Isosceles Right Triangle

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IsoscelesRightTriangle

A right triangle with the two legs (and their corresponding angles) equal. An isosceles right triangle therefore has angles of 45 degrees, 45 degrees, and 90 degrees. For an isosceles right triangle with side lengths a, the hypotenuse has length sqrt(2)a. The hypotenuse length for a=1 is called Pythagoras's constant.

Polyforms made up of isosceles right triangles are called polyaboloes.

The height of area of an isosceles right triangle with side length a is

[画像: h=sqrt(3/2)a ]

(1)

and the area is

A = 1/2a^2

(2)

= (h^2)/(sqrt(3))

(3)

= sqrt(3)(3+2sqrt(2))r^2

(4)

= sqrt(3)R^2,

(5)

where h is the height, r the inradius, and R the circumradius.

IsoscelesRightTriangleI

IsoscelesRightTriangleO

The inradius r and circumradius R are

r = 1/2(2-sqrt(2))a

(6)

R = 1/2sqrt(2)a.

(7)

Triangle line picking for points in an isosceles right triangle with edge lengths a, a, and sqrt(2)a gives a mean line segment length of

l^__(Delta(a,a,sqrt(2)a)) = 1/(30)[2+4sqrt(2)+(4+sqrt(2))sinh^(-1)1]a

(8)

= 1/(60)(4+8sqrt(2)+sqrt(2)cosh^(-1)3+8sinh^(-1)1)a

(9)

= 0.414293...a.

(10)

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Weisstein, Eric W. "Isosceles Right Triangle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IsoscelesRightTriangle.html

Isosceles Right Triangle

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