Right triangle

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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).

[edit] Terminology

The side opposite the right angle is called the hypotenuse (side [BC] in the figure above). The sides adjacent to the right angle are called legs or catheti (singular: cathetus). Side [AB] may be identified as the side adjacent to angle α and opposed to (or opposite) angle β, while side [AC] is the side adjacent to angle β and opposed to angle α. In relation to angle α, side [AB] may be called adjacent cathetus and [AC] the opposite cathetus. Where the legs differ in length, the longer leg can be called the major cathetus and the shorter called the minor cathetus.

[edit] Principal properties

[edit] Area

As with any triangle, to calculate the area, multiply the base and the corresponding height, and divide it by two. If ABC is a right triangle in A, each of the sides [AB] and [AC] can be considered as the height; the base is then the other side of the right angle ([AC] and [AB], respectively). The area S of the triangle is equal to $S = (AB\times AC)/2$ .

For example, we have a right triangle in A with [AB] = 4 cm, [AC] = 3 cm, and hypotenuse [BC] = 5 cm. Area S is calculated as $S = (4 \times 3)/2 = 6$ ; the area of the triangle is therefore 6 cm².

The area of the triangle could also be calculated by using the hypotenuse as the base. One would then have to calculate the height associated with the hypotenuse, as it would no longer be one of the sides.

[edit] Pythagorean theorem

Main article: 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 is stated in equation form as C² = A² + B².

This theorem is a consequence of the definition of the distance between two points from the inner square of the vector. Indeed

$BC^2 = \overrightarrow{BC}\cdot \overrightarrow{BC}=\left(\overrightarrow{BA}+ \overrightarrow{AC}\right)\cdot\left(\overrightarrow{BA}+ \overrightarrow{AC}\right)$

$BC^2=\overrightarrow{BA}\cdot\overrightarrow{BA}+2 \overrightarrow{BA}\cdot \overrightarrow{AC}+ \overrightarrow{AC}\overrightarrow{AC}=AB^2+AC^2$

since $\overrightarrow{BA}\cdot\overrightarrow{AC}=0$ as vectors $\overrightarrow{BA}$ and $\overrightarrow{AC}$ are orthogonal.

[edit] Trigonometric ratios in right triangles

Main article: Trigonometric functions

A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.

In right triangles, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. The sides of the triangle are known as follows:

The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
The opposite side is the side opposite to the angle we are interested in, in this case a.
The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.

[edit] Sine, cosine and tangent

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

$\sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}\,.$

Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar.

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case

$\cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}\,.$

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case

$\tan A = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {a} {b}\,.$

The acronym "SOHCAHTOA" is a useful mnemonic for these ratios.

[edit] Inverse functions

The inverse trigonometric functions can be used to calculate the internal angles for a right angled triangle with the length of any two sides.

Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse.

$\theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right)$

Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypontenuse.

$\theta = \arccos \left( \frac{\text{adjacent}}{\text{hypotenuse}} \right)$

Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side.

$\theta = \arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right)$

In introductory geometry and trigonometry courses, the notation sin⁻¹, cos⁻¹, etc., are often used in place of arcsin, arccos, etc. However, the arcsin, arccos, etc., notation is standard in higher mathematics where trigonometric functions are commonly raised to powers, as this avoids confusion between multiplicative inverse and compositional inverse.

[edit] Median theorem

Median of a right angle of a triangle

For a right triangle, the median theorem reads:

If M is the midpoint of the hypotenuse, then AM = ½ BC. One can also say that point A is located on the circle with diameter [BC]. Conversely, if A is any point of the circle with diameter [BC] (except B or C themselves) then angle A in triangle ABC is a right angle.

There are several demonstrations of this theorem. The direct effect may be purely geometric: by definition M is the midpoint of [BC]. The triangle ABC is half of a rectangle ABCD. A rectangle is a parallelogram, so its diagonals bisect in the center; therefore, M, which is the midpoint of [BC], is also the midpoint of [AD]. The diagonals of a rectangle are of equal length, so AD = BC = AD and AM = BC / 2.

One can also use the vector:

$\overrightarrow{AM} = \overrightarrow{AB} + \frac{1}{2}\overrightarrow{BC}$ and $\overrightarrow{BC} = \overrightarrow{BA} + \overrightarrow{AC}$ where $\overrightarrow{AM} = \frac{1}{2}(\overrightarrow{AB} + \overrightarrow{AC}).$

These last two vectors are orthogonal; thus, AM² = (AB² + AC²)/4.

On the other hand, applying the Pythagorean theorem to the triangle ABC yields the equation BC² = AB² + AC², and also AM = BC / 2.

One can also apply the theorem of the angle at the center, which also helps to demonstrate the converse. Consider the circle circumscribed about the triangle ABC and note its center, O. According to the theorem, the angle BOC is twice the angle BAC. Thus:

$\widehat{BOC} = 2\widehat{BAC} = \pi.$

The points B, O, and C are aligned. As BO = OC, O is the midpoint of [BC], so O = M.

Conversely, if we know that A is a point on a circle with diameter [BC], then according to the theorem of the angle at the center, angle BAC is half of angle BOC, so it is π / 2. Therefore, BAC is a right triangle in A.

[edit] Center of gravity

With the previous notations (ABC with a right angle on A and M as the midpoint of [BC]), the center of gravity G verifies

$\overrightarrow{AG} = \frac{2}{3}\overrightarrow{AM}.$

M is also the projected midpoint of [AB] and [AC] (ABM and ACM are isosceles triangles). Point G is therefore predicted to be one-third of [AB] and [AC]:

$\overrightarrow{AG} = \frac{1}{3}(\overrightarrow{AB} + \overrightarrow{AC}).$

Right triangle

From Wikipedia, the free encyclopedia

Contents

[edit] Terminology

[edit] Principal properties

[edit] Area

[edit] Pythagorean theorem

[edit] Trigonometric ratios in right triangles

[edit] Sine, cosine and tangent

[edit] Inverse functions

[edit] Median theorem

[edit] Center of gravity

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