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Properties of
Right Triangles

Master the Pythagorean theorem, special right triangles, and trigonometric ratios: the essential toolkit for SAT Math geometry questions.

SAT Math

What Is a Right Triangle?

α β A B C a b (hypotenuse) c

A right triangle has exactly one \(90^\circ\) angle. The side opposite the right angle is the hypotenuse: the longest side. The other two sides are called legs.

Key Vocabulary

Hypotenuse
The side opposite the \(90^\circ\) angle: always the longest side of the triangle.
Legs
The two sides that form the right angle. Often labeled \(a\) and \(b\).
Altitude
A perpendicular segment from the right-angle vertex to the hypotenuse, creating two smaller right triangles.

Why Right Triangles Matter on the SAT

Right triangles are the most tested geometry shape on the SAT. They appear directly and are hidden inside circles, rectangles, and coordinate-geometry problems.

1
Pythagorean Theorem questions: find a missing side.
2
Special right triangles (\(45^\circ\text{-}45^\circ\text{-}90^\circ\) and \(30^\circ\text{-}60^\circ\text{-}90^\circ\)) appear on the reference sheet.
3
Trigonometric ratios (sin, cos, tan) only apply to right triangles.
4
Distance formula on the coordinate plane is the Pythagorean Theorem in disguise.

The Pythagorean Theorem

a b c
\(a^2 + b^2 = c^2\)
(legs → hypotenuse)

Common Pythagorean Triples

A Pythagorean triple is a set of three positive integers \((a, b, c)\) satisfying \(a^2 + b^2 = c^2\). Memorizing these saves time on the SAT:

Triple Check Multiples
\(3\text{-}4\text{-}5\) \(9+16=25\) ✓ \(6\text{-}8\text{-}10\), \(9\text{-}12\text{-}15\), ...
\(5\text{-}12\text{-}13\) \(25+144=169\) ✓ \(10\text{-}24\text{-}26\), ...
\(8\text{-}15\text{-}17\) \(64+225=289\) ✓ \(16\text{-}30\text{-}34\), ...
\(7\text{-}24\text{-}25\) \(49+576=625\) ✓ \(14\text{-}48\text{-}50\), ...

The \(45^\circ\text{-}45^\circ\text{-}90^\circ\) Triangle

\(45^\circ\)
\(45^\circ\)
\(s\)
\(s\)
\(s\sqrt{2}\)
 Ratio
\(1 : 1 : \sqrt{2}\)

The \(30^\circ\text{-}60^\circ\text{-}90^\circ\) Triangle

\(30^\circ\)
\(60^\circ\)
\(s\sqrt{3}\)
\(s\)
\(2s\)
 Ratio
\(1 : \sqrt{3} : 2\)

Comparing Special Right Triangles

Property \(45^\circ\text{-}45^\circ\text{-}90^\circ\) \(30^\circ\text{-}60^\circ\text{-}90^\circ\)
Angles \(45^\circ, 45^\circ, 90^\circ\) \(30^\circ, 60^\circ, 90^\circ\)
Side ratio \(1 : 1 : \sqrt{2}\) \(1 : \sqrt{3} : 2\)
Derived from Square diagonal Equilateral triangle
Legs equal? Yes No
Hypotenuse from short side \(\times \sqrt{2}\) \(\times 2\)

SOH CAH TOA: Trig Ratios

SOH
\(\sin\theta = \dfrac{\text{Opposite}}{\text{Hypotenuse}}\)
CAH
\(\cos\theta = \dfrac{\text{Adjacent}}{\text{Hypotenuse}}\)
TOA
\(\tan\theta = \dfrac{\text{Opposite}}{\text{Adjacent}}\)

Complementary Angle Identity

If two acute angles in a right triangle are \(\alpha\) and \(\beta\), then \(\alpha + \beta = 90^\circ\). This gives a powerful identity tested on the SAT:

The SAT loves this: if \(\sin(x^\circ) = \cos(y^\circ)\), then \(x + y = 90\). No calculation needed: just set up the equation.

How to Solve Right Triangle Problems

1
Identify the right angle. Look for the square symbol or a stated \(90^\circ\) angle.
2
Label the sides. Mark hypotenuse (opposite right angle), then label opposite and adjacent relative to the angle in question.
3
Choose your tool. Need a side? Try the Pythagorean Theorem or special-triangle ratios first. Need an angle or a ratio? Use SOH CAH TOA.
4
Check for triples. Before computing, see if the given sides match a Pythagorean triple or multiple.
5
Verify. Confirm the hypotenuse is the longest side and angles sum to \(180^\circ\).

Example 1: Finding a Missing Side

6 8 c = ?

Example 2: \(45^\circ\text{-}45^\circ\text{-}90^\circ\) in Action

\(45^\circ\)
\(45^\circ\)
 ? leg ? leg 10

Example 3: \(30^\circ\text{-}60^\circ\text{-}90^\circ\) with Trig

A ladder leans against a wall, making a \(60^\circ\) angle with the ground. The ladder is 20 ft long. How high up the wall does it reach?

\(60^\circ\)
 20 ft h = ?

Common Mistakes

✕ Swapping Legs and Hypotenuse
Using the hypotenuse as a leg in \(a^2 + b^2 = c^2\). Remember: \(c\) is always the longest side, opposite the right angle.
✕ Wrong Special Triangle Ratio
Applying the \(\sqrt{2}\) multiplier to a \(30^\circ\text{-}60^\circ\text{-}90^\circ\) triangle or the \(\sqrt{3}\) to a \(45^\circ\text{-}45^\circ\text{-}90^\circ\). Match the ratio to the angles.
✕ Opposite vs. Adjacent Mix-Up
Using the wrong leg for sin or cos. Always label sides relative to the specific angle, not the triangle in general.

Question 1 of 5

A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?

Question 2 of 5

In a \(45^\circ\text{-}45^\circ\text{-}90^\circ\) triangle, one leg has length 7. What is the hypotenuse?

Question 3 of 5

In a \(30^\circ\text{-}60^\circ\text{-}90^\circ\) triangle, the hypotenuse is 16. What is the length of the side opposite the \(30^\circ\) angle?

Question 4 of 5

If \(\sin(x^\circ) = \cos(40^\circ)\), what is the value of \(x\)?

Question 5 of 5

In right triangle \(ABC\), angle \(C = 90^\circ\), \(AC = 9\), and \(BC = 12\). What is \(\tan(A)\)?