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# 10 Examples of Pythagorean Theorem

October 7, 2023
written by Rida Mirza

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 = âˆš16
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:
Î”x = 6 – 3 = 3
Î”y = 8 – 4 = 4
Distance = âˆš(Î”x^2 + Î”y^2) = âˆš(3^2 + 4^2) = âˆš(9 + 16) = âˆš25 = 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 = âˆš28
h â‰ˆ 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 â‰ˆ âˆš164
c â‰ˆ 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.

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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