Pythagorean Theorem and Applications

Triangles — Right Triangle Side Lengths

In this lesson:

• Apply to find missing sides
• Recognize Pythagorean triples for speed
• Use the converse to classify triangles

What You Will Learn Today

After this lesson, you will be able to:

1. Apply to find missing sides
2. Identify the hypotenuse opposite the right angle
3. Recognize common Pythagorean triples and multiples
4. Use the converse to classify triangles
5. Solve multi-step problems using the theorem

What Is the Longest Side Here?

The side opposite the right angle is always the longest side.

The Pythagorean Theorem States This

where = hypotenuse, and = legs

Finding the Hypotenuse From Two Legs

Legs are 6 and 8. Find the hypotenuse.

Step 1: Plug into the formula

Step 2: Compute

Step 3: Take the square root

Finding a Leg From Hypotenuse

Hypotenuse is 13, one leg is 5. Find the other leg.

Step 1: Rearrange the formula

Step 2: Compute

Step 3: Take the square root

Quick Check on Irrational Results

Legs are 5 and 7. Find the hypotenuse.

The answer stays as — ACT choices often use simplified radicals.

Does make sense? It's between 8 and 9, reasonable for legs 5 and 7.

Always Identify the Hypotenuse First

Before computing, follow this process:

1. Find the right angle in the diagram
2. Mark the hypotenuse — opposite the right angle
3. Label the legs — the two remaining sides
4. Then plug in — hypotenuse is always

Given sides 15, 9, 12 — hypotenuse is 15 (largest).

Common Pythagorean Triples Save Time

How to Recognize Triple Multiples Quickly

Sides 20 and 48 in a right triangle. Find the third.

Step 1: Find a common factor

Step 2: Recognize the triple: 5-12-13

Step 3: Scale back up:

The Converse Classifies Any Triangle

Let be the longest side. Compare to :

• → Right triangle
• → Acute triangle
• → Obtuse triangle

Memory aid: Sum bigger → a-cute. Sum smaller → ob-scenely big.

Converse Example With an Acute Triangle

Sides 7, 10, 12 — longest side is 12.

→ Sum is bigger, so the triangle is acute.

Converse Example With an Obtuse Triangle

Sides 5, 8, 11 — longest side is 11.

→ Sum is smaller, so the triangle is obtuse.

ACT Practice on Triangle Classification

Which set of side lengths forms a right triangle?

(A) 9, 12, 16
(B) 10, 24, 26
(C) 6, 7, 10
(D) 8, 14, 17

Check each — does equal ?

Answer to the Classification Problem

(B) 10, 24, 26 forms a right triangle.

Shortcut: is multiplied by 2.

The others: (A) , (C) , (D)

Rectangle Diagonal Uses Hidden Right Triangles

A TV screen is 36 in × 48 in. Find the diagonal.

Recognize: is → in

Coordinate Distance Is the Theorem

Find the distance from to .

Horizontal leg:

Vertical leg:

The distance formula is just the Pythagorean theorem on a grid.

Ladder Against a Wall Problem

A 17-foot ladder leans against a wall, base 8 feet away. How high does it reach?

The ladder is the hypotenuse (17 ft).
The ground distance is a leg (8 ft).

Recognize: 8-15-17 triple!

Finding Area Using the Theorem

An isosceles triangle has sides 10, 10, and base 12. Find the area.

Half-base:

Height:

Area:

When to Apply the Pythagorean Theorem

Look for these ACT signals:

• Right angle symbol in a diagram
• "Right triangle" stated in the problem
• Rectangles — diagonals create right triangles
• Coordinate distances — horizontal and vertical legs
• Walls, floors, ladders — implied right angles
• "Find the height" of a triangle

Key Takeaways and Common Mistakes

Remember:

• — right triangles only
• Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25

Watch out:

• Hypotenuse is opposite the right angle
• Leg? Subtract, then root
• means , not
• Sum ? Acute. Sum ? Obtuse.

Coming Up Next in Triangles

Up next: Special right triangles

• The 45-45-90 and 30-60-90 patterns
• Fixed side ratios you can memorize
• How these connect to the Pythagorean theorem