The right triangle similarity theorem is otherwise called as Pythagoras theorem. This right angle theorem was introduced by Philosopher and Greek Mathematician, Pythagoras. The right triangle is a part of geometrical figures. This theorem is used for finding the length of any sides of a right triangle. The right triangles are special triangles that contain only one right angle. Here the right triangle is a triangle that measuring an angle `90^o`
Statement for Right Triangle Theorem:
In terms of area, we can define this theorem. In any right triangle, the area of the square whose side is hypotenuse is equal to the sum of the areas of the other two sides. Here the hypotenuse refers to the side that opposite to the right triangle. The other two sides of a triangle meet at right angle.
Here Hypotenuse is the side opposite to the right angle. an adjacent side is the side adjacent to the given angle and the opposite side is the side that opposite to given .
According to this theorem, the equation can be given as
(Opposite)2+ (Adjacent)2 = (Hypotenuse)2
Here, an angle value lies between 0 and 90 degree. Here, 90 degree is one of the interior angle and the other two interior angles are complementary. Complementary means the angle value less than 90 degree. The other two interior angle sum should be equal to 90 degree. This right triangle similarity theorem can also be used in trigonometric functions. This right triangle forms six possible ratios in trigonometry.
Other Forms of the Equation:
Here consider a is opposite side, b is an adjacent side and c is hypotenuse. Therefore, an equation can be written as a2 + b2 = c2. To find c, an equation can be written as c= `sqrt(a^(2)+b^(2))` . If c is known value, the length of one side is given; the following equations can be used;
a =`sqrt(c^(2)+b^(2))` (or) b = `sqrt(c^(2)-c^(2))`
Proof:
This theorem proof is based on the proportionality of two similar triangles. It depends on the ratio of any two corresponding sides of similar triangles.
Let ABC represent a right triangle, with the right angle located at B. Here, H is the altitude drawn from B and it also intersects AC. The point H divides the hypotenuse c into two parts d and e. The new triangle ABH is similar to the triangle ABC because both the triangles have a right angle. Similarly, the triangle BCH is also similar to the triangle ABC. Thus the proof of similarity of the right triangles requires the Triangle postulate i.e., the sum of the angles in a triangle is 2 right angles, and is equivalent to the parallel postulate. Similarity of the right angle triangles leads to the equality of ratios of their corresponding sides:
a/c =e/a and b/c= d/b
In this the first result is equal to cosine angle and the second result is equal to sine angle.
These ratios can be written as:
a2= c*e and b2=c*d
Summing these two equations we get:
a2 + b2 = (c*e) + (c*d) = c* (e + d) = c*c = c2
Therefore a2+ b2 = c2
Hence, right triangle similarity theorem is proved.
Statement for Right Triangle Theorem:
In terms of area, we can define this theorem. In any right triangle, the area of the square whose side is hypotenuse is equal to the sum of the areas of the other two sides. Here the hypotenuse refers to the side that opposite to the right triangle. The other two sides of a triangle meet at right angle.
Here Hypotenuse is the side opposite to the right angle. an adjacent side is the side adjacent to the given angle and the opposite side is the side that opposite to given .
According to this theorem, the equation can be given as
(Opposite)2+ (Adjacent)2 = (Hypotenuse)2
Here, an angle value lies between 0 and 90 degree. Here, 90 degree is one of the interior angle and the other two interior angles are complementary. Complementary means the angle value less than 90 degree. The other two interior angle sum should be equal to 90 degree. This right triangle similarity theorem can also be used in trigonometric functions. This right triangle forms six possible ratios in trigonometry.
Other Forms of the Equation:
Here consider a is opposite side, b is an adjacent side and c is hypotenuse. Therefore, an equation can be written as a2 + b2 = c2. To find c, an equation can be written as c= `sqrt(a^(2)+b^(2))` . If c is known value, the length of one side is given; the following equations can be used;
a =`sqrt(c^(2)+b^(2))` (or) b = `sqrt(c^(2)-c^(2))`
Proof:
This theorem proof is based on the proportionality of two similar triangles. It depends on the ratio of any two corresponding sides of similar triangles.
Let ABC represent a right triangle, with the right angle located at B. Here, H is the altitude drawn from B and it also intersects AC. The point H divides the hypotenuse c into two parts d and e. The new triangle ABH is similar to the triangle ABC because both the triangles have a right angle. Similarly, the triangle BCH is also similar to the triangle ABC. Thus the proof of similarity of the right triangles requires the Triangle postulate i.e., the sum of the angles in a triangle is 2 right angles, and is equivalent to the parallel postulate. Similarity of the right angle triangles leads to the equality of ratios of their corresponding sides:
a/c =e/a and b/c= d/b
In this the first result is equal to cosine angle and the second result is equal to sine angle.
These ratios can be written as:
a2= c*e and b2=c*d
Summing these two equations we get:
a2 + b2 = (c*e) + (c*d) = c* (e + d) = c*c = c2
Therefore a2+ b2 = c2
Hence, right triangle similarity theorem is proved.
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