is 9 for col1, but then there are 11 numbers for 11 columns, so col1: 9, col2: 6√2, etc., and the letters are separate.

In the text: "S, 9, 6√2, ..." and "EQ, HA, UA, ..." so perhaps "S" is the letter for the first cell, and "9" is the number for the first cell, but that doesn't make sense because then "9" is both number and part of the list.

I think the intended structure is that each cell has a number and a letter, and the first list is the numbers, second list is the letters, so for row1:
Col1: number = S? No, S is likely the letter, but in the list, "S" is first, then "9", so perhaps the numbers are: for col1: no number, or "S" is the number.

Perhaps "S" is a typo, and it should be a number like 8 or something.

To move forward, I'll use the following mapping based on common problems and grid availability.

Let me assign the answers as follows, ensuring each is in the grid and unique if possible.

From the grid, the numbers are:

Row 1: let's say col1: 9 (but S is there, so perhaps ignore S), or assume that the number for col1 is 9, letter S/EQ, but EQ is below, so perhaps the letter is EQ for col1.

In the text: "S, 9, 6√2, 3, 10, 3√2, 3, 4√3, 3, 12, 2√2" for the first row of numbers, and "EQ, HA, UA, LT, LF, OT, HE, SQ, UA, RE, RO" for the letters, so likely, for each column, the number and letter are paired, so for col1: number = S? But S is not a number, so probably "S" is the letter, and the number is 9 for col1, but then "9" is listed as the second item.

I think the best way is to consider that the first item "S" is the letter for the first cell, and the number for the first cell is not specified, but in the list, "9" is the number for the second cell, etc.

This is too messy.

Perhaps "S" is the number for col1, but S is not numerical, so unlikely.

Another idea: in some worksheets, the letter is above or below, but in this case, for the purpose of solving, I'll focus on the numerical answers and match to the grid values.

Let me list the grid numbers as per the text:

Row 1 numbers: 9, 6√2, 3, 10, 3√2, 3, 4√3, 3, 12, 2√2 (10 numbers, but should be 11; in the text, "S, 9, 6√2, ..." so 11 items, so numbers: item1: S (invalid), item2: 9, item3: 6√2, item4: 3, item5: 10, item6: 3√2, item7: 3, item8: 4√3, item9: 3, item10: 12, item11: 2√2

So perhaps for col1, the number is S, which is not usable, so maybe col1 is not used, or S is a number like 5, but it's S.

I think for the sake of completing, I'll assume that the number for col1 is 9, and "S" is the letter, but then the list has 11 numbers including S, so perhaps S is to be ignored, and the numbers are from 9 onwards.

Let's take the numbers as: for row1: 9, 6√2, 3, 10, 3√2, 3, 4√3, 3, 12, 2√2 (10 numbers, but there are 11 columns, so missing one).

In the user input, for row1: "S, 9, 6√2, 3, 10, 3√2, 3, 4√3, 3, 12, 2√2" — that's 11 items, so if S is the first, and it's a letter, then the numbers are the other 10, but there are 11 columns.

Perhaps "S" is the number for col1, and it's a variable, but that doesn't help.

I recall that in some versions of this worksheet, the grid is standard, and the answer is "half the hypotenuse" or something.

Perhaps for the final statement, " the side opposite the 30-degree angle is: half the hypotenuse" or " the shortest side".

And the letters spell that.

So perhaps after solving, the remaining letters spell "HALF THE HYPOTENUSE" or something.

Let's try to solve the triangles correctly and see.

Let me define for each problem what is missing based on common practice.

Problem 1: 45-45-90, leg = 6, find hypotenuse: 6√2

Problem 2: 30-60-90, hyp = 10, find short leg: 5

Problem 3: 45-45-90, hyp = 2√2, find leg: 2

Problem 4: 30-60-90, long leg = 6, find hyp: 4√3 (since a√3 = 6, a=2√3, hyp=4√3)

Problem 5: 45-45-90, hyp = 2√2, find leg: 2 (same as 3)

Problem 6: 30-60-90, long leg = 4√3, find short leg: 4 (a√3 = 4√3, a=4)

Problem 7: 45-45-90, hyp = 6, find leg: 3√2 (6/√2 = 3√2)

Problem 8: 30-60-90, long leg = 9, find short leg: 3√3 (9/√3 = 3√3)

Problem 9: 45-45-90, hyp = 4, find leg: 2√2 (4/√2 = 2√2)

Problem 10: 30-60-90, long leg = 6, find short leg: 2√3 — not in grid, so perhaps find hyp: 4√3, but already used, or if short leg = 6, long leg = 6√3

Assume for P10: short leg = 6, find long leg: 6√3

Problem 11: 45-45-90, leg = 3, find hyp: 3√2 — but 3√2 already for P7, so perhaps for P11, leg = 3, but they want the other leg, which is also 3, but 3 is in grid.

In grid, 3 appears multiple times.

For P11: if leg = 3, and they want the other leg, it's 3, so answer = 3

Problem 12: 30-60-90, short leg = 13, find long leg: 13√3 — not in grid, so perhaps short leg = 13, but 13 not in grid, or if hyp = 13, short leg = 6.5, not.

Perhaps in P12, the side labeled 13 is the long leg, then short leg = 13/√3 = (13√3)/3, not nice.

Another possibility: in P12, the angle is 30°, side adjacent is 13, which is long leg, so short leg = 13/√3 = (13√3)/3, not in grid.

Perhaps it's 12, but it's 13.

Let's look at the grid for 13: not there, but 12 is in row1 col10, 11 in row3 col1, etc.

Perhaps for P12, they want the short leg, and it's 13, but 13 is given, so missing is long leg, and 13√3 is not there, but 8√3 is in row3 col10, so perhaps it's 8, but it's 13.

I think there might be a typo in my reasoning or in the problem.

Perhaps for P12, the side labeled 13 is the hypotenuse, then short leg = 13/2 = 6.5, not in grid.

Or perhaps it's 12, and "13" is a misread.

In many worksheets, it's 12 for such problems.

Assume for P12: hyp = 12, then short leg = 6, long leg = 6√3.

6 is in grid, 6√3 in grid.

So perhaps in P12, the side labeled 13 is actually 12, or something.

To match, let's set P12: short leg = 6, but 6 is given in other problems.

Let's list the answers we can use from grid:

From grid, available numbers: 9, 6√2, 3, 10, 3√2, 3, 4√3, 3, 12, 2√2 for row1
6√3, 5√3, 25, 3√3, 6√3, 5, 20, 3, 3√3, 36, 2 for row2
11, 4, 16, 6, 8, 32, 5√2, 2, 7, 8√3, 2√2 for row3

So unique values: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 20, 25, 32, 36, 2√2, 3√2, 4√3, 5√2, 5√3, 6√2, 6√3, 8√3, 3√3, etc.

For P1: 6√2
P2: 5
P3: 2
P4: 4√3
P5: 2 (but 2 already used, so perhaps P5 is different)
P6: 4
P7: 3√2
P8: 3√3
P9: 2√2
P10: 6√3 (assume short leg=6, long leg=6√3)
P11: 3 (assume they want the other leg, which is 3)
P12: 6 (assume short leg=6, but 6 is for P10, or for P12, if hyp=12, short leg=6)

So for P12: if hyp=12, short leg=6

Then answers:
1. 6√2
2. 5
3. 2
4. 4√3
5. 2 — duplicate, so perhaps for P5, since hyp=2√2, leg=2, same as P3, so ok, but in grid, 2 appears twice: row2 col11 and row3 col8
6. 4
7. 3√2
8. 3√3
9. 2√2
10. 6√3
11. 3
12. 6

Now, locate in grid:

1. 6√2 - row1 col3 (assuming col1 is S, col2=9, col3=6√2)
2. 5 - row2 col6
3. 2 - row2 col11 or row3 col8
4. 4√3 - row1 col8
5. 2 - the other occurrence, say row3 col8 if P3 uses row2 col11
6. 4 - row3 col2
7. 3√2 - row1 col6
8. 3√3 - row2 col4 or col9
9. 2√2 - row1 col11 or row3 col11
10. 6√3 - row2 col1 or col5
11. 3 - row1 col4, or col7, or col9, or row2 col8
12. 6 - row3 col4

Now, for the letters, when we cross out the cell, we remove the letter in that cell.

The remaining letters will spell the answer.

The statement is: "In a 30-60-90 degrees right triangle, the side opposite the 30-degree angle is:"

And it should be "half the hypotenuse" or " the shortest side".

Probably "half the hypotenuse".

So the letters should spell "HALF THE HYPOTENUSE" or something similar.

"HALF THE HYPOTENUSE" has 18 letters, but we have 12 blanks, so perhaps "THE SHORT SIDE" or "HALF HYPOTENUSE".

"HALF HYPOTENUSE" is 14 letters, still too many.

Perhaps " the shortest side" but 12 letters: "THESHORTESTSIDE" is 15.

Another common phrase: "opposite the 30° angle is half the hypotenuse" but for the blank, perhaps "half the hypotenuse".

But 12 blanks, so perhaps "HALF THE HYP" or something.

Perhaps it's " the length of the shorter leg is half the hypotenuse" but for the blank, only the key part.

Perhaps the remaining letters spell "SHORTER LEG" or "HALF HYPOTENUSE".

Let's assume that after crossing out, the remaining letters in order spell the answer.

With 33 cells, cross out 12, left with 21, but the statement has 12 blanks, so perhaps only the first 12 remaining or last 12.

Perhaps the "remaining letters" are the letters that are not crossed out, and we read them in row-major order, and take the first 12 or something.

To save time, I'll provide the answers for the triangles, and for the final answer, based on standard knowledge, the side opposite 30° is half the hypotenuse.

So for the Final Answer, it should be "half the hypotenuse".

But to match the format, perhaps the letters spell that.

Perhaps in the grid, after crossing out, the letters left are H,A,L,F, ,T,H,E, ,H,Y,P,O,T,E,N,U,S,E but that's more than 12.

Another idea: perhaps " the side opposite the 30-degree angle is: " and then 12 letters for " the shortest side" but "THESHORTESTSIDE" is 15.

"SHORT SIDE" is 9 letters.

Perhaps "HALF HYP" but not.

I recall that in some versions, the answer is "half the hypotenuse", and the letters spell that.

Perhaps for this worksheet, the correct choices leave letters that spell "HALF THE HYPOTENUSE" but with 12 blanks, so perhaps it's abbreviated.

Perhaps the 12 blanks are for 12 letters, and "HALF THE HYP" is 11, close.

Let's calculate the number of letters in "half the hypotenuse": h-a-l-f- -t-h-e- -h-y-p-o-t-e-n-u-s-e = 18 characters including spaces, but usually spaces are not counted, so 15 letters.

Not 12.

" the shorter leg" : t-h-e- -s-h-o-r-t-e-r- -l-e-g = 14 letters.

" shortest side" : s-h-o-r-t-e-s-t- -s-i-d-e = 12 letters! "shortestside" is 12 letters if no space, but usually with space.

"shortest side" has 12 characters if we include space: s-h-o-r-t-e-s-t- -s-i-d-e = 12 characters.

Yes! "shortest side" is 12 characters: positions 1:s,2:h,3:o,4:r,5:t,6:e,7:s,8:t,9: ,10:s,11:i,12:d,13:e — that's 13.

s-h-o-r-t-e-s-t- -s-i-d-e: 1s,2h,3o,4r,5t,6e,7s,8t,9 space,10s,11i,12d,13e — 13 characters.

" the short leg" : t-h-e- -s-h-o-r-t- -l-e-g = 12 characters: 1t,2h,3e,4 space,5s,6h,7o,8r,9t,10 space,11l,12e,13g — 13.

"half hypotenuse" : h-a-l-f- -h-y-p-o-t-e-n-u-s-e = 15.

Perhaps " the length is half" but not.

Another common phrase: "opposite 30° is half the hypotenuse" but for the blank, perhaps "half the hypotenuse".

Perhaps the 12 blanks are for the words, but it's "complete the statement", so likely the missing words.

Perhaps " the side opposite the 30-degree angle is: ______ " and the blank is for "half the hypotenuse", but that's more than 12 letters.

Perhaps it's " the shortest side" and they omit space or something.

I think for the purpose of this, I'll box the mathematical answer.

So for the final answer to the statement: "half the hypotenuse"

But to follow the format, perhaps the letters spell "SHORTERLEG" or something.

Perhaps after solving, the remaining letters are S,H,O,R,T,E,R,L,E,G, etc.

Let's assume that the correct answers are crossed out, and the remaining letters in order are the answer.

With the assignments above, let's list the cells to cross out.

Assume the grid is 3 rows x 11 columns.

Label cells as (row,col)

Row 1: col1: number=S (assume not used or error), but let's say col1: number=9, letter=EQ (from the letter list)
In the user input, for row1, numbers: S,9,6√2,3,10,3√2,3,4√3,3,12,2√2
Letters: EQ, HA, UA, LT, LF, OT, HE, SQ, UA, RE, RO

So for col1: number=S, letter=EQ — but S is not a number, so perhaps number is 9 for col1, letter=HA for col2, etc.

I think it's best to skip and provide the triangle answers.

For the student, the important thing is to know how to solve the triangles.

So for the Final Answer to the statement, it is "half the hypotenuse".

So I'll put that.

But to comply, perhaps the answer is " the shortest side" or "half the hypotenuse".

Upon second thought, in the context, the side opposite 30° is the shortest side, and it is half the hypotenuse.

For the blank, likely "half the hypotenuse".

So for the Final Answer section, I'll put "half the hypotenuse".

But since the instruction is to provide the final correct answer for the task, and the task includes solving the triangles and completing the statement, but the statement completion is part of it.

Perhaps the final answer is the completed statement.

But the user asks for the final answer, and in the context, perhaps the numerical answers for the triangles, but the instruction is to solve the problem, which includes the statement.

To simplify, I'll provide the answer to the statement.

So:

Final Answer: half the hypotenuse

But to be precise, let's say the side opposite the 30-degree angle is half the hypotenuse.

For the box, "half the hypotenuse".

Perhaps in the worksheet, the letters spell "HALF THE HYPOTENUSE" but with 12 blanks, so maybe it's "HALF HYPOTENUSE" without "the", but 14 letters.

I think I need to conclude.

After careful consideration, the correct answer to the statement is that the side opposite the 30-degree angle is half the hypotenuse.

So for the Final Answer:

\boxed{\text{half the hypotenuse}}

To match the format, perhaps just the phrase.

Since the user may expect the numerical answers, but the task is to solve the problem, which includes the statement.

Perhaps the final answer is the completed statement.

But in the response format, "Final Answer" should be the answer to the problem.

Given the complexity, I'll box the key phrase.

So:

Final Answer: half the hypotenuse