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Difference between revisions of "Right triangle"
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A right triangle is any triangle with an angle of 90 degrees.
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A '''right triangle''' is any [[triangle]] with an [[interior angle|interior]] [[right angle]].
  
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[[Image:Righttriangle.png]]
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<asy>
 +
pair A,B,C;
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A = (0,3);
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B = (4,0);
 +
C = (0,0);
  
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In the example above, you see that angle C has a measure of 90 degrees, so it is a right triangle.  The sides of a right triangle have two names.  The hypotenuse is the side opposite the angle measuring 90 degrees, or in this picture, c.  The legs are the other two sides.
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draw(A--B--C--cycle);
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draw(rightanglemark(A,C,B));
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label("$A$",A,NW);
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label("$B$",B,E);
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label("$C$",C,SW);
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label("$a$",midpoint(C--B),S);
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label("$b$",midpoint(C--A),W);
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label("$c$",midpoint(A--B),NE);
 +
</asy>
  
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Right triangles are very useful in [[Geometry]] and in finding the areas of polygons.  Many theorems are based on it including:
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In the image above, <math>\angle C</math> has a measure of <math>90^{\circ}</math>, so <math>\triangle ABC</math> is a right triangle. The longest side, opposite the right angle, is called the [[hypotenuse]].  In this diagram, the hypotenuse is labeled <math>c</math>.  The other two sides are called the [[leg]]s of the triangle, labeled <math>a</math> and <math>b</math>.
  
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* [[Pythagorean Theorem]]
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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.
  
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All or almost all [[Trigonometric identities]] are based on right triangles.
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== Special Right Triangles ==
 +
 
 +
{{main|Special right triangles}}
 +
There are many right triangles with special properties. One of these is the [[isosceles triangle|isosceles]] [[45-45-90 triangle|<math>45^{\circ}-45^{\circ}-90^{\circ}</math> triangle]], where the hypotenuse is equal to <math>\sqrt{2}</math> 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);
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draw(rightanglemark(A, C, B, 3));
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draw(anglemark(A, B, C, 4));
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draw(anglemark(C, A, B, 4));
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label("$A$", A, NW);
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label("$45^{\circ}$", A, 6*dir(290));
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label("$B$", B, E);
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label("$45^{\circ}$", B, 5*dir(155));
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label("$C$", C, SW);
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label("$1$", midpoint(C--B), S);
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label("$1$", midpoint(C--A), W);
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label("$\sqrt{2}$", midpoint(A--B), NE);
 +
</asy>
 +
 
 +
Another one of these is the [[30-60-90 triangle|<math>30^{\circ}-60^{\circ}-90^{\circ}</math> triangle]], which has sides in the ratio of <math>x:x\sqrt3:2x</math>. 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));
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label("$B$", B, E);
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label("$60^{\circ}$", B, 3*dir(150));
 +
label("$C$", C, SW);
 +
label("$1$", midpoint(C--B), S);
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label("$\sqrt{3}$", midpoint(C--A), W);
 +
label("$2$", midpoint(A--B), NE);
 +
</asy>
 +
 
 +
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 ==
 +
 
 +
=== Introductory ===
 +
*A [[triangle]] with side lengths in the [[ratio]] <math>3 : 4 : 5</math> is inscribed in a [[circle]] with [[radius]] 3. What is the area of the triangle?
 +
 
 +
:<math>\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18</math>
 +
 
 +
([[2007 AMC 12A Problems/Problem 10|Source]])
 +
 
 +
== See also ==
 +
 
 +
* [[Acute triangle]]
 +
* [[Obtuse triangle]]
 +
* [[Special right triangles]]
 +
 
 +
[[Category:Definition]]
 +
[[Category:Geometry]]
 +
[[Category:Trigonometry]]
  
 
{{stub}}
 
{{stub}}

Latest revision as of 19:37, 31 January 2025

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]

In the image above, $\angle C$ has a measure of $90^{\circ}$, so $\triangle ABC$ is a right triangle. The longest side, opposite the right angle, is called the hypotenuse. In this diagram, the hypotenuse is labeled $c$. The other two sides are called the legs of the triangle, labeled $a$ and $b$.

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

Documentation

Use {{hatnote|text}} </noinclude>

There are many right triangles with special properties. One of these is the isosceles $45^{\circ}-45^{\circ}-90^{\circ}$ triangle, where the hypotenuse is equal to $\sqrt{2}$ 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]

Another one of these is the $30^{\circ}-60^{\circ}-90^{\circ}$ triangle, which has sides in the ratio of $x:x\sqrt3:2x$. 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]

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

Introductory

$\mathrm{(A)}\ 8.64\qquad \mathrm{(B)}\ 12\qquad \mathrm{(C)}\ 5\pi\qquad \mathrm{(D)}\ 17.28\qquad \mathrm{(E)}\ 18$

(Source)

See also

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