The Pythagorean Theorem and Special Right Triangles, SAT math lesson

A ladder leans against a wall. Its foot is 6 feet from the wall and its top touches the wall 8 feet up. How long is the ladder? The wall, the ground and the ladder form a right triangle, and one equation about right triangles answers the question without a tape measure. The same equation, together with two triangles whose sides you can write down from their angles alone, sits behind a large share of the geometry questions on the SAT.

See also Special Right Triangles in Precalculus.

Key terms

Right triangle

Definition A right triangle is a triangle with one angle that measures exactly $90^\circ$ . Because the three angles of a triangle add up to $180^\circ$ , the other two angles add up to $90^\circ$ , so each of them is acute and a triangle can have at most one right angle. In a figure the right angle is marked with a small square at its vertex.

In plain words One corner is square, like the corner of a sheet of paper. The other two corners are smaller than that and share the remaining $90^\circ$ between them.

Example A triangle has angles $90^\circ$ , $35^\circ$ and $x^\circ$ . Then

x = 180 - 90 - 35 = 55,

and indeed $35 + 55 = 90$ . On the SAT a figure is drawn to scale unless the question says otherwise, but you still need the right angle to be marked or stated before you treat a corner as $90^\circ$ . A corner that only looks square is not enough.

Hypotenuse and legs

Definition In a right triangle the hypotenuse is the side opposite the right angle. The other two sides, which meet at the right angle, are the legs. The hypotenuse is always the longest side of a right triangle, since it lies opposite the largest angle.

In plain words Find the little square first. The side that does not touch it is the hypotenuse, and the two sides that form it are the legs.

Example In triangle $PQR$ the right angle is at $Q$ . The side opposite $Q$ is $PR$ , so $PR$ is the hypotenuse and $PQ$ and $QR$ are the legs. If someone tells you $PQ = 12$ and $PR = 9$ , something is wrong, because a leg can never be longer than the hypotenuse.

Pythagorean theorem

Definition If a right triangle has legs of lengths $a$ and $b$ and hypotenuse of length $c$ , then

a^2 + b^2 = c^2.

The theorem is about right triangles only. For any other triangle the equation is false. Side lengths are positive, so when you solve for a side you keep only the positive square root,

c = \sqrt{a^2 + b^2}, \qquad a = \sqrt{c^2 - b^2}.

The SAT reference sheet states this relationship with a labeled figure, but in your own triangle you have to decide which side plays the role of $c$ . It must be the side opposite the right angle.

In plain words Build a square on each side of the triangle. The two squares on the legs together have exactly the same area as the square on the hypotenuse. One way to see why is to place four copies of the triangle inside a square of side $a + b$ . They leave a tilted square of side $c$ in the middle, so

(a + b)^2 = 4 \cdot \frac{ab}{2} + c^2,

and expanding the left side to $a^2 + 2ab + b^2$ and cancelling $2ab$ leaves $a^2 + b^2 = c^2$ .

Example For the ladder, the legs are $6$ and $8$ and the ladder is the hypotenuse.

\begin{aligned} c &= \sqrt{6^2 + 8^2} = \sqrt{100} \\[0.4em] &= 10 \text{ feet.} \end{aligned}

Now find a leg. A right triangle has hypotenuse $13$ and one leg $5$ . The other leg is

\begin{aligned} b &= \sqrt{13^2 - 5^2} = \sqrt{169 - 25} \\[0.4em] &= \sqrt{144} = 12. \end{aligned}

The most common mistake is adding when you should subtract. Adding gives $\sqrt{194}$ , which would make a leg longer than the hypotenuse.

Converse of the Pythagorean theorem

Definition If a triangle has side lengths $a$ , $b$ and $c$ with

a^2 + b^2 = c^2,

then the triangle is a right triangle and its right angle is opposite the side of length $c$ . If the squares of the two shorter sides do not add up to the square of the longest side, the triangle is not a right triangle.

In plain words The equation works in both directions. Three lengths that fit it always make a right angle, and three lengths that do not fit it never do.

Example The lengths $7$ , $24$ and $25$ give

7^2 + 24^2 = 49 + 576 = 625 = 25^2,

so they form a right triangle with hypotenuse $25$ . The lengths $6$ , $7$ and $9$ give $36 + 49 = 85$ , while $9^2 = 81$ , so that triangle has no right angle.

Pythagorean triples and exact answers

Definition A Pythagorean triple is a set of three positive integers $a$ , $b$ , $c$ with $a^2 + b^2 = c^2$ . Multiplying all three numbers of a triple by the same positive integer $k$ gives another triple, since

\begin{aligned} (ka)^2 + (kb)^2 &= k^2(a^2 + b^2) \\[0.4em] &= (kc)^2. \end{aligned}

Most right triangles do not have whole number sides, and then the exact length is a square root, written in simplest radical form when an answer choice asks for it.

In plain words A few side sets such as $3, 4, 5$ and $5, 12, 13$ and $8, 15, 17$ show up again and again, along with their multiples. Recognizing them saves time, but the theorem itself always works.

Example A right triangle has legs $15$ and $20$ . These are $5 \cdot 3$ and $5 \cdot 4$ , so the hypotenuse is $5 \cdot 5 = 25$ . As a check, $225 + 400 = 625$ . A right triangle with hypotenuse $10$ and leg $4$ has no whole number leg, and the other leg is

\sqrt{100 - 16} = \sqrt{84} = 2\sqrt{21}.

45-45-90 triangle

Definition A 45-45-90 triangle is a right triangle whose acute angles both measure $45^\circ$ . Its legs are equal, and if each leg has length $x > 0$ , the hypotenuse has length $x\sqrt{2}$ . So the sides are in the ratio

x : x : x\sqrt{2}.

The reason is the Pythagorean theorem, since $x^2 + x^2 = 2x^2$ and $\sqrt{2x^2} = x\sqrt{2}$ for $x > 0$ . The SAT reference sheet shows this triangle with its side lengths.

In plain words Cut a square along its diagonal and you get two of these triangles. The diagonal is the hypotenuse, and it is about $1.414$ times as long as a side.

Example A square has side $7$ . Its diagonal is $7\sqrt{2}$ . Going the other way, a 45-45-90 triangle has hypotenuse $10$ . Each leg is

x = \frac{10}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}.

As a check, the two legs give $50 + 50 = 100 = 10^2$ .

30-60-90 triangle

Definition A 30-60-90 triangle is a right triangle whose acute angles measure $30^\circ$ and $60^\circ$ . If the leg opposite the $30^\circ$ angle has length $x > 0$ , then the leg opposite the $60^\circ$ angle has length $x\sqrt{3}$ and the hypotenuse has length $2x$ . So, listed opposite $30^\circ$ , $60^\circ$ and $90^\circ$ in that order,

x : x\sqrt{3} : 2x.

The ratio only works when each length is matched with the angle opposite it. The shortest side is always opposite the smallest angle, $30^\circ$ . The SAT reference sheet shows this triangle with its side lengths too.

In plain words Take an equilateral triangle and cut it in half from a vertex straight down to the middle of the opposite side. Each half is a 30-60-90 triangle. The hypotenuse is a full side of the equilateral triangle and the short leg is half a side, which is why the hypotenuse is twice the short leg.

Example An equilateral triangle has side $6$ . Its height splits it into two 30-60-90 triangles with hypotenuse $6$ and short leg $3$ , so the height is $3\sqrt{3}$ . Check with the theorem,

3^2 + (3\sqrt{3})^2 = 9 + 27 = 36 = 6^2.

A harder case starts from the long leg. Suppose the side opposite the $60^\circ$ angle is $9$ . Then $x\sqrt{3} = 9$ , so

x = \frac{9}{\sqrt{3}} = 3\sqrt{3},

and the hypotenuse is $2x = 6\sqrt{3}$ . Putting the $9$ opposite the $30^\circ$ angle instead would give the wrong triangle.

Check yourself

A right triangle has a hypotenuse of length $25$ and one leg of length $7$ . What is the length of the other leg? (A) $18$ (B) $\sqrt{674}$ (C) $24$ (D) $32$

Show the answer

The correct choice is (C). The unknown side is a leg, so subtract squares,

\begin{aligned} \sqrt{25^2 - 7^2} &= \sqrt{625 - 49} \\[0.4em] &= \sqrt{576} = 24. \end{aligned}

Choice (B) treats $25$ as a leg and adds the squares, which would make a leg longer than the hypotenuse. Choices (A) and (D) subtract or add the lengths themselves instead of their squares.

In right triangle $PQR$ the right angle is at $R$ , angle $P$ measures $60^\circ$ , and $PR = 5$ . What is the length of $PQ$ ?

Show the answer

The answer is $10$ . Angle $Q$ measures $90^\circ - 60^\circ = 30^\circ$ , and side $PR$ is opposite angle $Q$ , so $PR$ is the short leg with $x = 5$ . Side $PQ$ is opposite the right angle, so it is the hypotenuse, $2x = 10$ . Reading $PR$ as the leg opposite $60^\circ$ would give the wrong value $\frac{10}{\sqrt{3}}$ .

A square has a diagonal of length $12$ . What is the area of the square? (A) $36$ (B) $72$ (C) $144$ (D) $36\sqrt{2}$

Show the answer

The correct choice is (B). The diagonal cuts the square into two 45-45-90 triangles with hypotenuse $12$ , so each side is $\frac{12}{\sqrt{2}} = 6\sqrt{2}$ and the area is $(6\sqrt{2})^2 = 72$ . Choice (C) squares the diagonal instead of the side, choice (A) uses half the diagonal as the side, and choice (D) forgets that squaring $\sqrt{2}$ gives $2$ .

The Pythagorean Theorem and Special Right Triangles

Theory, part 97

Practice, part 98

Key terms

Right triangle

Hypotenuse and legs

Pythagorean theorem

Converse of the Pythagorean theorem

Pythagorean triples and exact answers

45-45-90 triangle

30-60-90 triangle

Check yourself

How this lesson connects

Builds on

Used later in