# 10 Examples of Pythagorean Theorem

Pythagorean Theorem is a fundamental concept in **mathematics** that describes the relationship between the sides of a right triangle. In a right triangle, where ‘a’ and ‘b’ are the lengths of the legs, and ‘c’ is the length of the hypotenuse, theorem is expressed as a^2 + b^2 = c^2.

In this article, we will explore ten examples of Pythagorean Theorem mathematics.

**Examples of Pythagorean Theorem**

These are 10 examples of Pythagorean theorem.

**1: Determining the Length of a Missing Side**

Suppose we have a right triangle with one leg measuring 3 units and the hypotenuse measuring 5 units. Using the Pythagorean Theorem, we find the length of the other leg:

a^2 + 3^2 = 5^2

a^2 + 9 = 25

a^2 = 25 – 9

a^2 = 16

a = √16

a = 4 units

**2: Checking for a Right Triangle**

Given the sides of a triangle with lengths 6, 8, and 10 units, we can apply the Pythagorean Theorem to check if it’s a right triangle:

6^2 + 8^2 = 10^2

36 + 64 = 100

100 = 100

Since the equation holds true, this is a right triangle.

**3: Finding Distance in 2D Space**

Imagine two points in a coordinate system, (3, 4) and (6, 8). The distance between them is a straight line, and we can use the Pythagorean Theorem to calculate it:

Δx = 6 – 3 = 3

Δy = 8 – 4 = 4

Distance = √(Δx^2 + Δy^2) = √(3^2 + 4^2) = √(9 + 16) = √25 = 5 units

**4: Measuring Roof Height**

When determining the height of a gabled roof, we can use the Pythagorean Theorem. If the base is 6 meters long, and the roof’s slope is 8 meters long, the height is:

h^2 = 8^2 – 6^2

h^2 = 64 – 36

h^2 = 28

h = √28

h ≈ 5.29 meters

**5: Building Staircases**

In construction, staircases often involve right triangles. If we have a staircase with a horizontal run of 10 feet and a vertical rise of 8 feet, we find the length of the diagonal:

c^2 = 10^2 + 8^2

c^2 = 100 + 64

c^2 = 164

c ≈ √164

c ≈ 12.81 feet

**6: Calculating Cable Lengths**

When installing diagonal cables in a square-shaped frame, such as for a suspension bridge, the length of the cables are found using the Pythagorean Theorem.

**7: Finding the Diagonal of a Rectangle**

In a rectangle with sides measuring 3 meters and 4 meters, the diagonal is found using the theorem to be 5 meters.

**8: Navigational Applications**

In navigation, the theorem is used to calculate distances and routes, especially in situations involving right angles, like determining distances between waypoints.

**9: Art and Design**

Artists and designers use the Pythagorean Theorem to create balanced compositions and determine proportions in their work.

**10: Surveying Land**

Surveyors use the theorem to measure distances and ensure accurate boundary markings in land surveying.

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