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SPECIAL RIGHT TRIANGLES - Printable Mathematics Worksheets - Wayground

SPECIAL RIGHT TRIANGLES

15 questions

9th - 9th Grade

Mathematics

Special Right Triangles is a Grade 9 mathematics worksheet that helps students identify and solve problems involving 45-45-90 and 30-60-90 triangles. Across 15 questions—3 multiple-select and 12 multiple-choice—students determine missing side lengths, recognize short legs, long legs, and hypotenuses, identify triangle types, and find missing angle measures. The problems require students to select correct values, apply special-triangle relationships, and distinguish between common radical and whole-number results. This free printable worksheet is useful for guided practice, independent work, review, or a quick assessment after introducing special right-triangle ratios. Students practice multiplying a leg by √2 in a 45-45-90 triangle, using equal legs in that triangle, and applying the 1:√3:2 relationship in a 30-60-90 triangle. The worksheet includes a complete answer key so teachers can check both selected responses and students’ understanding of the underlying relationships.

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Worksheets

SPECIAL RIGHT TRIANGLES

Total questions: 15

Name
Class
Date
1.

Choose the correct values for x and y in the right triangle.

a)

x=3x=3

b)

x=32x=3\sqrt{2}

c)

y = 6y\ =\ 6

d)

y=62y=6\sqrt{2}

2.

Choose the correct values for x and y in the right triangle.

a)

x=562x=\frac{5\sqrt{6}}{2}

b)

x=5x=5

c)

y = 10y\ =\ 10

d)

y=103y=10\sqrt{3}

3.

Choose the correct values for x and y in the right triangle.

a)

x=1133x=\frac{11\sqrt{3}}{3}

b)

x=11x=11

c)

y = 22y\ =\ 22

d)

y=113y=11\sqrt{3}

4.
What is the length of x in this 45-45-90 triangle?
a)
4√2
b)
4
c)
8
d)
4√3
5.

What is the measure of side b?

a)

5

b)

10√2

c)

5√2

d)

102\frac{10}{\sqrt{2}}

6.
Find x.
a)
45°
b)
90°
c)
30°
d)
60°
7.
What type of special right triangle is this?
a)
45-45-90
b)
30-60-90
c)
not a special right triangle
8.
In this 45-45-90 triangle, I have been given the length of a leg.  How do I find the length of the hypotenuse?
a)
It is the same length as the given leg.
b)
Multiply that leg's length by √2.
c)
Multiply that leg's length by 2.
d)
Divide that leg's length by √2.
9.
In this 45-45-90 triangle, I have been given a leg, so to find the other leg I...
a)
Multiply that leg by 2
b)
Use the same length for the second leg
c)
Multiply that leg by √2
d)
Divide that leg by √2
10.
I have been given the short leg in this 30-60-90 triangle.  How do I find the length of the hypotenuse?
a)
Multiply 4 by 2
b)
Multiply 4 by √3
c)
Multiply 4 by √2
d)
Divide 4 by √3
11.
Which side is the long leg in this 30-60-90 triangle?
a)
4
b)
u
c)
v
12.

How long is side c?

a)

1

b)

2

c)

2\sqrt{2}

d)

3\sqrt{3}

13.
What is the measure of side c?
a)
6
b)
6√2
c)
6√3
d)
12
14.
Find x.
a)
30°
b)
60°
c)
90°
d)
180°
15.
Find x.
a)
10
b)
20
c)
10√3
d)
√3
Answer Key

SPECIAL RIGHT TRIANGLES

Total questions: 15

1.
b)

x=32x=3\sqrt{2}

, c)

y = 6y\ =\ 6

2.
b)

x=5x=5

, c)

y = 10y\ =\ 10

3.
b)

x=11x=11

, d)

y=113y=11\sqrt{3}

4.
a) 4√2
5.
d)

102\frac{10}{\sqrt{2}}

6.
b) 90°
7.
b) 30-60-90
8.
b) Multiply that leg's length by √2.
9.
b) Use the same length for the second leg
10.
a) Multiply 4 by 2
11.
c) v
12.
c)

2\sqrt{2}

13.
d) 12
14.
b) 60°
15.
a) 10

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FAQs

What does the Special Right Triangles worksheet cover?

This Grade 9 worksheet uses 3 multiple-select and 12 multiple-choice questions to assess 45-45-90 and 30-60-90 triangle relationships. Students identify triangle types and side roles, find missing side lengths and angle measures, and choose expressions such as multiplying a 45-45-90 leg by √2 or using the 30-60-90 ratio 1:√3:2.

How can I use this worksheet to teach special right triangles?

Introduce the two triangle patterns before assigning the worksheet: equal legs and a leg-to-hypotenuse factor of √2 for 45-45-90 triangles, and short leg, long leg, and hypotenuse in the 30-60-90 pattern. Model one side-length problem and one angle-identification problem, then have students explain why each selected value or operation fits the triangle type. Use the multiple-select items to discuss why both values must be correct, rather than treating them as single-answer questions.

What mistakes should I watch for on this special right triangles worksheet?

Students may confuse the 45-45-90 rule with the 30-60-90 rule, especially by multiplying a 30-60-90 short leg by √2 instead of 2 to find the hypotenuse. They may also mistake the long leg for the hypotenuse, assume unequal legs in a 45-45-90 triangle, or select only one correct value in the multiple-select questions. Check that students match each operation to the identified triangle type before simplifying their answers.

How do I assign and grade this Special Right Triangles worksheet?

On Wayground, assign it as a digital quiz or download the printable PDF for students to complete on paper. The worksheet includes a complete answer key, and printable submissions can be graded by scanning student work with the Wayground for Teachers app. The printable format can also support schools that are reducing screen time.

How can I differentiate this special right triangles worksheet for my students?

Use the worksheet’s Advanced Settings to adjust font spacing and font size so the triangle labels, radical expressions, and answer choices are easier to read. You can also enable a dyslexia-friendly font or translate the worksheet into another language when that better supports access to the side-ratio and angle questions.

Where can I find more worksheets like this on Wayground?

Wayground offers a comprehensive collection of free printable worksheets and practice problems across all subjects, with downloadable PDFs and answer keys to help students master essential skills and concepts. Browse more resources at https://wayground.com/en-us/worksheets.

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