Pythagorean Theorem

📐 The Pythagorean Theorem

Part 1 of 5 — Right Triangles & the Theorem

Topics in This Part

🔑 Key Concept: The Pythagorean Theorem connects the three sides of a right triangle. It is one of the most-used relationships in all of mathematics — from carpentry to GPS to computer graphics.

Parts of a Right Triangle

A right triangle has exactly one $90^\circ$ angle (the right angle, marked with a small square).

• The two sides that form the right angle are the legs. We usually call them $a$ and $b$.
• The side across from the right angle is the hypotenuse, called $c$.

🔑 The hypotenuse is always the longest side, and it is always the one opposite the right angle. The two legs can be in either order — $a$ and $b$ are interchangeable.

Concept Check 🎯

The Theorem

For any right triangle with legs $a$ and $b$ and hypotenuse $c$:

$a^2 + b^2 = c^2$

In words: the square of the hypotenuse equals the sum of the squares of the two legs.

Why It's True — The Area Picture

Build a square on each side of the triangle. The theorem says the area of the square on the hypotenuse ($c^2$) is exactly equal to the combined areas of the squares on the two legs ($a^2 + b^2$).

⚠️ Order matters: $c$ (the hypotenuse) is always by itself on one side of the equation. The two legs are the ones that get added. Putting a leg where $c$ belongs is the #1 mistake.

Check the Theorem 🧮

For the classic $3$-$4$-$5$ right triangle (legs $3$ and $4$, hypotenuse $5$), fill in each square's value.

1) $3^2 = \,?$ 2) $4^2 = \,?$ 3) $3^2 + 4^2 = \,?$ (this should equal $5^2$)

Setting Up the Equation Correctly

Before solving anything, get the equation in the right shape. Three habits keep you out of trouble:

1. Spot the hypotenuse first — it's opposite the right angle and is the longest side. Its square ($c^2$) stands alone.
2. The two legs get added, never subtracted, when you build the basic equation.
3. Write it the same way every time: $a^2 + b^2 = c^2$.

Lock that pattern in with the check below.

Set Up the Equation 🔽

A right triangle has legs $a$ and $b$ and hypotenuse $c$. Pick the correct piece for each blank.

Recap

• A right triangle has one $90^\circ$ angle.
• The legs ($a$, $b$) form the right angle; the hypotenuse ($c$) is opposite it and is the longest side.
• The theorem: $a^2 + b^2 = c^2$.

In Part 2 we'll use this equation to find a missing hypotenuse.

📐 The Pythagorean Theorem

Part 2 of 5 — Finding the Hypotenuse

🔑 The Goal: When you know both legs and want the hypotenuse, plug into $a^2 + b^2 = c^2$, add, then take the square root.

The Steps

To find the hypotenuse $c$ when you know legs $a$ and $b$:

1. Write $a^2 + b^2 = c^2$.
2. Square each leg and add.
3. Take the square root to get $c$: $\;c = \sqrt{a^2 + b^2}$.

Worked Example: legs $6$ and $8$

$c^2 = 6^2 + 8^2 = 36 + 64 = 100$ $c = \sqrt{100} = 10$

So the hypotenuse is $10$.

✅ Check: $6^2 + 8^2 = 36 + 64 = 100 = 10^2$ ✓

Worked Example: legs $5$ and $12$

$c^2 = 5^2 + 12^2 = 25 + 144 = 169$ $c = \sqrt{169} = 13$

Worked Example: legs $1$ and $1$ (a non-perfect square)

$c^2 = 1^2 + 1^2 = 1 + 1 = 2 \;\Rightarrow\; c = \sqrt{2} \approx 1.41$

Not every answer is a whole number! When $a^2 + b^2$ isn't a perfect square, leave it in exact radical form ($\sqrt{2}$) or round to a decimal if asked.

💡 Pythagorean triples are whole-number trios that satisfy $a^2+b^2=c^2$. Memorizing a few — $3$-$4$-$5$, $5$-$12$-$13$, $8$-$15$-$17$, $7$-$24$-$25$ — lets you spot answers instantly.

Concept Check 🎯

Your Turn — Build the Reflex

The fastest way to internalize "add, then root" is repetition. Each triangle below is a Pythagorean triple, so every hypotenuse is a clean whole number. Square both legs, add, and take the root.

💡 If you recognize the triple (like $9$-$12$-$15$, which is $3$-$4$-$5$ tripled), you can write the answer without a calculator.

Find the Hypotenuse 🧮

Each triangle below is a right triangle. Find the hypotenuse $c$. All three are perfect squares, so answers are whole numbers.

1) legs $9$ and $12$ $\;\Rightarrow\; c = \,?$ 2) legs $7$ and $24$ $\;\Rightarrow\; c = \,?$ 3) legs $20$ and $21$ $\;\Rightarrow\; c = \,?$

When the Answer Isn't "Nice"

The drills above were all Pythagorean triples, so the square roots came out whole. Most real triangles aren't so tidy. When $a^2 + b^2$ is not a perfect square, you have two honest ways to report the answer:

• Exact form: leave the radical, e.g. $c = \sqrt{13}$.
• Decimal approximation: round, e.g. $c \approx 3.6$ (use the $\approx$ symbol, not $=$).

The next check walks through one of these step by step.

Exact vs. Rounded 🔽

Legs are $2$ and $3$. Work through to the hypotenuse.

📐 The Pythagorean Theorem

Part 3 of 5 — Finding a Missing Leg

🔑 The Twist: When the hypotenuse is known but a leg is missing, you subtract instead of add: rearrange to $a^2 = c^2 - b^2$.

Solving for a Leg

Start from $a^2 + b^2 = c^2$ and isolate the missing leg. If $a$ is unknown:

$a^2 = c^2 - b^2 \quad\Longrightarrow\quad a = \sqrt{c^2 - b^2}$

⚠️ Watch which number is biggest. The hypotenuse $c$ is always the largest side, so $c^2$ must come first in the subtraction. If you accidentally compute $b^2 - c^2$ you'll get a negative number under the root — a sign you mixed up the hypotenuse and a leg.

Worked Example: hypotenuse $13$, one leg $5$

$a^2 = 13^2 - 5^2 = 169 - 25 = 144$ $a = \sqrt{144} = 12$

The missing leg is $12$. (This is the $5$-$12$-$13$ triple.)

Worked Example: hypotenuse $17$, one leg $8$

$a^2 = 17^2 - 8^2 = 289 - 64 = 225$ $a = \sqrt{225} = 15$

Worked Example: hypotenuse $10$, one leg $6$

$a^2 = 10^2 - 6^2 = 100 - 36 = 64$ $a = \sqrt{64} = 8$

✅ Check: $6^2 + 8^2 = 36 + 64 = 100 = 10^2$ ✓ — the recovered leg fits.

Concept Check 🎯

Walk Through the Order of Operations

It's easy to set up a leg problem wrong by subtracting in the wrong direction. The next check makes you commit to each step in order, so you can see exactly where the hypotenuse-square must lead.

⚠️ The biggest number (the hypotenuse) is squared first and everything else is subtracted from it. If your result under the root is negative, you've swapped a leg with the hypotenuse.

Order the Work 🔽

Hypotenuse is $15$, one leg is $9$. Find the other leg.

Recognizing the Subtraction Setup

Every "missing leg" problem follows the same rhythm: biggest square minus known square, then root.

$\text{leg} = \sqrt{(\text{hypotenuse})^2 - (\text{known leg})^2}$

Many of these are just familiar triples in disguise — if you see a $13$ hypotenuse with a $12$ leg, you can almost feel the $5$ coming. Try the next set without a calculator where you can.

Find the Missing Leg 🧮

Each is a right triangle with the hypotenuse given. Find the missing leg (all whole numbers).

1) hypotenuse $13$, leg $12$ $\;\Rightarrow\;$ other leg $= \,?$ 2) hypotenuse $26$, leg $10$ $\;\Rightarrow\;$ other leg $= \,?$ 3) hypotenuse $41$, leg $9$ $\;\Rightarrow\;$ other leg $= \,?$

📐 The Pythagorean Theorem

Part 4 of 5 — Applications, Distance & the Converse

🔑 Big Payoff: The theorem powers real-world distance problems, the distance formula on the coordinate plane, and a test (the converse) for whether a triangle is right at all.

Word Problems

The trick is to find the right triangle hiding in the situation. The hypotenuse is the slanted/longest distance; the legs are usually horizontal and vertical.

Example: The Ladder

A $13$-ft ladder leans against a wall with its base $5$ ft from the wall. How high up the wall does it reach?

The ladder is the hypotenuse ($13$); the ground distance is one leg ($5$); the wall height is the missing leg.

$h^2 = 13^2 - 5^2 = 169 - 25 = 144 \;\Rightarrow\; h = 12 \text{ ft}$

💡 The ladder is the longest side, so it must be the hypotenuse. Spotting the hypotenuse first keeps you from subtracting in the wrong direction.

Application Check 🎯

The Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is just the Pythagorean Theorem in disguise:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

The horizontal gap $(x_2 - x_1)$ and the vertical gap $(y_2 - y_1)$ are the two legs; the straight-line distance $d$ is the hypotenuse.

Example: distance from $(1, 2)$ to $(4, 6)$

$d = \sqrt{(4-1)^2 + (6-2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

🔑 You don't have to memorize a "new" formula — it's $a^2 + b^2 = c^2$ where the legs are the run and the rise between the points.

Distance on the Plane 🧮

Use $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. All answers come out whole.

1) distance from $(0,0)$ to $(6,8)$ $= \,?$ 2) distance from $(-1, 2)$ to $(2, 6)$ $= \,?$ 3) distance from $(1, 3)$ to $(9, 18)$ $= \,?$

The Converse: Is It a Right Triangle?

The converse of the theorem runs the logic backward:

If $a^2 + b^2 = c^2$ (with $c$ the longest side), then the triangle IS a right triangle.

You can also classify triangles that aren't right:

Always let $c$ be the longest side before comparing.

Example: sides $9$, $12$, $15$

Longest is $15$: $\;9^2 + 12^2 = 81 + 144 = 225$ and $15^2 = 225$. Equal → right triangle. ✓

Classify the Triangle 🔽

For each side-trio, the longest side is $c$. Compare $a^2 + b^2$ to $c^2$.

📐 The Pythagorean Theorem

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) name the parts of a right triangle, (2) find the hypotenuse, (3) find a missing leg, and (4) apply the theorem to word problems, distance, and the converse. Let's put it all together.

Quick Reference

Triples Worth Knowing

⚠️ Remember: the hypotenuse is always the longest side and stands alone as $c^2$. Add to find the hypotenuse; subtract to find a leg.

Mixed Practice 🎯

Decide Before You Compute

The hardest part of a mixed set is choosing the right move. A two-second sort saves you every time:

• Given two legs and want the long side? → add, then root (hypotenuse).
• Given the hypotenuse and one leg? → subtract, then root (missing leg).
• Given two points? → subtract coordinates to get the legs, then it's a hypotenuse problem.

Apply that sort to each item below.

Final Drill 🧮

1) Legs $10$ and $24$: hypotenuse $= \,?$ 2) Hypotenuse $37$, leg $12$: other leg $= \,?$ 3) Distance from $(2, 3)$ to $(14, 8)$ $= \,?$

One Last Look Before the Quiz

You've covered all four skills. The exit quiz pulls one question from each major idea:

• finding a hypotenuse from two legs,
• using the converse to classify a triangle,
• and the distance formula on the coordinate plane.

Keep the triples handy and remember: hypotenuse alone, legs added. Ready when you are.

Exit Quiz ✅

Answer all three to finish the lesson.