Verifying Right Triangles Using the Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that helps us determine whether a triangle is a right triangle. According to this theorem, in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. This can be represented mathematically as:

Mathematical Representation

Let's denote the sides of the triangle as (a), (b), and (c), where (c) is the hypotenuse. The Pythagorean theorem states:

[a^2 b^2 c^2]

In this article, we will explore how to apply this theorem to verify if a triangle with sides of lengths 7 yards, 12 yards, and 15 yards is a right triangle.

Example 1: Triangle with Sides 7, 12, and 15 Yards

Let's identify the longest side (hypotenuse) first:

The longest side is 15 yards. The other two sides are 7 yards and 12 yards.

To determine if this is a right triangle, we will check if the square of the longest side equals the sum of the squares of the other two sides:

Step-by-Step Calculation

(a^2 b^2 c^2)

(7^2 12^2 15^2)

Calculating the squares:

(7^2 49) (12^2 144) (15^2 225)

Adding the squares of the two shorter sides:

49 144 193

Comparing this to the square of the hypotenuse:

193 ≠ 225

Since the equation does not hold true, the triangle with sides 7 yards, 12 yards, and 15 yards is not a right triangle.

Example 2: Triangle with Sides 8, 15, and 17 Yards

Now, let's take another example where the sides of the triangle are 8 yards, 15 yards, and 17 yards. Again, we will check the Pythagorean theorem:

The longest side (hypotenuse) is 17 yards. The other two sides are 8 yards and 15 yards.

Applying the theorem:

(8^2 15^2 17^2)

Calculating the squares:

(8^2 64) (15^2 225) (17^2 289)

Adding the squares of the two shorter sides:

64 225 289

Comparing this to the square of the hypotenuse:

289 289

Since the equation holds true, the triangle with sides 8 yards, 15 yards, and 17 yards is a right triangle.

Example 3: Triangle with Sides 8, 15, and 17 Centimeters

Let's consider the same example as the previous one but with measurements in centimeters:

The longest side (hypotenuse) is 17 cm. The other two sides are 8 cm and 15 cm.

Applying the theorem:

(8^2 15^2 17^2)

Calculating the squares:

(8^2 64) (15^2 225) (17^2 289)

Adding the squares of the two shorter sides:

64 225 289

Comparing this to the square of the hypotenuse:

289 289

Since the equation holds true, the triangle with sides 8 cm, 15 cm, and 17 cm is a right triangle.

Additional Considerations

It's important to note that the converse of the Pythagorean theorem is also useful. It states that if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. For instance, considering if:

(sqrt{213 - 513 - 12} 512 - 13)

(sqrt{2 times 8 times 1} 17 - 13)

(sqrt{16} 4)

The last statement verifies that a triangle with sides 5 yards, 12 yards, and 13 yards is a right triangle. Moreover, it is notable that 4 is the diameter of the incircle of the triangle, providing additional insight into the geometric properties of the triangle.

In conclusion, the Pythagorean theorem is a powerful tool for verifying if a triangle is a right triangle. By systematically applying this theorem and performing accurate calculations, we can confirm the presence or absence of a right angle within a triangle, which is essential for various applications in mathematics and real-world problem-solving scenarios.