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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.
A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle . A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). [ 1] For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not.
The (4, 5, 6) triangle and its multiples are the only triangles with one angle being twice another and having integer sides in arithmetic progression. [39] The (3, 5, 7) triangle and its multiples are the only triangles with a 120° angle and having integer sides in arithmetic progression. [39]
Certain practical results from Euclidean geometry (such as the right-angle property of the 3-4-5 triangle) were used long before they were proved formally. [21] The fundamental types of measurements in Euclidean geometry are distances and angles, both of which can be measured directly by a surveyor.
Interior angle Δθ = θ 1 −θ 2. The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines, which states that where is the angle between sides and . [ 45] When is radians or 90°, then , and the formula reduces to the usual Pythagorean theorem.
It is based on the Pythagorean triple (3, 4, 5) and the rule of 3-4-5. From the angle in question, running a straight line along one side exactly three units in length, and along the second side exactly four units in length, will create a hypotenuse (the longer line opposite the right angle that connects the two measured endpoints) of exactly ...
Triangles have many types based on the length of the sides and on the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular forming a right angle ( 1⁄4 turn or 90 degrees ). The side opposite to the right angle is called the hypotenuse (side in the figure). The sides adjacent to the right angle are called ...