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# Solutions of Right Triangles

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 Sub Topics Solving right Triangles means we need to find the measures of other two angles (as one angle is of 900) or the measures of sides of the Right Triangle. Figure shown below is a right triangle, right angled at ‘C’ i.e. angle ACB. The side opposite to the Right Angle is called the hypotenuse, the side opposite to the angle ‘p’ is opposite side and the side ‘b’ is adjacent side. Figure:- Right triangle, right angled at ‘C’. To solve right triangles we can use Pythagorean Theorem and the six Trigonometric Functions as per the requirement. According to the Pythagorean Theorem, in any right triangle, the Square of the hypotenuse is equal to the sum of the squares of the adjacent and opposite sides. i.e. c2 = a2 + b2 where a, b and c are sides of a right triangle. Solutions of right triangles can be understood by going through the process given below. Here we have to find the measure of side ‘b’. So according to the Pythagorean Theorem, c2 = a2 + b2, 52 + b2 = 202, 25 + b2 = 400, Take 20 from both sides: b2 = 375, b = √375, b = 19.36, Solving right triangles Trigonometry: When we know one of the acute angles of a right triangle and the length of one of the sides then one can solve for the length of the other two sides. In Right Triangles Trigonometry Sin A = opposite side / hypotenuse, Cos A = adjacent side / hypotenuse, Tan A = opposite side / adjacent, Let us see how we can find the side of the triangle ABC, here we have A = 33° and c = 15. Solving right triangles Using Trigonometry Sin 330 = a/b = a/15, So a = sin330 * 15, a = 0.544 * 15, a = 8.16.

## Pythagorean Theorem

A right angled triangle is a triangle whose one angle is a Right Angle. The side opposite the right angle is called the hypotenuse and is the longest side of the triangle. The side with angle 900 is called the perpendicular and other side is the base.
Pythagorean Theorem states that Square of the hypotenuse are equal to the sum of squares of the base and perpendicular.

AC=hypotenuse AB=perpendicular CB=base
(Hypotenuse)2 = (Base)2 + (Perpendicular)2
If we know the measure of any two sides we can find the other side easily.
Hypotenuse = √ [(Base)2 + (Perpendicular)2]
Base = √ [(Hypotenuse)2 − (Perpendicular)2]
Perpendicular =√ [(Hypotenuse)2−(Base)2]

It is used to find the length of different sides of right angled triangle. It can be defined by geometric proof and algebraic proof. It is not only defined for Euclidean figures but also for n – dimensional shapes that are Solid in form.

As per Pythagorean Theorem Trigonometry:

Sin2 ø + cos2 ø = 1

By Trigonometry Pythagorean Theorem identity we get,
Sin ø = opposite side/hypotenuse,
Cos ø = adjacent/ hypotenuse,

The Pythagorean identity we get,
Opposite2+ adjecent2 = 1
Hypotenuse2

If we take an example in which value of base is equal to 12 units (B = 12) and the value of hypotenuse is equal to 13 units (C = 13) which is a right angled triangle then find the value of perpendicular (A = ?) .
Solution: According to the Pythagorean Theorem:

(13)2 = (12)2 + (perpendicular)2

=> (A)2 = (C)2 - (B)2
=> (A) 2 = (13)2 - (12)2
=> (A) = √ [(13)2 - (12)2]
=> (A) = √ [(169) - (144)]
=> (A) = √ 25
=> (A) = 5 unit.