Special Triangles & Triple Recognition Drill

What It Targets

• Right Triangles
• Pythagorean Triples
• 30-60-90 Triangles
• 45-45-90 Triangles
• Trigonometry
• Geometry Speed Questions

The Core Idea

The SAT loves right triangles.

Fortunately, the SAT also loves using the same right triangles over and over again.

The fastest students do not repeatedly use the Pythagorean Theorem.

They recognize patterns instantly.

Think:

Don't calculate it.

Recognize it.

The Four Triangle Patterns You Must Know

Pattern #1: The 3-4-5 Triangle

Memorize:

3, 4, 5

Every multiple also works:

• 6, 8, 10
• 9, 12, 15
• 12, 16, 20
• 15, 20, 25

SAT Shortcut:

If you see two sides that look like a scaled 3-4-5 triangle, stop calculating.

Recognize the pattern.

Pattern #2: The 5-12-13 Triangle

Memorize:

5, 12, 13

Common scaled versions:

• 10, 24, 26
• 15, 36, 39
• 20, 48, 52

SAT Shortcut:

This triple appears surprisingly often.

Many SAT questions disguise it inside word problems.

Pattern #3: The 45-45-90 Triangle

Always remember:

x : x : x√2

SAT Shortcut:

If you know the hypotenuse:

Divide by √2

If you know a leg:

Multiply by √2

Pattern #4: The 30-60-90 Triangle

Always remember:

x : x√3 : 2x

SAT Shortcut:

The shortest side controls everything.

Find x first.

Then build the rest of the triangle.

How Do I Know This Is a Special Triangle Question?

Look for:

Clue #1

The problem explicitly says:

• 30-60-90
• 45-45-90
• Isosceles Right Triangle

Use the ratio immediately.

Clue #2

The side lengths look familiar.

Examples:

• 6 and 8
• 9 and 12
• 12 and 16
• 5 and 12

These are usually scaled triples.

Clue #3

The answer choices contain:

• √2
• √3

This is often a special triangle clue.

Clue #4

The SAT seems to want a lot of computation.

Pause.

Ask:

Is this secretly a special triangle?

Often the answer is yes.

Guided Practice

Question 1

A right triangle has legs of length 6 and 8.

What is the length of the hypotenuse?

Answer: 10

Why?

This is a scaled 3-4-5 triple.

Multiply everything by 2:

3-4-5

becomes

6-8-10

Question 2

A 30-60-90 triangle has a hypotenuse of length 12.

What is the length of its shortest side?

Answer: 6

Why?

In every 30-60-90 triangle:

Short Side = Half of Hypotenuse

12 ÷ 2 = 6

Question 3

An isosceles right triangle has a hypotenuse of 5√2.

What is the length of one leg?

Answer: 5

Why?

45-45-90 triangles follow:

x : x : x√2

If:

x√2 = 5√2

then

x = 5

Question 4

A right triangle has a leg of 5 and a hypotenuse of 13.

What is the length of the other leg?

Answer: 12

Why?

This is the classic:

5-12-13

triple.

Question 5

In a 30-60-90 triangle, the side opposite the 60° angle is 9√3.

What is the length of the hypotenuse?

Answer: 18

Why?

The long leg equals:

x√3

So:

x = 9

Hypotenuse:

2x = 18

Question 6

The perimeter of a square is 32.

What is the length of its diagonal?

Answer: 8√2

Why?

Square side:

32 ÷ 4 = 8

A diagonal creates two 45-45-90 triangles.

Diagonal:

8√2

Question 7

A right triangle has legs of 9 and 12.

What is its perimeter?

Answer: 36

Why?

This is:

3-4-5 × 3

giving:

9-12-15

Perimeter:

9 + 12 + 15 = 36

Question 8

In a right triangle,

sin(θ) = 3/5

If the hypotenuse is 20, what is the length of the side adjacent to θ?

Answer: 16

Why?

3/5 signals a 3-4-5 triangle.

Scale factor:

20 ÷ 5 = 4

Scaled triangle:

12-16-20

Adjacent side:

16

Question 9

A 30-60-90 triangle has a shortest side of 15 cm.

What is its exact perimeter?

Answer:

45 + 15√3

Why?

Sides:

• 15
• 15√3
• 30

Perimeter:

15 + 15√3 + 30

Question 10

The hypotenuse of a 30-60-90 triangle is 72 units.

What is its exact perimeter?

Answer:

108 + 36√3

Why?

Short side:

36

Long side:

36√3

Perimeter:

72 + 36 + 36√3

Instant Recognition Drill

Do not calculate.

Identify the pattern first.

1

A right triangle has side lengths 15 and 20.

What is the hypotenuse?

2

A 45-45-90 triangle has a leg length of 9.

What is its hypotenuse?

3

A 30-60-90 triangle has a short side of 11.

What is the hypotenuse?

4

A right triangle has side lengths 10 and 24.

What is the hypotenuse?

5

A square has side length 12.

What is its diagonal?

6

A 30-60-90 triangle has hypotenuse 50.

What is its long leg?

7

A right triangle has side lengths 12 and 16.

What is its hypotenuse?

8

A 45-45-90 triangle has hypotenuse 14√2.

What is the length of each leg?

Answer Key

1

25

(3-4-5 × 5)

2

9√2

3

22

4

26

(5-12-13 × 2)

5

12√2

6

25√3

7

20

(3-4-5 × 4)

8

14

Memory Card

3-4-5

3, 4, 5

5-12-13

5, 12, 13

45-45-90

x, x, x√2

30-60-90

x, x√3, 2x