Right Angled Triangle

A right angled triangle is a triangle with one of the angles as 90 degrees. A 90-degree angle is called a right angle, and hence the triangle with a right angle is called a right triangle. In this triangle, the relationship between the various sides can be easily understood with the help of the Pythagoras rule. The side opposite to the right angle is the largest side and is referred to as the hypotenuse. Further, based on the other angle values, the right triangles are classified as an isosceles right triangle and a scalene right triangle. Also, the lengths of the sides of the right triangle, such as 3, 4, 5 are referred to as Pythagorean triples.

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 simply, a right triangle. In the given image, triangle ABC is a right triangle, where we have the base, the altitude, and the hypotenuse. Here AB is the base, AC is the altitude, and BC is the hypotenuse. The hypotenuse is the important side of a right triangle which is the largest side and is opposite to the right angle within the triangle.

Here we can have understood the distinct features of a right triangle. The features of triangle ABC are as follows:

• 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.

The great Greek philosopher, Pythagoras, derived an important formula for a right triangle. The formula states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two legs. It was named after him as Pythagoras theorem. 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.

In a right triangle we have: (Hypotenuse)² = (Base)² + (Altitude)²

Pythagorean Triplet: The three numbers which satisfy the above equation are 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 These three numbers satisfying this condition are called the Pythagorean triplet. Some of the other examples of Pythagorean triples are (6, 8, 10), and (12, 5, 13).

The right triangle perimeter 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. Here, for the below right triangle, the perimeter is equal to the sum of the sides BC + AC + AB = (a + b + c) units. The perimeter is a linear value and has a unit of length.

The area of a right triangle gives the spread or 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 only two sides needed to find the right-angled triangle area are the base and the altitude.

Applying the right triangle definition, the area of a right triangle is given by the formula: Area of a right triangle = (1/2 × base × height) square units.

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 or are acute angles. The right triangle properties are listed below:

• The largest angle is always 90º.
• The largest side 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 namely the isosceles right triangles and the scalene right triangles. The triangle having both the other two angles equal is referred to as an isosceles right triangle, and the triangle with the other two angles having different values is called a scalene right triangle.

Isosceles Right Triangle

An isosceles right triangle is called a 90º-45º- 45º triangle. In triangle ABC, angle A = 90º; so by right triangle definition, triangle ABC is a right triangle. Also AB = AC; since two sides are equal, the triangle is also an isosceles triangle. Since AB = AC, the base angles are equal. 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, angles will always be 90º-45º- 45º.

Scalene Right Triangle

A scalene right triangle is a triangle where one angle is 90° and the other two angles that up to 90º are of different measurements. In the triangle PQR, ∠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 relating to a right triangle.

• The measurements of the side lengths will always satisfy the Pythagoras theorem.
• In a right triangle, the hypotenuse is the side opposite to the right angle and it 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 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 degree measures.

Related Topics

Check these articles related to the concept of the right-angled triangle.

• Hypotenuse
• Pythagorean Triplet
• 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 as 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 used for a right-angled triangle is the Pythagoras formula. It states the square of the hypotenuse is equal to the sum of the squares of the other two sides. The Pythagoras formula is (Hypotenuse)² = (Base)²< + (Altitude)². This formula has given the Pythagoras triplets such as 3, 4, 5.

How do you 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.

Area of a right triangle = 1/2 × Base × Altitude square units

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 from the measure of the other two sides. The Pythagoras rule is helpful to find the value of the missing side. As per the Pythagoras rule, we have the square of the hypotenuse 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 (a being the hypotenuse), then we have the relationship as 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 one other angle of the triangle is known, then the missing angle can be easily calculated by using the angle sum formula which states that the sum of the angles of a triangle is always equal to 180º.