Right Triangle

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Table of contents
  1. Definition of a Right Triangle
  2. Properties of a Right Triangle
    1. Right Angle
    2. Pythagorean Theorem
    3. Converse of the Pythagorean Theorem
    4. Altitude and Similarity
  3. Types of a Right Triangle
    1. Isosceles Right Triangle
    2. Scalene Right Triangle
  4. Right Triangle Formulas
    1. Pythagorean Theorem
    2. Perimeter of a Right Triangle
    3. Area of a Right Triangle
  5. Solved Examples on Right Triangle
  6. FAQs on Right Triangle

The right triangle is one of the most fundamental shapes in geometry that holds a significant place in mathematics, physics, engineering, and everyday life applications. Defined by its unique property of having one angle equal to 90°, the right triangle offers a host of fascinating characteristics that make it a crucial concept for understanding various mathematical principles. Let's explore the properties, types, formulas and examples of the right triangle.

Definition of a Right Triangle: Right triangle is a triangle in which one of the interior angles is 90° (right angle). The longest side of the right triangle, which is also the side opposite the right angle, is called the hypotenuse and the other two sides are called the height and the base of the triangle.

ABC

In the figure above, ∠B is right angle in △ABC, hence △ABC is right triangle.

Properties of a Right Triangle

Below are some characteristics and properties of a right triangle:

  • Right Angle

    : One of the angles in a right triangle measures 90°. The right angle is always the largest angle in a right triangle.

  • Pythagorean Theorem

    : The Pythagorean Theorem states that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem can be expressed as below:
    c2=a2+b2
    where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

  • Converse of the Pythagorean Theorem

    : The converse of the Pythagorean Theorem states that if the square of the length of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

  • Altitude and Similarity

    : The altitude drawn from the right angle of a right triangle to its hypotenuse forms two smaller triangles that are similar to the original triangle. This property is useful in various geometric proofs and calculations.

Understanding these properties is crucial not only for solving problems related to right triangles but also for comprehending the broader principles of geometry, trigonometry, and various fields that heavily rely on these concepts.

Types of a Right Triangle

Right triangles can be categorized as:

  • Isosceles Right Triangle

    In an isosceles right triangle, two sides have the same length, making the two base angles also equal measuring 45° each. The third side, the hypotenuse, has a different length. The two identical sides are the legs, and the angle opposite the hypotenuse is the right angle. This type of triangle is also known as a 45°-45°-90° triangle.

  • Scalene Right Triangle

    In a scalene right triangle, all three sides have different lengths and one interior angle measures 90°, while the other two angles have different measures.

Right Triangle Formulas

Below are some of the essential formulas associated with right triangles:

Pythagorean Theorem

The Pythagorean Theorem, named after the ancient Greek mathematician Pythagoras, is the most famous theorem associated with right triangles. It states that in any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. Mathematically, it is represented as below:
c2=a2+b2
where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

Perimeter of a Right Triangle

The perimeter of a right triangle is the sum of the lengths of all three sides, which can be expressed as:
P=a+b+c
where a, b and c are the lengths of the sides of the right triangle.

Area of a Right Triangle

The area of a triangle is the region occupied by it in a two-dimensional plane. The area of a right triangle can be calculated using below formula:
A=12×base×height

Solved Examples on Right Triangle

Example 1: Find the perimeter of a right triangle if base, height and hypotenuse are 5cm, 12cm and 13cm respectively.
Given:a=5cmb=12cmc=13cmSolution:∵P=a+b+c∴P=5+12+13∴P=30cm
Therefore, the perimeter of the given right triangle is 30cm.
Example 2: Find the area of a right-angled triangle whose base is 15inches and height is 8inches.
Given:base=15inchesheight=8inchesSolution:∵A=12×base×height∴A=12×15×8∴A=15×4∴A=60inch2
Therefore, the area of the given right triangle is 60inch2.
Example 3: Find the length of the hypotenuse of a right triangle if the height and the base of the triangle are 4cm and 3cm respectively.
Given:height (a)=4cmbase (b)=3cmSolution:∵c2=a2+b2∴c2=42+32∴c2=16+9∴c2=25∴c2=52∴c=5cm
Hence, the length of the hypotenuse of the given right triangle is 5cm.

FAQs on Right Triangle

  1. What is a right triangle?

    A right triangle is a type of triangle that has one angle measuring exactly 90°.

  2. What are the properties of a right triangle?

    Properties of right triangles include:

    • One angle is exactly 90°.
    • The sum of the other two angles is 90°, making them acute.
    • The side opposite the right angle is called the hypotenuse, which is the longest side.

  3. How do you identify a right triangle?

    You can identify a right triangle by checking for a 90° angle using a protractor or by applying the Pythagorean theorem.

  4. What is the Pythagorean theorem?

    The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Mathematically, it is represented as below:

    c2=a2+b2

  5. Can a right triangle be isosceles?

    Yes, a right triangle can be isosceles if the two legs (the sides forming the right angle) are of equal length. This would mean both acute angles are also equal.

  6. What is the difference between a right triangle and an acute triangle?

    A right triangle contains one angle that measures exactly 90°, while an acute triangle has all angles measuring less than 90°.

  7. Are there different types of right triangles?

    Yes, right triangles can be categorized based on their side lengths:

    • Isosceles right triangle: Two legs are equal in length.
    • Scalene right triangle: All sides are of different lengths.

  8. How do I calculate the area of a right triangle?

    The area of a right triangle can be calculated using the formula:

    A=12×base×height
    where the base and height correspond to the two legs of the triangle.

  9. What are special right triangles?

    Special right triangles include:

    • 45-45-90 triangle: An isosceles right triangle where the angles are 45°, 45° and 90°. The ratio of the sides is 1:1:2.
    • 30-60-90 triangle: A triangle with angles measuring 30°, 60° and 90°. The ratio of the sides is 1:3:2.

  10. Can the Pythagorean theorem be applied to non-right triangles?

    No, the Pythagorean theorem specifically applies only to right triangles. For other types of triangles, different formulas such as the Law of Cosines may be used.

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