Pythagorean Theorem

The **Pythagorean Theorem** describes the relationship between the three sides of a right triangle: $$a^2 + b^2 = c^2$$ Where: - $a$ and $b$ are the lengths of the two **legs** (the sides that form the right angle) - $c$ is the length of the **hypotenuse** (the side opposite the right angle) The hypotenuse is always the longest side of a right triangle. **Key insight**: The sum of the squares of the two shorter sides equals the square of the longest side.

The Classic 3-4-5 Triangle

A right triangle has legs of length 3 and 4. Find the hypotenuse.

Write the Pythagorean theorem: $a^2 + b^2 = c^2$ = Formula ready

Substitute the known values: $3^2 + 4^2 = c^2$ = Values plugged in

Calculate the squares: $9 + 16 = c^2$ = $25 = c^2$

Take the square root: $c = \sqrt{25}$ = $c = 5$

Answer: The hypotenuse is 5 units long.

Finding a Missing Leg

A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg.

Write the theorem with unknowns: $a^2 + b^2 = c^2$ where $a = 5$, $c = 13$ = $5^2 + b^2 = 13^2$

Calculate known squares: $25 + b^2 = 169$ = Simplify left and right

Isolate $b^2$: $b^2 = 169 - 25$ = $b^2 = 144$

Take the square root: $b = \sqrt{144}$ = $b = 12$

Answer: The missing leg is 12 units long. This is a 5-12-13 right triangle!

Real-World Application: Ladder Safety

A 10-meter ladder leans against a wall. The base is 6 meters from the wall. How high up the wall does the ladder reach?

Identify the triangle parts: Ladder = hypotenuse = 10m, Ground = leg = 6m, Wall height = unknown leg = $6^2 + h^2 = 10^2$

Calculate known squares: $36 + h^2 = 100$ = Simplified equation

Solve for $h^2$: $h^2 = 100 - 36 = 64$ = $h^2 = 64$

Find $h$: $h = \sqrt{64}$ = $h = 8$ meters

Answer: The ladder reaches 8 meters up the wall.

Checking If a Triangle Is Right

Does a triangle with sides 7, 24, and 25 form a right triangle?

Identify the longest side: The longest side (25) would be the hypotenuse if it's a right triangle = $c = 25$, $a = 7$, $b = 24$

Check if $a^2 + b^2 = c^2$: $7^2 + 24^2 = 49 + 576 = 625$ = $a^2 + b^2 = 625$

Calculate $c^2$: $25^2 = 625$ = $c^2 = 625$

Compare the values: $625 = 625$ ✓ = They are equal!

Answer: Yes! Since $7^2 + 24^2 = 25^2$, this is a right triangle. The 7-24-25 triangle is another Pythagorean triple!

The Pythagorean theorem is one of the most useful tools in mathematics: - **Construction**: Builders use it to ensure corners are perfectly square - **Navigation**: Calculate the shortest distance between two points - **Sports**: Determine diagonal distances on fields and courts - **Architecture**: Design stable structures with precise measurements - **Video games**: Calculate distances and movements in 2D and 3D spaces This 2,500-year-old theorem is used daily by engineers, architects, and scientists worldwide!

Mistake: Using the formula on non-right triangles

Why: The Pythagorean theorem ONLY works for right triangles (triangles with a 90° angle).

Correct: Always check that the triangle has a right angle before applying $a^2 + b^2 = c^2$.

Mistake: Putting the hypotenuse in the wrong position

Why: Students sometimes add the hypotenuse with a leg: $a^2 + c^2 = b^2$. This gives wrong answers.

Correct: The hypotenuse ($c$) is ALWAYS alone on one side: $a^2 + b^2 = c^2$. The hypotenuse is opposite the right angle.

Mistake: Forgetting to take the square root

Why: After finding $c^2 = 25$, students write $c = 25$ instead of $c = 5$.

Correct: Always take the square root as the final step: if $c^2 = 25$, then $c = \sqrt{25} = 5$.

Mistake: Adding instead of squaring first

Why: Writing $3 + 4 = 7$ then squaring, instead of $3^2 + 4^2$.

Correct: Square EACH leg individually first: $3^2 + 4^2 = 9 + 16 = 25$, not $(3+4)^2 = 49$.

Construction and Carpentry

Builders use the 3-4-5 rule to check if corners are perfectly square (90°).

Measure 3 feet along one wall, 4 feet along the other. If the diagonal is exactly 5 feet, the corner is a perfect right angle.

Navigation and GPS

Finding the shortest (straight-line) distance between two locations.

If you walk 400 meters east and then 300 meters north, the direct distance back is $\sqrt{400^2 + 300^2} = 500$ meters.

Sports Field Measurements

Calculate diagonal distances across rectangular fields and courts.

A soccer field is 100 meters long and 70 meters wide. The diagonal distance is $\sqrt{100^2 + 70^2} \approx 122$ meters.

The Pythagorean theorem states that in a right triangle, $a^2 + b^2 = c^2$

The legs ($a$ and $b$) are the two sides that form the right angle

The hypotenuse ($c$) is the longest side, opposite the right angle

Use this theorem to find any missing side when you know the other two

Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25

Q: Who was Pythagoras?

A: Pythagoras was a Greek mathematician who lived around 500 BCE. While the theorem bears his name, similar knowledge existed in Babylon and India even earlier. Pythagoras (or his school) is credited with the first formal proof.

Q: What is a Pythagorean triple?

A: A Pythagorean triple is a set of three whole numbers that satisfy the theorem. Examples: 3-4-5, 5-12-13, 8-15-17. Any multiple of a triple also works (6-8-10, 9-12-15, etc.).

Q: Can the theorem give decimal answers?

A: Yes! If the sides are $1$ and $1$, the hypotenuse is $\sqrt{2} \approx 1.414$. Not all right triangles have whole-number sides.

Q: Does the theorem work in reverse?

A: Yes! If three sides satisfy $a^2 + b^2 = c^2$, the triangle MUST be a right triangle. This is called the converse of the Pythagorean theorem.

Exit Presentation

Pythagorean Theorem

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Geometry / Triangles

Pythagorean Theorem

Learn the famous relationship between the sides of a right triangle and use it to solve real-world problems.

Intermediate25 min

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