(Redirected from Special right triangles)

Position of some special triangles in an Euler diagram of types of triangles, using the definition that isosceles triangles have at least two equal sides, i.e. equilateral triangles are isosceles.

Therefore, this is an isosceles right triangle with the ratio of sides x: x: x Because one leg is 10, the other must also be 10, and the hypotenuse is 10, soy = 10 and z = 10. 30°− 60°− 90° right triangle. A 30°− 60°− 90° right triangle has a unique ratio of its sides.
Special Right Triangles Date Period Find the missing side lengths. Leave your answers as radicals in simplest form. 1) a 2 2 b 45° 2) 4 x y 45° 3) x y 3 2 2 45° 4) x y 3 2 45° 5) 6 x y 45° 6) 2 6 y x 45° 7) 16 x y 60° 8) u v 2 30°-1.

A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an 'angle-based' right triangle. A 'side-based' right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.

Angle-based[edit]

Right Triangles Test Review Multiple Choice Identify the choice that best completes the statement or answers the question. Find the length of the missing side. The triangle is not drawn to scale. Triangle ABC has side lengths 9, 40, and 41. Do the side lengths form a.

Special angle-based triangles inscribed in a unit circle are handy for visualizing and remembering trigonometric functions of multiples of 30 and 45 degrees.

'Angle-based' special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or π/2radians, is equal to the sum of the other two angles.

The side lengths are generally deduced from the basis of the unit circle or other geometric methods. This approach may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, and 60°.

Special Right Triangles 2

Special triangles are used to aid in calculating common trigonometric functions, as below:

degrees	radians	gons	turns	sin	cos	tan	cotan
0°	0	0^g	0	√0/2 = 0	√4/2 = 1	0	undefined
30°	π/6	33+1/3^g	1/12	√1/2 = 1/2	√3/2	1/√3	√3
45°	π/4	50^g	1/8	√2/2 = 1/√2	√2/2 = 1/√2	1	1
60°	π/3	66+2/3^g	1/6	√3/2	√1/2 = 1/2	√3	1/√3
90°	π/2	100^g	1/4	√4/2 = 1	√0/2 = 0	undefined	0

45°–45°–90°

30°–60°–90°

The 45°–45°–90° triangle, the 30°–60°–90° triangle, and the equilateral/equiangular (60°–60°–60°) triangle are the three Möbius triangles in the plane, meaning that they tessellate the plane via reflections in their sides; see Triangle group.

45°–45°–90° triangle[edit]

The side lengths of a 45°–45°–90° triangle

In plane geometry, constructing the diagonal of a square results in a triangle whose three angles are in the ratio 1 : 1 : 2, adding up to 180° or π radians. Hence, the angles respectively measure 45° (π/4), 45° (π/4), and 90° (π/2). The sides in this triangle are in the ratio 1 : 1 : √2, which follows immediately from the Pythagorean theorem.

Of all right triangles, the 45°–45°–90° degree triangle has the smallest ratio of the hypotenuse to the sum of the legs, namely √2/2.^[1]^:p.282,p.358 and the greatest ratio of the altitude from the hypotenuse to the sum of the legs, namely √2/4.^[1]^:p.282

Triangles with these angles are the only possible right triangles that are also isosceles triangles in Euclidean geometry. However, in spherical geometry and hyperbolic geometry, there are infinitely many different shapes of right isosceles triangles. Anno 1800 item editor.

30°–60°–90° triangle[edit]

Set square

The side lengths of a 30°–60°–90° triangle

This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° (π/6), 60° (π/3), and 90° (π/2). The sides are in the ratio 1 : √3 : 2.

The proof of this fact is clear using trigonometry. The geometric proof is:

Draw an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30°–60°–90° triangle with hypotenuse of length 2, and base BD of length 1.

The fact that the remaining leg AD has length √3 follows immediately from the Pythagorean theorem.

The 30°–60°–90° triangle is the only right triangle whose angles are in an arithmetic progression. The proof of this fact is simple and follows on from the fact that if α, α + δ, α + 2δ are the angles in the progression then the sum of the angles 3α + 3δ = 180°. After dividing by 3, the angle α + δ must be 60°. The right angle is 90°, leaving the remaining angle to be 30°.

Side-based[edit]

Right triangles whose sides are of integer lengths, with the sides collectively known as Pythagorean triples, possess angles that