Right triangles

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A right triangle or right-angled triangle 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.

The side opposite the right angle is called the hypotenuse (side c in the figure). The sides adjacent to the right angle are called legs.

If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.

 

A right angled triangle with sides a, b and hypotenuse c

 

Pythagoras Theorem

In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

 

 

Pythagoras Triplets

Following is the list of Pythagorean triples, hypotenuse <100.

Side 1 Side 2 Hypotenuse
3 4 5
5 12 13
8 15 17
7 24 25
20 21 29
12 35 37
9 40 41
28 45 53
11 60 61
16 63 65
33 56 65
48 55 73
13 84 85
36 77 85
39 80 89
65 72 97

 

Properties of right triangles

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.

Area=(½)xaxb

where a and b are the legs of the triangle.

 

Altitudes

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

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

 

Altitude CD or f

 

In equations,

f2=de, (this is sometimes known as the right triangle altitude theorem)

b2=ce,

a2=cd

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

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

The altitude from either leg coincides with the other leg. Since these intersect at the right-angled vertex, the right triangle's orthocenter;the intersection of its three altitudes coincides with the right-angled vertex.

 

Inradius

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

 

Circumradius

If A is any point on the circle with diameter BC (except B or C themselves) ABC is a right triangle where A is 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.This is also called Thales' theorem.

A right angled triangle and its circumcircle

 

The radius of the circumcircle is half the length of the hypotenuse,

 

Euler line

In a right triangle, the Euler line contains the median on the hypotenuse—that is, it goes through both the right-angled vertex and the midpoint of the side opposite that vertex. This is because the right triangle's orthocenter, the intersection of its altitudes, falls on the right-angled vertex while its circumcenter, the intersection of its perpendicular bisectors of sides, falls on the midpoint of the hypotenuse.

 


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