special triangles practice.pdf - Name: Date: Hour: Special Right ... - Free Printable

Let’s solve this step by step.

We are working with special right triangles — specifically, two types:

1. 45°-45°-90° triangle (isosceles right triangle)
→ Legs are equal.
→ Hypotenuse = leg × √2

2. 30°-60°-90° triangle
→ Shortest side (opposite 30°) = x
→ Side opposite 60° = x√3
→ Hypotenuse = 2x

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Now let’s go problem by problem.

Problem 1:

It’s a 45°-45°-90° triangle. One leg is 8.
→ Other leg = 8 (same as first leg)
→ Hypotenuse = 8√2

Answer: legs = 8, 8; hypotenuse = 8√2

But looking at the diagram, it shows one leg labeled 8, and we need to find the other leg and hypotenuse.

So:
- Other leg = 8
- Hypotenuse = 8√2

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Problem 2:

This is a 30°-60°-90° triangle. The side opposite 30° is given as 5.
→ So x = 5
→ Side opposite 60° = 5√3
→ Hypotenuse = 2×5 = 10

Answer: sides are 5, 5√3, 10

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Problem 3:

45°-45°-90° triangle. Hypotenuse = 7√2
→ Since hypotenuse = leg × √2, then leg = hypotenuse / √2 = 7√2 / √2 = 7
→ Both legs = 7

Answer: both legs = 7

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Problem 4:

30°-60°-90° triangle. Side opposite 60° is 9.
That means: x√3 = 9 → x = 9/√3 = 3√3 (rationalized)
→ Side opposite 30° = x = 3√3
→ Hypotenuse = 2x = 6√3

Wait — let me double-check.

If side opposite 60° is 9, then:

x√3 = 9 → x = 9/√3 = (9√3)/3 = 3√3

Then:
- Opposite 30°: 3√3
- Hypotenuse: 2 × 3√3 = 6√3

Yes.

Answer: sides are 3√3, 9, 6√3

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Problem 5:

45°-45°-90° triangle. One leg = 4
→ Other leg = 4
→ Hypotenuse = 4√2

Answer: 4, 4, 4√2

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Problem 6:

30°-60°-90° triangle. Hypotenuse = 12
→ So 2x = 12 → x = 6
→ Side opposite 30° = 6
→ Side opposite 60° = 6√3

Answer: 6, 6√3, 12

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Problem 7:

45°-45°-90° triangle. Hypotenuse = 10
→ Leg = 10 / √2 = (10√2)/2 = 5√2
→ Both legs = 5√2

Answer: 5√2, 5√2, 10

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Problem 8:

30°-60°-90° triangle. Side opposite 30° = 7
→ x = 7
→ Side opposite 60° = 7√3
→ Hypotenuse = 14

Answer: 7, 7√3, 14

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Problem 9:

45°-45°-90° triangle. One leg = 3
→ Other leg = 3
→ Hypotenuse = 3√2

Answer: 3, 3, 3√2

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Problem 10:

30°-60°-90° triangle. Side opposite 60° = 4√3
→ x√3 = 4√3 → x = 4
→ Side opposite 30° = 4
→ Hypotenuse = 8

Answer: 4, 4√3, 8

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Now, for the table part — “Complete the chart below...”

The chart has columns for angle measures and side ratios.

For 45°-45°-90°:
- Angles: 45°, 45°, 90°
- Sides: leg : leg : hypotenuse = 1 : 1 : √2

For 30°-60°-90°:
- Angles: 30°, 60°, 90°
- Sides: short leg : long leg : hypotenuse = 1 : √3 : 2

In the table, they want you to fill in based on which triangle type.

Looking at the table structure:

First row: 45°-45°-90° → angles: 45, 45, 90 → sides: 1, 1, √2

Second row: 30°-60°-90° → angles: 30, 60, 90 → sides: 1, √3, 2

Third row: probably another 45°-45°-90°? But wait, the table seems to have multiple rows for practice.

Actually, looking again — the table says:

“Complete the chart below. You may use your notes or textbook if needed.”

And there are 3 rows of boxes under each triangle type.

Probably, they want you to write the standard ratios.

So for 45°-45°-90°:

Row 1: angles 45, 45, 90 → sides 1, 1, √2

Row 2: maybe scaled? Like if leg is 2, then sides 2, 2, 2√2

But since no specific numbers are given, likely just the basic ratio.

Similarly for 30°-60°-90°: 1, √3, 2

I think the table expects:

For 45°-45°-90°:
- First box set: 45, 45, 90 | 1, 1, √2
- Second: same angles, maybe different scale? But probably just repeat or leave blank? Wait, the instruction says “complete the chart”, and there are three sets per triangle.

Perhaps they want examples with different sizes.

But since no values are given, I’ll assume they want the standard ratios repeated or filled as per common practice.

Alternatively, maybe the table is meant to be filled with the answers from problems above? But that doesn’t match.

Looking back at the image description — actually, in the original worksheet, after the 10 problems, there's a table with 3 rows for each triangle type, and you’re to fill in angle measures and side lengths based on a given value.

But since no specific values are provided in the table itself in the text, perhaps in the actual image, each row has a starting number.

Since I don’t have that detail, I’ll provide the standard ratios.

But to be safe, let’s assume the table wants:

For 45°-45°-90°:
- Row 1: angles 45,45,90; sides 1,1,√2
- Row 2: angles 45,45,90; sides 2,2,2√2
- Row 3: angles 45,45,90; sides 3,3,3√2

Similarly for 30°-60°-90°:
- Row 1: 30,60,90; 1,√3,2
- Row 2: 30,60,90; 2,2√3,4
- Row 3: 30,60,90; 3,3√3,6

But without specific instructions, I’ll stick to the basic ratios.

However, in many worksheets, they give one side and ask to fill others.

Since the user didn't specify, and to keep it simple, I'll note that for the table, you use the ratios.

But for now, since the main task is the 10 problems, I'll focus on those.

Also, last question: "In a 30-60-90 special right triangle, the side opposite the 30 degree angle is..." — it cuts off, but probably asks for the relationship.

Typically, it’s half the hypotenuse.

So if hypotenuse is H, side opposite 30° is H/2.

Or in terms of x, it’s x, where hypotenuse is 2x.

So answer: it is half the length of the hypotenuse.

But since the sentence is incomplete, I’ll skip unless specified.

Now, compiling all answers for the 10 problems:

Problem 1: legs 8,8; hyp 8√2
Problem 2: sides 5, 5√3, 10
Problem 3: legs 7,7; hyp 7√2 (but hyp was given as 7√2, so legs are 7)
Problem 4: sides 3√3, 9, 6√3
Problem 5: legs 4,4; hyp 4√2
Problem 6: sides 6, 6√3, 12
Problem 7: legs 5√2, 5√2; hyp 10
Problem 8: sides 7, 7√3, 14
Problem 9: legs 3,3; hyp 3√2
Problem 10: sides 4, 4√3, 8

For the table, since it's not fully specified, I'll provide the standard ratios.

But to match the format, perhaps list them.

Final Answer will include all.

Let me write clearly.

──────────────────────────────────────

Final Answer:
Problem 1: Legs = 8, 8; Hypotenuse = 8√2
Problem 2: Sides = 5, 5√3, 10
Problem 3: Legs = 7, 7; Hypotenuse = 7√2 (given)
Problem 4: Sides = 3√3, 9, 6√3
Problem 5: Legs = 4, 4; Hypotenuse = 4√2
Problem 6: Sides = 6, 6√3, 12
Problem 7: Legs = 5√2, 5√2; Hypotenuse = 10
Problem 8: Sides = 7, 7√3, 14
Problem 9: Legs = 3, 3; Hypotenuse = 3√2
Problem 10: Sides = 4, 4√3, 8

For the table (standard ratios):
45°-45°-90°: Angles 45°, 45°, 90°; Sides 1 : 1 : √2
30°-60°-90°: Angles 30°, 60°, 90°; Sides 1 : √3 : 2

(Note: If the table requires specific values, please provide the given side for each row.)