Right Angled Triangle

A right angled triangle is a triangle in which one of the angles is 90°. A 90-degree angle is called a right angle, and hence the triangle with a right angle is called a right triangle. Further, based on the other angle values, the right triangles are classified as isosceles right triangles and scalene right triangles. Let us learn more about the properties of a right angled triangle, the parts of a right angled triangle along with some right triangle examples in this article.

A right triangle is a triangle in which one angle is 90°. In this triangle, the relationship between the various sides can be easily understood with the help of the Pythagoras theorem. The side opposite to the right angle is the longest side and is referred to as the hypotenuse. Observe the right-angled triangle ABC given below which shows the base, the altitude, and the hypotenuse. Here AB is the base, AC is the altitude, and BC is the hypotenuse.

Right Angled Triangle Definition

The definition for a right triangle states that if one of the angles of a triangle is a right angle - 90º, the triangle is called a right-angled triangle or a right triangle.

Now, let us understand the distinct features of a right triangle referring to the triangle ABC given above.

• AC is the height, altitude, or perpendicular
• AB is the base
• AC ⊥ AB
• ∠A = 90º
• The side BC opposite to the right angle is called the hypotenuse and it is the longest side of the right triangle.

Some of the examples of right triangles in our daily life are the triangular slice of bread, a square piece of paper folder across the diagonal, or the 30-60-90 triangular scale in a geometry box.

According to the Pythagoras theorem, in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two legs. Using this rule, the right triangle formula can be represented in the following way: The square of the hypotenuse is equal to the sum of the square of the base and the square of the altitude.

(Hypotenuse)² = (Base)² + (Altitude)²

Pythagorean Triplets: The three numbers which satisfy the above equation are known as the Pythagorean triplets. For example, (3, 4, 5) is a Pythagorean triplet because we know that 3² = 9, 4² = 16, and 5² = 25 and, 9 + 16 = 25. Therefore, 3² + 4² = 5². Any three numbers that satisfy this condition are called Pythagorean triplets like 3, 4, and 5. Some of the other examples of Pythagorean triples are (6, 8, 10), and (12, 5, 13).

The perimeter of a right triangle is the sum of the measures of all the 3 sides. It is the sum of the base, altitude, and hypotenuse of the right triangle. Observe the right triangle shown below in which the perimeter is equal to the sum of the sides BC + AC + AB = (a + b + c). The perimeter is a linear value and is represented with linear units like, cm, inches, yards, and so on.

The area of a right triangle is the space occupied by the triangle. It is equal to half of the product of the base and the height of the triangle. It is a two-dimensional quantity and therefore represented in square units. The two sides that are required to find the right-angled triangle area are the base and the altitude.

The area of a right triangle is calculated using the formula: Area of a right triangle = (1/2 × base × height)

The first property of a right triangle is that it has one of its angles as 90º. The 90º angle is a right angle and the largest angle of a right triangle. Also, the other two angles are lesser than 90º and are called acute angles. The right triangle properties are listed below:

• The largest angle of a right angle triangle is always 90º.
• The largest side of a right triangle is called the hypotenuse which is always the side opposite to the right angle.
• The measurements of the sides follow the Pythagoras rule.
• It cannot have any obtuse angle.

We have learned that one of the angles in a right triangle is 90º. This implies that the other two angles in the triangle will be acute angles. There are a few special right triangles such as the isosceles right triangles and the scalene right triangles. A triangle in which one angle is 90º and the other two angles are equal is referred to as an isosceles right triangle, and the triangle in which the other two angles have different values is called a scalene right triangle.

Isosceles Right Triangle

An isosceles right triangle is called a 90º-45º- 45º triangle. Observe the triangle ABC given below in which angle A = 90º, and we can see that AB = AC. Since two sides are equal, the triangle is also an isosceles triangle. We know that the sum of the angles of a triangle is 180º. Hence, the base angles add up to 90º which implies that they are 45º each. So in an isosceles right triangle, the angles are always 90º-45º- 45º.

Scalene Right Triangle

A scalene right triangle is a triangle where one angle is 90° and the other two angles are of different measurements. In the triangle PQR given below, ∠Q = 90º, hence, it is a right triangle. PQ is not equal to QR, hence, it is a scalene triangle. There is also a special case of a scalene triangle 30º-60º-90º which is also a right triangle where the ratio of the triangle's longest side to its shortest side is 2:1. The side opposite to the 30º angle is the shortest side.

Tips & Tricks

Listed here are some of the important tips and tricks related to a right triangle.

• The side lengths of a right-angled triangle always satisfy the Pythagoras theorem.
• In a right triangle, the hypotenuse is the side opposite to the right angle and is the longest side of the triangle.
• The other two legs are perpendicular to each other; one is the base and the other is the height.

Important Notes

1. In a right triangle, (Hypotenuse)² = (Base)² + (Altitude)²
2. The area of a right triangle is calculated using the formula, Area of a right triangle = 1/2 × base × height
3. The perimeter of a right triangle is the sum of the measures of all three sides.
4. Isosceles right triangles have 90º, 45º, 45º as their angles.

☛ Related Topics

• Hypotenuse
• Pythagorean Triples
• Hypotenuse Formula

What is a Right Angled Triangle in Geometry?

A triangle in which one of the measures of the angles is 90 degrees is called a right-angled triangle or right triangle.

What are the Different Types of Right Triangles?

The triangles are classified based on the measurement of the sides and the angles. The three types of right triangles are mentioned below.

• An isosceles right triangle is a triangle in which the angles are 90º, 45º, and 45º.
• A scalene right triangle is a triangle in which one angle is 90º and the other two acute angles are of different measurements.
• 30º - 60º - 90º triangle is another interesting right triangle where the ratio of the triangle's longest side to its shortest side is 2:1.

What is the Measure of the Angles in a Right Triangle?

A right triangle has one of its angles as 90º. The other two angles are acute angles. And all three angles of the right triangle add up to 180° like any other triangle.

What is the Formula for a Right-Angled Triangle?

The formula which is used for a right-angled triangle is the Pythagoras theorem. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This means, (Hypotenuse)² = (Base)² + (Altitude)².

How to Find the Area of a Right-Angled Triangle?

The area of a right-angled triangle is the space occupied by the triangle and it is equal to half of the product of the base and altitude of the triangle. It is two-dimensional and represented in square units. The formula which is used to find the area of a right-angled triangle is: Area of a right triangle = 1/2 × Base × Altitude

Can a Right Triangle have Two Equal Sides?

Yes, a right triangle can have two equal sides. The longest side is called the hypotenuse and the other two sides may or may not be equal to each other. A right triangle that has two equal sides is called an isosceles right triangle.

How to Find the Missing Side of a Right Triangle?

The missing side of a right triangle can be found if the measure of the other two sides is given. The Pythagoras theorem is helpful to find the value of the missing side. As per the Pythagoras theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides of a right triangle. For example, if a, b, and c are the three sides of the right-angled triangle, where 'a' is the hypotenuse, then as per the theorem, a² = b² + c².

How to Find the Angle of a Right Triangle?

The calculation of angles of a right triangle is very simple. One of the angles of a right triangle is a right angle or 90^º. Now, if the other angle of the triangle is known, then the missing angle can be easily calculated by using the angle sum property which states that the sum of the angles of a triangle is always equal to 180º.

What does a Right Triangle Look Like?

A right triangle looks like a triangle in which two sides form the alphabet 'L' and the ends of the letter 'L' are joined by a line which is the hypotenuse and the longest side of the triangle.

What are the Parts of a Right Triangle?

The parts of a right triangle can be discussed as follows:

• A right triangle has one angle equal to 90° and the other two angles are acute angles.
• The side opposite to the right angle is the largest side and is referred to as the hypotenuse.
• The other two sides of a right triangle are called the legs and are specifically referred to as the 'base and 'height' of the triangle. The height is also known as the altitude of this triangle.