# Right Triangles

## The Fascinating Geometry of Right Triangles: A Closer Look at Their Properties

Most of us have a natural tendency to take the diagonal route when walking across a rectangular or square lawn, believing it to be the shortest distance. However, this path actually forms a right triangle. In this article, we will delve into the concept of right triangles, their types, and how we can calculate their area and perimeter.

### What is a Right Triangle?

A right triangle is a type of triangle with one 90-degree angle, known as a right angle. You can easily identify one by the presence of a square drawn on the vertex of its right angle, as shown below.

**Types of Right Triangles**

Right triangles can be classified into two types: isosceles and scalene.

### Isosceles Right Triangle

An isosceles right triangle has two equal sides, with the remaining interior angles both measuring 45 degrees.

### Scalene Right Triangle

On the other hand, a scalene right triangle has none of its sides equal, with one 90-degree angle and the other two angles not equal but adding up to 90 degrees.

Scalene right triangles are particularly useful when calculating the sine, cosine, and tangent of special angles, such as 30 degrees and 60 degrees.

**Geometry of Right Triangles**

A right triangle is made up of three sides, two complementary angles, and one right angle. The longest side of the triangle is called the hypotenuse, which is opposite the right angle. The other two sides are referred to as the base and the altitude (or height).

**Properties of Right Triangles**

There are two main criteria for identifying a right triangle:

- One of its angles must measure 90 degrees.
- The other two angles must be acute, measuring less than 90 degrees each.

Let's classify the following angles labeled I to III.

- Figure I - Right Triangle: One angle measures 90 degrees, but its sides are not equal, making it a scalene right triangle.
- Figure II - Non-right Triangle: None of its angles measures 90 degrees.
- Figure III - Isosceles Right Triangle: One angle measures 90 degrees, and the other two angles are equal, making it an isosceles right triangle.

### Perimeter of Right Triangles

The perimeter of a two-dimensional figure is the distance around its outer edge. For a right triangle, the perimeter is the sum of all three sides - the height, the base, and the hypotenuse.

To find the perimeter of any right triangle with sides a, b, and c, we use the following formula:

P = a + b + c

**Area of Right Triangles**

The area of a right triangle can be calculated by multiplying its base by its height and then dividing the result by two.

For an isosceles right triangle, the base and height are of equal length, so we can use either measurement to find the area.

For example, let's say we have a right triangular cement block with sides measuring 5 cm, 12 cm, and 13 cm. How many blocks would we need to cover a square lawn with a side length of 30 cm?

First, we need to calculate the surface area of the square lawn, l, which is equal to 30 cm. Then, we can find the area of each right triangle that would fit within the square and convert it from square centimeters to square meters by using the conversion factor.

Therefore, we would need 300,000 right triangles with dimensions of 5 cm by 12 cm by 13 cm to cover the lawn.

## Solving Right Triangle Problems

In this article, we will explore different examples to gain a clear understanding of how to solve problems involving right triangles.

For instance, let's consider the joined right triangles shown below. If the larger triangle's hypotenuse measures 15 cm, what is the ratio of its area to that of the smaller triangle?

**Solution:** We can use the formula provided in the article to find the area of the larger triangle (Ab). Similarly, by using its specific formula, we can determine the smaller triangle's area (As). Finally, we can calculate the ratio of the two areas by dividing Ab by As.

Another example involves finding the perimeter of a right triangle with side lengths of 11 cm, 15.6 cm, and 11 cm.

**Solution:** Based on the given measurements, we can determine that it is an isosceles right triangle. Therefore, we can find the perimeter by adding the three sides' lengths.

## Key Takeaways

- Right triangles always have one angle measuring 90 degrees.
- There are two types of right triangles: scalene and isosceles.
- The perimeter of a right triangle is calculated by adding all three sides' lengths.
- The area of a right triangle equals half of the base multiplied by the height.
- The angles of right triangles can be determined using SOHCAHTOA when one side length is given.
- The Pythagorean theorem is used to find the hypotenuse of a right triangle.

## In Conclusion

In conclusion, understanding the properties of right triangles is crucial in both mathematics and real-life applications. By being able to compute their area and perimeter, we can solve a variety of problems and gain a more in-depth understanding of these geometric shapes.