

# Equilateral triangles: definition and properties

An equilateral triangle is a triangle with all three sides having the same length. This is also known as an isosceles triangle, as two sides are equal in length. The angles in an equilateral triangle are all 60 degrees.

The area of an equilateral triangle can be calculated using the formula:

$$A = \frac{\sqrt{3}}{4}a^2$$

where $a$ is the length of one side.

The perimeter of an equilateral triangle can be calculated using the formula:

$$P = 3a$$

where $a$ is the length of one side.

Consider an equilateral triangle with side length 4 units. The area of this triangle is:

$$A = \frac{\sqrt{3}}{4}(4)^2 = \frac{16\sqrt{3}}{4} = 8\sqrt{3}$$

The perimeter of this triangle is:

$$P = 3(4) = 12$$

## Exercise
Calculate the area and perimeter of an equilateral triangle with side length 5 units.

# Isosceles triangles: definition and properties

An isosceles triangle is a triangle with two sides having the same length. The angles in an isosceles triangle can be of any size, as long as they sum up to 180 degrees.

The area of an isosceles triangle can be calculated using the formula:

$$A = \frac{1}{2}bh$$

where $b$ is the base of the triangle and $h$ is the height.

The perimeter of an isosceles triangle can be calculated using the formula:

$$P = 2a + b$$

where $a$ and $b$ are two equal sides of the triangle.

Consider an isosceles triangle with sides 5, 5, and 7 units. The area of this triangle is:

$$A = \frac{1}{2}(7)(5) = 21.5$$

The perimeter of this triangle is:

$$P = 2(5) + 7 = 14$$

## Exercise
Calculate the area and perimeter of an isosceles triangle with sides 6, 6, and 8 units.

# Scalene triangles: definition and properties

A scalene triangle is a triangle with all three sides having different lengths. The angles in a scalene triangle can be of any size, as long as they sum up to 180 degrees.

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

$$A = \frac{1}{2}ab\sin(C)$$

where $a$, $b$, and $C$ are the lengths of two sides and the angle between them, respectively.

The perimeter of a scalene triangle can be calculated using the formula:

$$P = a + b + c$$

where $a$, $b$, and $c$ are the lengths of the three sides.

Consider a scalene triangle with sides 4, 6, and 8 units. The area of this triangle is:

$$A = \frac{1}{2}(4)(6)\sin(60) = 12\sqrt{3}$$

The perimeter of this triangle is:

$$P = 4 + 6 + 8 = 18$$

## Exercise
Calculate the area and perimeter of a scalene triangle with sides 5, 7, and 9 units.

# Similar triangles: definition and properties

Two triangles are similar if they have the same shape, but may have different sizes. This means that the corresponding sides of the triangles are in the same ratio as the sides of the original triangle.

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

$$A = kA$$

where $k$ is the ratio of the corresponding sides of the triangles.

The perimeter of a similar triangle can be calculated using the formula:

$$P = kP$$

where $k$ is the ratio of the corresponding sides of the triangles.

Consider two similar triangles with sides 4, 5, and 6 units, and sides 8, 10, and 12 units. The area of the second triangle is:

$$A = kA = 2(4) = 8$$

The perimeter of the second triangle is:

$$P = kP = 2(4 + 5 + 6) = 24$$

## Exercise
Calculate the area and perimeter of a similar triangle with sides 3, 4, and 5 units, and sides 6, 8, and 10 units.

# Congruent triangles: definition and properties

Two triangles are congruent if they have the same shape and size. This means that the corresponding sides of the triangles are the same length.

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

$$A = A$$

where $A$ is the area of the original triangle.

The perimeter of a congruent triangle can be calculated using the formula:

$$P = P$$

where $P$ is the perimeter of the original triangle.

Consider two congruent triangles with sides 4, 5, and 6 units, and sides 4, 5, and 6 units. The area of the second triangle is:

$$A = 6$$

The perimeter of the second triangle is:

$$P = 15$$

## Exercise
Calculate the area and perimeter of a congruent triangle with sides 3, 4, and 5 units, and sides 3, 4, and 5 units.

# The Pythagorean theorem: definition and properties

The Pythagorean theorem states that in a right-angled 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 lengths of the other two sides.

The Pythagorean theorem can be written as:

$$a^2 + b^2 = c^2$$

where $a$ and $b$ are the lengths of the two shorter sides, and $c$ is the length of the hypotenuse.

Consider a right-angled triangle with sides 3, 4, and 5 units. The Pythagorean theorem states that:

$$3^2 + 4^2 = 5^2$$

## Exercise
Calculate the length of the hypotenuse of a right-angled triangle with sides 6, 8, and 10 units.

# Applications of the Pythagorean theorem in triangles

The Pythagorean theorem has numerous applications in various fields, including geometry, physics, and engineering. Some of its applications include:

- In geometry, it can be used to calculate the length of the hypotenuse of a right-angled triangle.
- In physics, it can be used to calculate the speed of a projectile in two dimensions.
- In engineering, it can be used to calculate the distance between two points in a Cartesian coordinate system.

Consider a projectile launched at an angle of 45 degrees with an initial velocity of 200 m/s. The distance the projectile travels can be calculated using the Pythagorean theorem:

$$d = \sqrt{(u^2\sin^2(\theta) + u^2\cos^2(\theta))}$$

where $u$ is the initial velocity and $\theta$ is the launch angle.

## Exercise
Calculate the distance traveled by a projectile launched at an angle of 30 degrees with an initial velocity of 100 m/s.

# Relationships between triangle properties and relationships

The properties of triangles can be related to each other. Some of these relationships include:

- The area of a triangle is equal to half the product of the length of one side and the length of the perpendicular dropped from the opposite vertex.
- The perimeter of a triangle is equal to the sum of the lengths of its three sides.
- The Pythagorean theorem can be used to calculate the length of the hypotenuse of a right-angled triangle.

Consider a triangle with sides 4, 5, and 6 units. The area of this triangle is:

$$A = \frac{1}{2}(4)(6\sin(60)) = 12\sqrt{3}$$

The perimeter of this triangle is:

$$P = 4 + 5 + 6 = 15$$

## Exercise
Calculate the area and perimeter of a triangle with sides 3, 4, and 5 units.

# Examples and exercises



# Summary and conclusion

In this textbook, we have covered the properties and relationships of triangles, including equilateral, isosceles, scalene, similar, congruent, and right-angled triangles. We have also discussed the Pythagorean theorem and its applications in various fields.

By studying this textbook, you should now have a solid understanding of the properties and relationships of triangles, and be able to calculate their properties using the formulas provided. Remember to practice your skills by working through the exercises provided, and to apply what you have learned in real-world situations.

## Exercise
Review the examples and exercises provided in this textbook, and practice calculating the properties of triangles using the formulas and relationships discussed.