Right Angled Triangle
A right 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 a right triangle, the relationship between the various sides can be easily understood with the help of the Pythagoras rule. The side opposite to the right triangle is the largest side and is referred to as the hypotenuse.
Further, based on the other angle values in a right triangle, the triangles are classified as an isosceles right triangle and scalene right triangle. Also, the lengths of the sides of the right triangle, such as 3, 4, 5 are referred to as Pythagorean triples. Here, we shall explore and learn more about the features and properties of a right triangle in this lesson.
| 1. | What is Right Triangle? |
| 2. | Right Triangle Formula |
| 3. | Perimeter of a Right Triangle |
| 4. | Right Triangle Area |
| 5. | Properties of Right Triangle |
| 6. | Types of Right Triangles |
| 7. | Tips and Tricks |
What is Right Triangle?
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. 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 a right 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 right triangular-shaped triangular scale in a geometry box.
Right Triangle Formula
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 hypotenuse and the square of the altitude.
In a right triangle we have: (Hypotenuse)2 = (Base)2 + (Altitude)2
Right Triangle Area Formula: The area of a right triangle is the space occupied by the triangle. The area is equal to the product of the half of the base and the altitude of the right triangle. The area of a right triangle = ½ × Base × Altitude.
Right Triangle Perimeter Formula: The perimeter of a right triangle is the sum of the lengths of the three sides of the right triangle. It gives the total length of the boundary of the right triangle and is a linear quantity. For a right triangle having lengths of measure a units, b units, and c units, the perimeter of the triangle is equal to (a + b + c) units.
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. Further we know that 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 triplets are 6, 8, 10, and 12, 5, 13.
Perimeter of a Right Triangle
The perimeter of a right triangle is the sum of the measures of all the 3 sides. The perimeter of a right triangle 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 AB + BC + AC = a + b + c. The perimeter is a linear value and has a unit of length.
Right Triangle Area
The area of a right triangle gives the spread or the space occupied by the right triangle. The area of a right triangle is equal to half of the product of the base and the height of the right triangle. The area of a right triangle is a two-dimensional quantity and has units of square units. The only two sides needed to find find the area of a right triangle is the base and the altitude. The hypotenuse is not used to find the area of a right triangle.
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.
Properties of Right Triangle
The first property of a right triangle is that it has one of its angles as 90º. The angle of 90º is a right angle and is the largest angle of a right triangle. Also the other two angles of the right triangle are lesser than 90º or are acute angles. The properties of a right triangle can be listed as follows.
- The largest angle in a right triangle is always 90º.
- The largest side of a right triangle is called the hypotenuse.
- The measures of the sides of the right triangle follow Pythagoras rule.
- The sides holding the right angle are the base and the height of the right triangle.
Types of Right Triangles
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 scalene right triangle. The triangle having both the other angles equal is referred to as an isosceles right triangle, and the triangle with the other two angles with different values is called a scalene right triangle. Let us learn more about each of the right triangles.
Isosceles Right Triangle
An isosceles right triangle is called a 90º-45º- 45º triangle. In triangle ABC,A = 90º; by right triangle definition, triangle ABC is a right triangle. 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 add up to 90º. 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 example of a scalene triangle 30º-60º-90º 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 measures of a right triangle 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
- In a right triangle, (Hypotenuse)2 = (Base)2 + (Altitude)2
- Area of a right triangle is ½ × base × height.
- The perimeter of a right triangle is the sum of the measures of all the three sides.
- Isosceles right triangles have 90º, 45º, 45º as their degree measures.
FAQs on Right Triangle
What is a Right 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 measure of the sides and the measure of the angles of the triangle. The three types of right triangles are as mentioned below.
- An isosceles right triangle is a right triangle where the angles of the triangle are 90º, 45º, 45º
- A scalene right triangle is a right triangle where one angle is 90º and the other two angles add up to 180º
- 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 of a right triangle are acute angles. And all three angles of the right triangle add up to 180°.
What is the Formula for a Right-Angled Triangle?
The formula used for a right-angled triangle is the Pythagoras formula. The Pythagoras formula for the right triangle is the square of the hypotenuse equals the sum of the squares of the other two sides. The pythagoras formula is (Hypotenuse)2 = (Base)2 + (Altitude)2. This Pythagoras 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 right triangle. The area of a right triangle is a two-dimensional value and has square units as its units. Area of a right triangle = ½ × Base × Altitude
Can a Right Triangle have Two Equal Sides?
Yes, a right triangle can have two equal sides. The longest side in a right triangle is called the hypotenuse and the other two sides may or may not be equal to each other. A right triangle 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.
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 90º. Now if one other angle of the triangle is known, then the other angle of the triangle can be easily calculated. The sum of the angles of a triangle is always equal to 180º. This idea is helpful to find the angles of a right triangle.