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#### 45-45-90 Triangles and Their Properties

Referencing Styles : APA | Pages : 1

A triangle is the figure that has 3 sides those meet at the point that is called as vertex. Each vertex forms an angle and when they are added up it forms a triangle that is equal to 180°. 45-45-90 triangle is the special type of isosceles right triangle under which 2 legs are congruent to one another. As the 2 angles of the triangle are equal to 45°, the triangle is the isosceles triangle. Here, both non-right angles are equal to 45 degrees. Special relationship is there among lengths of sides for a 45-45-90 triangle that can be used for determining the lengths of other 2 sides. If the triangle is 45-45-90 triangle, ratio of the sides will be leg: leg: hypotenuse or 1:1: Generally Pythagorean Theorem is used for finding out the missing leg or for hypotenuse of 45-45-90 triangle. Ratio of sides to hypotenuse always is 1:1: square root of 2. Formula for Pythagorean Theorem is a2 + b2 = c2. Rule for using the formula is c shall stand for hypotenuse irrespective of which side is called as ‘a’ and which side is called as ‘b’. The same is explained through the example below –

Leg of above presented 45-45-90 triangle measures 5 units. Pythagorean Theorem states that for right triangle, a2 + b2 = c2, where ‘a’ and ‘b’ are lengths of legs and ‘c’ is length of hypotenuse. As the triangle is 45-45-90 triangle, the legs of this triangle is congruent and value of ‘a’ and ‘b’ both is 5. Using the formula of a2 + b2 = c2, ‘c’ can be solved as follows –

Hence, the length of hypotenuse is equal to length of a leg multiplied by  or  units.

Properties of a 45-45-90 triangle –

Each 45-45-90 triangle has some special properties. When it is talked about a 45-45-90 triangle, the numbers represent measures of angles of the triangle. Hence, it means that the triangle has two 45° angles and one 90° angle. For any isosceles right triangle when the legs are ‘x’ units in length, hypotenuse will always be x. Hypotenuse of the 45-45-90 triangle is longest side of the triangle and is across from right angle. The legs of the 45-45-90 triangle are 2 shorter sides of the triangle and are adjacent to right angle. Most important property of this triangle is that this triangle has 1 right angle and 2 other angles those are equal to 45°. It signifies that 2 sides – legs – equal in the length and hypotenuse can be computed easily. Another interesting property is that it is the only possible triangle that is also the isosceles triangle. Further, it has smallest ratio of hypotenuse to sum of legs and has greatest ratio of altitude from the hypotenuse to sum of legs.

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