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Right Angle

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Introduction to Right Angle

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In geometry, an angle is formed when two rays are joined together at a common point. Here the common point is called the node or vertex and two rays are called arms of the angle. An angle is a form of geometrical shape, which is constructed by joining two rays to each other at their end-points. Majorly there are six types of angles in geometry. In this article, we will learn one of these angles i.e right angle definition, triangle and its properties .


Right Angle Definition

If the measure of the angle between two rays is exactly equal to 90 degrees, then the angle is called a right angle. Angles whose measure is less than 90° such as 87°, 56°, 77°, 42°, etc. are known as acute angles and the angles whose measure is more than 90° such as 91°, 98°, 102°, 150°, 167°, etc., are known as obtuse angles.

The measure of right angle is written in terms of degrees i.e 90° and in terms of radians as π/2 (= 1.5708) radians.

The below figure shows the shape of the angle formed by two rays.



Right Angle Triangle

A right-angled triangle is a triangle having angle between the base and the perpendicular is 90°. It has  three sides named as, “base” “hypotenuse” and “perpendicular”. A right-angled triangle is one of the basic shapes in geometry and it is the foundation of trigonometry.



In the right triangle, the hypotenuse is the longest side and is opposite to  the right angle of the triangle.


Right Angle Triangle Formula

The formula used to determine whether the given triangle is the right triangle or not, is known as  Pythagoras theorem. The theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.


(Hypotenuse)2 = (Base)2 + (Perpendicular)2


Right Angle Triangle Properties

Following are the some of the important properties of a right angle triangle 

  • In a right angle triangle measure of  one angle is exactly equal to 90 degrees.

  • The angles other than the right angle must be acute angles, i.e. less than 90 degrees.

  • The side opposite to vertex of 90 degrees is called the hypotenuse of the right triangle and it the longest side of the triangle

  • The other two sides which are adjacent to the right angle are called base and perpendicular.

  • The circumcircle of the right angle triangle passes through all three vertices, and the radius of this circle is equal to half of the length of the hypotenuse.

  • If one of the angles is 90° and the measure of other two angles is 45o each, then the triangle is called an Isosceles Right Angled Triangle, where the adjacent sides to 90° are equal in length.

Area of a Right Triangle

The area of a right triangle is the area enclosed by three sides of the triangle in a plane. The formula to find the area of a right triangle is given below:

Area (A) = ½ x Base x Height

Here, Height = Perpendicular


Right Angle Isosceles Triangle

When two sides other than hypotenuse, i.e. base and perpendicular are congruent in a right triangle, then it is called a right angle isosceles triangle or simply isosceles right triangle. In a triangle, the angles made by the base and perpendicular with the hypotenuse are congruent, i.e. measure of both the angles is 45 degrees.


Right Angle Example

We can observe right angles shapes in many daily life objects such as edges of a book meeting at right angles at the vertices, sides of a rectangular table and boards in classrooms forms right angles at the corners. The below figure shows the rectangular board that has right angles at its corners.



As we know all the interior angles of a square are right angles, i.e. equal to 90 degrees as shown in the below figure.



Also, the angle formed by the x-axis and y-axis in the coordinate plane at the centre (intersection of axes) is the right angle.


 Facts

  • All right angles correspond to a quarter of a complete turn.

  •  All triangles which have one angle right are called right-angled triangles.

  • Formula of right angle triangle is also known as Pythagoras Theorem


Solved Example:

Question: Find is the Value of the Hypotenuse of the Right-Angled Triangle if the Value of the Adjacent and Opposite Sides are 20 cm and 15 cm Respectively.

Solution: Given

Adjacent side = 20 cm

Opposite side = 15 cm

According to the right angled triangle formula 

(Hypotenuse)2 = (Adjacent side)2 + (Opposite side)2

                             = 202+152

                             = 400 + 225

                              = 625 cm

\[\Rightarrow Hypotenuse=\sqrt{625}\]cm = 25 cm

Hence the hypotenuse is 25 cm.

FAQ (Frequently Asked Questions)

1. What is an Example of a Right Angle?

Ans: There are many real-life examples that contain right angles such as corners of notebooks, tables, boards in classrooms, doors and windows of a house, which have their corners in the shape of a right angle, and so on.

2. Can a Triangle Have Two Right Angles? Explain Your Answer.

Ans: No, a triangle can never have two right angles. A triangle has exactly three sides and the sum of interior angles equal to 180o. So, if a triangle has two right angles then the measure of the third angle will have to be 0 degrees which means the third side will overlap with the other side. Therefore, it is not possible to have a triangle with two right angles.

3. What is the Formula of the Right Angle Triangle?

Ans: The three sides of the right angle triangle are called: Perpendicular, Base(Adjacent) and Hypotenuse(Opposite). The formula of right angle triangle is

(Hypotenuse)2 = (Adjacent side)2 + (Opposite side)2