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Right triangle

A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90 degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry.

## Terminology

The side opposite the right angle is called the hypotenuse(side c in the figure above). The sides adjacent to the right angle are called legs (or catheti, singular:cathetus). Side a may be identified as the side adjacent to angle B and opposed to (or opposite) angle A, while side b is the side adjacent to angle A and opposed to angle B.

## Principal properties

### Area

As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula:

\text{Area}=\tfrac{1}{2}ab

where a and b are the legs of the triangle.

If the incircle is tangent to the hypotenuse AB at point P, then PA = s&ndash; a and PB = s&ndash; b, and the area is given by

\text{Area}=\text{PA} \cdot \text{PB} = (s-a)(s-b).

### Altitude

If an altitude is drawn from the vertex with the right angle to the hypotenuse then the triangle is divided into two smaller triangle which are both similar to the original and therefore similar to each other. From this:

• The altitude is the mean proportional of the two segments of the hypotenuse.
• Each leg of the triangle is the mean proportional of the hypotenuse and the adjacent segment.

In equations,

f^2=de,\,b^2=ce,\,a^2=cd

where a, b, c, d, e, f are as shown in the diagram. Thus fc = ab.

Moreover, the altitude to the hypotenuse is related to the legs of the right triangle by

\frac{1}{a^2} + \frac{1}{b^2} = \frac{1}{ f^2}.

### Pythagorean theorem

The Pythagorean theorem states that:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

This can be stated in equation form as c2 = a2 + b2 where c is the length of the hypotenuse and a and b are the lengths of the remaining two sides.

## Trigonometric ratios

The trigonometric functions for acute angles can be defined as ratios of the sides of a right triangle. For a given angle, a right triangle may be constructed with this angle, and the sides labeled opposite, adjacent and hypotenuse with reference to this angle according to the definitions above. These ratios of the sides do not depend on the particular right triangle chosen, but only on the given angle, since all triangles constructed this way are similar. If, for a given angle Î±, the opposite side, adjacent side and hypotenuse are labeled O, A and H respectively, then the trigonometric functions are

\sin\alpha =\frac {O}{H},\,\cos\alpha =\frac {A}{H},\,\tan\alpha =\frac {O}{A},\,\sec\alpha =\frac {H}{A},\,\cot\alpha =\frac {A}{O},\,\csc\alpha =\frac {H}{O}.

## Special right triangles

The values of the trigonometric functions can be evaluated exactly for certain angles using right triangle with special angles. These include the 30-60-90 triangle which can be used to evaluate the trigonometric functions for any multiple of Ï€/6, and the 45-45-90 triangle which can be used to evaluate the trigonometric functions for any multiple of Ï€/4.

## Thales' theorem

Thales' theorem states that if A is any point of the circle with diameter BC (except B or C themselves) â–³ABC is a right triangle with A the right angle. The converse states that if a right triangle is inscribed in a circle then the hypotenuse will be a diameter of the circle. A corollary is that the length of the hypotenuse is twice the distance from the right angle vertex to the midpoint of the hypotenuse. Also, the center of the circle that circumscribes a right triangle is the midpoint of the hypotenuse and its radius is one half the length of the hypotenuse.

## Medians

The following formulas hold for the medians of a right triangle:

m_a^2 + m_b^2 = 5m_c^2 = \frac{5}{4}c^2.

The median on the hypotenuse of a right triangle divides the triangle into two isosceles triangles, because the median equals one-half the hypotenuse.

## Relation to various means and the golden ratio

Let H, G, and A be the harmonic mean, the geometric mean, and the arithmetic mean of two positive numbers a and b with a> b. If a right triangle has legs H and G and hypotenuse A, then

\frac{A}{H} = \frac{A^{2}}{G^{2}} = \frac{G^{2}}{H^{2}} = \phi \,

and

\frac{a}{b} = \phi^{3}, \,

where \phi is the golden ratio \tfrac{1+ \sqrt{5}}{2}. \,

## Other properties

The radius of the incircle of a right triangle with legs a and b and hypotenuse c is

r = \frac{1}{2}(a+b-c) = \frac{ab}{a+b+c}.

If segments of lengths p and q emanating from vertex C trisect the hypotenuse into segments of length c/3, then

p^2 + q^2 = 5(c/3)^2.

Solution of triangles

In trigonometry, to solve a triangle is to find the three angles and the lengths of the three sides of the triangle when given some, but not all of that information. In particular:

• If one knows two of the angles one can find the third by using the fact that the sum of the three must be a half-circle, or&nbsp;180&deg;.
• (SSS) If one knows the lengths of the three sides, one can find the three angles by using the law of cosines.
• (SAS) If one knows the lengths of two of the sides and the angle between them, one can find the length of the third side by using the law of cosines.
• (SSA) If one knows the lengths of two sides and an angle between one of those and the third side, one can find the third length and the other angles by using the law of sines, in some cases up to a choice between two possible solutions.
• (SAA) If one knows the length of one side and at least two of the angles, one can find the lengths of the other sides by using the law of sines.
• (HL) If the corresponding hypotenuse and leg of two right triangles are congruent, then the triangles are congruent.

In some cases, the law of tangents can also be used.

Mollweide's formula can be used to check solutions.

The half-side formulae are used in solving spherical triangles.

Triangle

A triangle is one of the basic shape s of geometry: a polygon with three corners or vertices and three sides or edges which are line segment s. A triangle with vertices A, B, and C is denoted . In Euclidean geometry any three non- collinear points determine a unique triangle

Question:

Answers:isosales triangle has the same size sides and all the angle are 45 deg right angle triangle has two angle the same and the one at 90 deg. Gee been awhile since I did geometry

Question:a. triangle b. pentagon c. hexagon d. quadrilateral

Answers:tri- means "three" penta- means "five" hex- means "six" quad- means "four" Answer: b. pentagon

Question:I need to know how to tell what corner of a triangle is A, B, and C. I know each side of a triangle is labeled according to the opposite corner. For example, I have a math problem which goes, "Side a=9, side b=10, and side c=15. Find the angle of C." How do I know which corner of the triangle is C?

Answers:hint: each angle of a triangle is labled according to the side opposite.

Question:math ? and toatally do not get the sooner the help the better thanx