Right triangle
A right triangle is any triangle with an interior right angle.
![[asy] pair A,B,C; A = (0,3); B = (4,0); C = (0,0); draw(A--B--C--cycle); draw(rightanglemark(A,C,B)); label("$A$",A,NW); label("$B$",B,E); label("$C$",C,SW); label("$a$",midpoint(C--B),S); label("$b$",midpoint(C--A),W); label("$c$",midpoint(A--B),NE); [/asy]](http://latex.artofproblemsolving.com/a/c/b/acbb699c20d4eac19f6db1a45de1791fadf27ab1.png)
In the image above, 
has a measure of 
, so 
is a right triangle. The longest side, opposite the right angle, is called the hypotenuse. In this diagram, the hypotenuse is labeled 
. The other two sides are called the legs of the triangle, labeled 
and 
.
Right triangles are very useful in geometry. One of the most important theorems about right triangles is the Pythagorean Theorem. Aside from this, the field of trigonometry arises from the study of right triangles and nearly all trigonometric identities can be deduced from them.
Special Right Triangles
- Main article: Special right triangles
There are many right triangles with special properties. One of these is the isosceles 
triangle, where the hypotenuse is equal to 
times the length of either of the legs. This triangle is analogous to a square cut in half along its diagonal.
![[asy] pair A, B, C; A = (0, 1); B = (1, 0); C = (0, 0); draw(A--B--C--cycle); draw(rightanglemark(A, C, B, 3)); draw(anglemark(A, B, C, 4)); draw(anglemark(C, A, B, 4)); label("$A$", A, NW); label("$45^{\circ}$", A, 6*dir(290)); label("$B$", B, E); label("$45^{\circ}$", B, 5*dir(155)); label("$C$", C, SW); label("$1$", midpoint(C--B), S); label("$1$", midpoint(C--A), W); label("$\sqrt{2}$", midpoint(A--B), NE); [/asy]](http://latex.artofproblemsolving.com/e/b/2/eb20d26c6abf555f810c0b19d828949a5e360d02.png)
Another one of these is the 
triangle, which has sides in the ratio of 
. This triangle is analogous to an equilateral triangle cut in half down the middle.
![[asy] pair A, B, C; A = (0, sqrt(3)); B = (1, 0); C = (0, 0); draw(A--B--C--cycle); draw(rightanglemark(A, C, B, 4)); label("$A$", A, NW); label("$30^{\circ}$", A, 10*dir(283)); label("$B$", B, E); label("$60^{\circ}$", B, 3*dir(150)); label("$C$", C, SW); label("$1$", midpoint(C--B), S); label("$\sqrt{3}$", midpoint(C--A), W); label("$2$", midpoint(A--B), NE); [/asy]](http://latex.artofproblemsolving.com/6/c/d/6cd037bb7cd1c93044333214d21e6e29f11bce76.png)
If the lengths of the legs and hypotenuse are integral, they are said to form a Pythagorean triple.
Some Pythagorean triples include (3, 4, 5), (5, 12, 13), and (7, 24, 25).
Properties
- The area of the triangle is equal to half of the product of the lengths of the legs. It can also be calculated using half of the product of the median to the hypotenuse and the hypotenuse. Using similarity, it is possible to derive several formulas relating the sides, the hypotenuse, and the median.
- The circumradius of a right triangle is equal to half of the hypotenuse, or the median to the hypotenuse.
Problems
See also
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