Determine if a Triangle is a Right Triangle: A Comprehensive Guide

Is a triangle with side lengths of 24 centimeters, 26 centimeters, and 10 centimeters a right triangle? To answer this question, we can use the Pythagorean theorem, which is a fundamental concept in geometry. This theorem helps us determine whether a triangle is a right triangle or not by checking if the relationship between the sides satisfies the equation (a^2 b^2 c^2), where (c) is the longest side.

Understanding the Pythagorean Theorem

The Pythagorean theorem is a powerful tool in mathematics. It states that in a right 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. Mathematically, this is represented as (a^2 b^2 c^2), where (c) is the hypotenuse and (a) and (b) are the other two sides.

Applying the Theorem to Our Triangle

First, let's identify the longest side of the triangle, which is the hypotenuse. By comparing the side lengths given (24 cm, 26 cm, and 10 cm), we can see that the longest side is 26 cm. Let's label the sides as follows:

a 24 cm b 10 cm c 26 cm

We can now plug these values into the Pythagorean theorem equation and check if the relationship holds:

Step 1: Calculate (a^2 b^2)

First, we calculate the squares of the two shorter sides:

(a^2 24^2 576) (b^2 10^2 100)

Adding these two values together, we get:

(a^2 b^2 576 100 676)

Step 2: Calculate (c^2)

Next, we calculate the square of the longest side, which is the hypotenuse:

(c^2 26^2 676)

Step 3: Compare the Results

Now we compare the results from the two calculations:

(a^2 b^2 676) (c^2 676)

Since (a^2 b^2 c^2), we can conclude that the triangle with side lengths 24 cm, 26 cm, and 10 cm is indeed a right triangle. This is because the Pythagorean theorem is satisfied.

Conclusion

Using the Pythagorean theorem, we determined that a triangle with side lengths of 24 cm, 26 cm, and 10 cm is a right triangle. This conclusion is based on the fact that the sum of the squares of the two shorter sides (a and b) equals the square of the longest side (c). Thus, the triangle with these given side lengths is a right-angled triangle with 26 cm as the hypotenuse.

It's always useful to recognize that certain sets of numbers can represent primitive Pythagorean triplets, like the triplet 5, 12, 13 (which is doubled to 10, 24, 26). Such triplets simplify the process of verification.

If you need to verify if other triangles are right triangles, you can apply the Pythagorean theorem in the same manner.