Apply the Pythagorean theorem (a² + b² = c²) and its converse to find missing sides of right triangles and determine if a triangle is right, acute, or obtuse. Apply to distance on the coordinate plane.
A lesson of Algebra 1, in Radical Expressions & Equations, about 35 minutes.
By the end of this lesson, you will be able to find any missing side of a right triangle, use the converse to test whether a triangle is right, apply it to distance on a grid, and avoid the one error that produces a side longer than the hypotenuse.
The Relationship
$$a^2 + b^2 = c^2$$
In a right triangle, the two shorter sides squared and added give the longest
side squared. It holds for every right triangle and for no other kind.
$a$ and $b$ are the legs, the two sides meeting at the right angle
$c$ is the hypotenuse, opposite the right angle, and always the longest side
$c$ is always alone on its own side of the equation. Putting a leg where $c$
belongs is the error that produces impossible triangles, and it is easy to check
for: the hypotenuse must come out larger than either leg.
Why the longest side is opposite the right angle. A bigger angle always faces
a longer side, and the right angle is the biggest in the triangle, since the three
angles add to $180°$ and only one of them can reach $90°$.
Sort It: Which Side Is the Hypotenuse
Example 1: Finding the Hypotenuse
Here is every step on one triangle, worked all the way through.
Legs of $6$ and $8$.
What is 6 squared? 36
What is 8 squared? 64
Add them. What does c squared equal? 100
Take the square root. What is the hypotenuse? 10
Finding a Leg: Subtract Instead
When the hypotenuse is known and a leg is missing, the same equation is
rearranged rather than reused blindly.
Subtract, do not add. Adding $169 + 25$ gives $\sqrt{194} \approx 13.9$, which
is longer than the hypotenuse and therefore impossible. The impossible answer is
the tell.
Identify the hypotenuse before writing anything down. It is the side opposite
the right angle, and in a word problem it is usually the ladder, the rope, the
diagonal or the direct distance, whatever spans the corner rather than forming
it.
Sort It: Add or Subtract
Worked Example 1: Finding a Leg
Hypotenuse $17$, one leg $8$.
When the Answer Is Irrational
Most triangles do not give whole numbers, and the exact radical is the answer.
$34$ has no square factor, so $\sqrt{34}$ is fully simplified and that is the
final answer. Give $5.83$ only if the problem asks for a decimal, which it will
when the thing being measured is physical.
$$\text{legs } 2 \text{ and } 4 \quad \Rightarrow \quad c = \sqrt{20} = 2\sqrt{5}$$
Simplify the radical, using the previous two lessons. $\sqrt{20}$ is not a
finished answer.
A few triples come out whole and are worth recognizing, because they appear
constantly in problems: $3\text{-}4\text{-}5$, $5\text{-}12\text{-}13$,
$8\text{-}15\text{-}17$, $7\text{-}24\text{-}25$, and every multiple of them, so
$6\text{-}8\text{-}10$ and $9\text{-}12\text{-}15$ as well.
Worked Example 2: An Irrational Answer
Legs of $4$ and $6$.
The Converse: Testing for a Right Angle
The theorem runs backwards, and that makes it a test rather than only a formula.
$$\text{if } a^2 + b^2 = c^2 \text{ then the triangle IS right}$$
Take the three sides, square them, and compare the two smaller squares against
the largest.
The two smaller squares falling short means the longest side has stretched
past a right angle, opening the corner opposite it. Falling over means the
opposite: the corner has closed up, and every angle is acute.
Always square the LONGEST side on its own. Comparing the wrong pair gives the
wrong verdict, and the longest side is the one facing the angle being tested.
This converse is genuinely useful outside mathematics: builders check that a
corner is square by measuring $3$ and $4$ along the two edges and confirming the
diagonal is exactly $5$.
Worked Example 3: Using the Converse
Sides $9$, $12$, $15$.
Distance on a Grid
Two points on a coordinate plane make a right triangle with the axes, and the
theorem gives the direct distance.
$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$
That is not a new formula to memorize. The horizontal gap is one leg, the vertical
gap is the other, and $d$ is the hypotenuse.
$$(1, 2) \text{ to } (4, 6)$$
Across $3$, up $4$, so $d = \sqrt{9 + 16} = 5$.
The differences get squared, so their signs do not matter. Subtracting in
either order gives the same distance, which is as it should be: the distance from
A to B is the distance from B to A.
Plot It: The Triangle Behind the Distance
Common Mistakes
Mistake 1: Adding when you should subtract
Hypotenuse known means $c^2 - b^2$. Adding gives a leg longer than the
hypotenuse.
Mistake 2: Putting a leg in the $c$ position
$c$ is opposite the right angle, and always the longest.
Mistake 3: Forgetting the square root
$c^2 = 100$ means $c = 10$, not $100$.
Mistake 4: Leaving a radical unsimplified
$\sqrt{20}$ should be $2\sqrt{5}$.
Mistake 5: Using it on a non-right triangle
The theorem needs a right angle. Without one, use the converse to find out.
Where This Goes Next
The theorem generalizes in two directions, and both are worth knowing exist.
Into more dimensions. The diagonal of a box with edges $a$, $b$ and $c$ is
$\sqrt{a^2+b^2+c^2}$, which is the theorem applied twice. The pattern continues
into any number of dimensions, and it is how distance is DEFINED in higher
dimensions rather than something proved there. Machine learning measures the
distance between two data points with a hundred features using exactly this sum.
Into non-right triangles. The law of cosines is:
$$c^2 = a^2 + b^2 - 2ab\cos(C)$$
When $C = 90°$, $\cos(90°) = 0$ and the last term vanishes, leaving
$a^2 + b^2 = c^2$. So Pythagoras is the special case where the correction term
disappears, and that term is exactly what accounts for the acute and obtuse cases
in the converse table above.
There are over three hundred published proofs of this theorem, including one by
a sitting US president. It is around 2,500 years old, was known to Babylonian
mathematicians a thousand years before Pythagoras, and it is still the single
most-used result in all of applied mathematics.
Check Your Understanding
Five of them, no instructions this time.
Summary
$$a^2 + b^2 = c^2$$
$c$ is the hypotenuse, opposite the right angle and always the longest side.
If your answer breaks that, you added when you should have subtracted.
Two legs known: add. Hypotenuse known: subtract.
Simplify the radical, and give a decimal only when asked.
The converse tests a triangle: equal means right, greater means acute, less
means obtuse, comparing against the longest side squared.
Distance on a grid is this theorem, with the horizontal and vertical gaps as
the legs.