Right Triangle

Precise definition

A right triangle has one $90^\circ$ angle. The side opposite it is the hypotenuse and is longest; the other sides are legs. For leg lengths $a,b$ and hypotenuse $c$, the Pythagorean theorem gives $a^2+b^2=c^2$, and its converse identifies a right triangle from side lengths.

Notation and mathematical language

For acute angle $\theta$, $\sin\theta=\text{opposite}/\text{hypotenuse}$, $\cos\theta=\text{adjacent}/\text{hypotenuse}$, and $\tan\theta=\text{opposite}/\text{adjacent}$. Labels opposite and adjacent depend on the chosen angle; hypotenuse does not.

Conceptual picture

Similarity explains why trigonometric ratios depend only on angle, not triangle size. Pythagoras encodes Euclidean distance and connects to the unit circle. The two acute angles are complementary.

Conditions and key results

These rules require a right triangle. Inverse trig gives a principal acute angle for positive side ratios, and calculator mode must match degrees or radians. Measurements and diagrams may be approximate or not to scale.

A reliable strategy

1. Mark the right angle, hypotenuse, reference angle, known sides, and units.
2. Choose Pythagoras for two-side relationships or a trig ratio containing the known and requested sides.
3. Solve symbolically, then calculate with correct angle mode and postpone rounding.
4. Check longest side, angle sum, magnitude, units, and the original ratio.

Fully worked example

Interpretation and application

Right triangles model surveying, construction, vectors, navigation, and coordinate distance. The perpendicular and straight-line assumptions must be validated; real slopes and surfaces may require measurement uncertainty.

Common mistakes

Verification and reasonableness

• Confirm the hypotenuse is longest.
• Use a second trig ratio or Pythagorean calculation.
• Check the two acute angles sum to $90^\circ$ and keep units.

Practice

1. Find hypotenuse for legs 5 and 12.
2. If opposite=3 and hypotenuse=5, find sine.
3. What theorem checks whether sides 6,8,10 form a right triangle?

Further deduction

Special triangles provide exact values. In a $45$–$45$–$90$ triangle, sides are proportional to $1:1:\sqrt2$; in a $30$–$60$–$90$ triangle, they are $1:\sqrt3:2$ opposite those angles. Exact ratios should be retained unless the application asks for a decimal measurement.

The altitude from the right angle to the hypotenuse creates three similar triangles and useful length relations. If hypotenuse $c$ is split into segments $p$ and $q$, with legs adjacent to them $a$ and $b$, similarity gives $a^2=cp$, $b^2=cq$, and altitude $h$ satisfying $h^2=pq$. For $c=25$, $p=9$, and $q=16$, the legs are $a=\sqrt{225}=15$ and $b=\sqrt{400}=20$, while $h=\sqrt{144}=12$. These agree with $15^2+20^2=25^2$. The formulas require the specified right-triangle configuration; assigning $p$ to the wrong adjacent leg swaps the relations but not the underlying similarity. A labelled diagram and a Pythagorean check are safer than choosing a memorized proportion from appearance alone.

Related topics

• Pythagorean Theorem
• Trigonometry Review
• Vectors in Two Dimensions