Question

Is a triangle with measures 30,40 , and 50 a right triangle? Explain.

Short answer

Yes, it is a right triangle because it satisfies the Pythagorean theorem.

Step-by-step solution

Understand the Problem

We need to determine if a triangle with sides measuring 30, 40, and 50 units is a right triangle. A triangle is a right triangle if it follows the Pythagorean theorem.

Recall the Pythagorean Theorem

The Pythagorean theorem states that for a right triangle with legs 'a' and 'b', and hypotenuse 'c', we have \( a^2 + b^2 = c^2 \).

Assign Side Lengths

Let's assign the sides as \( a = 30 \), \( b = 40 \), and \( c = 50 \) because 50 is the largest number and therefore should be the hypotenuse if it is a right triangle.

Apply the Pythagorean Theorem

Calculate \( a^2 + b^2 \) and \( c^2 \): - \( a^2 = 30^2 = 900 \) - \( b^2 = 40^2 = 1600 \) - \( a^2 + b^2 = 900 + 1600 = 2500 \)- \( c^2 = 50^2 = 2500 \)

Compare the Results

Since \( a^2 + b^2 = c^2 \), or \( 2500 = 2500 \), the triangle meets the condition of the Pythagorean theorem.

Conclusion

Since the equation \( a^2 + b^2 = c^2 \) holds true, the triangle with sides 30, 40, and 50 is indeed a right triangle.

Key concepts

Right Triangle

A right triangle is a type of triangle that has one of its angles equal to 90 degrees. This special angle is known as the right angle. In any right triangle, the side across from the right angle is the longest side of the triangle and is called the hypotenuse. The other two sides are known as the legs of the triangle. These geometric properties make right triangles very useful in many fields including architecture, engineering, and various branches of mathematics.
A right triangle has some important characteristics that differentiate it from other types of triangles:

• One angle is exactly 90 degrees.
• It has two shorter sides, known as legs, which form the right angle.
• The longest side, called the hypotenuse, is always opposite the right angle.

These characteristics provide the foundation for many geometric calculations and are crucial to understanding and applying the Pythagorean Theorem.

Triangle Sides

Triangles are polygons with three sides and three angles. In a right triangle specifically, these sides play a crucial role because of the Pythagorean Theorem, which relates the lengths of the sides. Understanding the names and roles of these sides helps in solving many mathematical problems.
The three sides in a triangle are:

• The hypotenuse, which is the longest side and is always opposite the right angle in a right triangle.
• The legs, which are the other two sides forming the right angle.

In applying the Pythagorean Theorem, these sides must be carefully identified to correctly determine if a triangle is a right triangle. For example, if you are given a triangle with sides 30, 40, and 50, the largest side, 50, should be considered as the hypotenuse when checking the Pythagorean condition.

Hypotenuse

The hypotenuse is a unique side of any right triangle, owing to its status as the longest side. It plays a critical role in geometrical calculations involving right triangles. According to the Pythagorean Theorem, the length of the hypotenuse can be calculated if the lengths of the other two sides are known.
When you're working with a right triangle, you need to remember:

• The hypotenuse is always opposite the right angle.
• It's the longest side in the triangle.
• The Pythagorean Theorem is used to find its length, expressed as \( c \), against the other sides \( a \) and \( b \), through the equation \( a^2 + b^2 = c^2 \).

In our example of a triangle with sides 30, 40, and 50, the value 50 is assigned as the hypotenuse, since 50 is the largest number, and it confirms the triangle's status as a right triangle because \( 30^2 + 40^2 = 50^2 \), satisfying the Pythagorean condition.