Right-Angled Triangle

A right-angled triangle (or simply right triangle) is one of the most important shapes in geometry. It has one angle that measures 90 degrees. This type of triangle is used in many fields, such as architecture, engineering, and physics. In this article, we will discuss many aspects related to right-angled triangles in detail. 

1.0What is a Right-Angled Triangle

A right-angled triangle has one angle that is 90 degrees. The side opposite the right angle is called the hypotenuse. The other two sides are called the base and height, or the perpendicular.

Right-angled triangles follow special rules and properties. These rules help us solve problems and calculate unknown values.

2.0Properties of Right-Angled Triangle

Let’s look at the main properties of right-angled triangle:

3.0Pythagoras Theorem

One of the most famous rules in geometry is the Pythagorean theorem. It applies only to right-angled triangles.

Statement of Pythagoras Theorem

The Pythagoras Theorem states that “In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.” 

Mathematically, it can represented as: (Hypotenuse)² = (Base)² + (Height)²

Formula of Pythagoras Theorem

Let the sides be:

• Hypotenuse = c
• Base = a
• Height = b

Then: c² = a² + b²

This formula helps to find the length of any side if the other two sides are known.

Example

If base = 3 units and height = 4 units,

Then hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5 units

So, the triangle has sides of 3, 4, and 5 units.

4.0Area of Right-Angled Triangle

To find the area of right-angled triangle, use a simple formula:

Formula: Area = 12 × base × height

This formula works because the two sides forming the right angle are perpendicular to each other.

Example

If base = 6 cm and height = 8 cm,

Then area = 12 × 6 × 8 = 24 cm²

It is very easy to use when you know the base and height.

5.0Trigonometry in Right Triangles

Trigonometry in right triangles helps to relate the angles and sides of a triangle.

These are written as:

• Opposite: side opposite to angle θ
• Adjacent: side next to angle θ
• Hypotenuse: the longest side of a triangle

These ratios are useful to find missing sides or angles. 

Example

In a right triangle,

• Hypotenuse = 10 units
• Opposite side = 5 units

Then, sin(θ) = 510 = 12
Hence, θ = 30°

6.0Real-Life Examples of Right-Angled Triangle

Right-angled triangles are used in many practical ways. Here are some common examples:

• Ladders against a wall: A ladder leaning against a wall forms a right triangle.
• Roof slope: The triangle formed by the shortest path often forms a right triangle with the known distances.

7.0Examples and Word Problems

Let us go through some examples and word problems using the concepts we learned.

Example 1: Finding the Hypotenuse

Problem: A triangle has a base = 5 m and a height = 12 m. Find the hypotenuse.

Solution:

Use Pythagoras theorem:

Hypotenuse² = 5² + 12² = 25 + 144 = 169

Hypotenuse = √169 = 13 m

Example 2: Finding Area

Problem: A right triangle has a base of 9 cm and a height of 6 cm. Find the area.

Solution:

Area = (1/2) × base × height

Area = (1/2) × 9 × 6 = 27 cm²

Example 3: Using Trigonometry

Problem: In a right triangle, the angle is 30°, and the hypotenuse is 10 m. Find the opposite side.

Solution:

Use $sin(\theta )=\frac{Opposite}{Hypotenuse}$

$sin(30°)=\frac{Opposite}{10}$

$0.5=\frac{Opposite}{10}$

Opposite = 0.5 × 10 = 5 m

Example 4: Real-world Word

Problem: A ramp is placed at a 45° angle. If the length of the ramp is 10 ft, how high is the ramp?

Solution:

Use $sin(45°)=\frac{Opposite}{10}$

$\frac{1}{\sqrt{2}}=\frac{Opposite}{10}$

Opposite = $\frac{10}{\sqrt{2}}=5\sqrt{2}ft$

Example 5: Missing Side Using Tangent

Problem: A surveyor stands 30 meters away from the base of a building and measures the angle of elevation from her eyes, which are 1.6 meters above the ground, to the top of the building to be 60°. Find the total height of the building?

Solution: Height of building from her eye level: 

$tan(\theta )=\frac{Opposite}{Adjacent}$

$tan(60)=\frac{Opposite}{30}$

$\sqrt{3}=\frac{Opposite}{30}$

Opposite = $30\sqrt{3}$ meters

Total height of the building = $30\sqrt{3}+1.6=31.6\sqrt{3}m$

8.0Word Problems for Practice on Right Angled Triangle

• A pole is 15 m tall. A wire is tied from the top of the pole to the ground, 9 m from the base. Find the length of the wire.
• A building casts a shadow of 20 m. The angle of elevation of the sun is 45°. What is the height of the building?
• A right triangle has sides of 7 m and 24 m. Find the third side and the area.
• The base of a triangle is 10 cm. The angle between the base and hypotenuse is 60°. Find the height.
• A right triangle has an area of 30 cm². One side is 5 cm. Find the other side.